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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES

Managing Editor: Professor M. Reid, Mathematics Institute,University of Warwick, Coventry CV4 7AL, United Kingdom

The titles below are available from booksellers, or from Cambridge University Press atwww.cambridge.org/mathematics

347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds)348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds)349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON,

A. PILLAY & A. WILKIE (eds)350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON,

A. PILLAY & A. WILKIE (eds)351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds)353 Trends in stochastic analysis, J. BLATH, P. MORTERS & M. SCHEUTZOW (eds)354 Groups and analysis, K. TENT (ed)355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI356 Elliptic curves and big Galois representations, D. DELBOURGO357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds)358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER &

I.J. LEARY (eds)359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCIA-PRADA &

S. RAMANAN (eds)360 Zariski geometries, B. ZILBER361 Words: Notes on verbal width in groups, D. SEGAL362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERON & R. ZUAZUA363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds)364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds)365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds)366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds)367 Random matrices: High dimensional phenomena, G. BLOWER368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZALEZ-DIEZ & C. KOUROUNIOTIS (eds)369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIE370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH371 Conformal fractals, F. PRZYTYCKI & M. URBANSKI372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds)373 Smoothness, regularity and complete intersection, J. MAJADAS & A. G. RODICIO374 Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds)376 Permutation patterns, S. LINTON, N. RUSKUC & V. VATTER (eds)377 An introduction to Galois cohomology and its applications, G. BERHUY378 Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds)379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds)380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds)381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA &

P. WINTERNITZ (eds)382 Forcing with random variables and proof complexity, J. KRAJICEK383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS,

J. NICAISE & J. SEBAG (eds)384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS,

J. NICAISE & J. SEBAG (eds)385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN &

T. WEISSMAN (eds)386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds)388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds)389 Random fields on the sphere, D. MARINUCCI & G. PECCATI390 Localization in periodic potentials, D.E. PELINOVSKY391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER392 Surveys in combinatorics 2011, R. CHAPMAN (ed)393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds)394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds)395 How groups grow, A. MANN396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds)

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398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI399 Circuit double cover of graphs, C.-Q. ZHANG400 Dense sphere packings: a blueprint for formal proofs, T. HALES401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds)403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS &

A. SZANTO (eds)404 Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds)405 Torsors, etale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed)406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds)407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT &

C.M. RONEY-DOUGAL408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds)409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds)410 Representation theory and harmonic analysis of wreath products of finite groups,

T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCIA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds)412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds)414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds)415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds)416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTATA & M. POPA (eds)418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA &

R. SUJATHA (eds)419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER &

D.J. NEEDHAM420 Arithmetic and geometry, L. DIEULEFAIT et al (eds)421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds)422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds)423 Inequalities for graph eigenvalues, Z. STANIC424 Surveys in combinatorics 2015, A. CZUMAJ et al (eds)425 Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL &

J.-F. LAFONT (eds)426 Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds)427 Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds)428 Geometry in a Frechet context, C. T. J. DODSON, G. GALANIS & E. VASSILIOU429 Sheaves and functions modulo p, L. TAELMAN430 Recent progress in the theory of the Euler and Navier–Stokes equations, J.C. ROBINSON, J.L. RODRIGO,

W. SADOWSKI & A. VIDAL-LOPEZ (eds)431 Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL432 Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO433 Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA434 Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA435 Graded rings and graded Grothendieck groups, R. HAZRAT436 Groups, graphs and random walks, T. CECCHERINI-SILBERSTEIN, M. SALVATORI &

E. SAVA-HUSS (eds)437 Dynamics and analytic number theory, D. BADZIAHIN, A. GORODNIK & N. PEYERIMHOFF (eds)438 Random walks and heat kernels on graphs, M.T. BARLOW439 Evolution equations, K. AMMARI & S. GERBI (eds)440 Surveys in combinatorics 2017, A. CLAESSON et al (eds)441 Polynomials and the mod 2 Steenrod algebra I, G. WALKER & R.M.W. WOOD442 Polynomials and the mod 2 Steenrod algebra II, G. WALKER & R.M.W. WOOD443 Asymptotic analysis in general relativity, T. DAUDE, D. HAFNER & J.-P. NICOLAS (eds)444 Geometric and cohomological group theory, P.H. KROPHOLLER, I.J. LEARY, C. MARTINEZ-PEREZ &

B.E.A. NUCINKIS (eds)445 Introduction to hidden semi-Markov models, J. VAN DER HOEK & R.J. ELLIOTT446 Advances in two-dimensional homotopy and combinatorial group theory, W. METZLER &

S. ROSEBROCK (eds)447 New directions in locally compact groups, P.-E. CAPRACE & N. MONOD (eds)448 Synthetic differential topology, M.C. BUNGE, F. GAGO & A.M. SAN LUIS449 Permutation groups and cartesian decompositions, C.E. PRAEGER & C. SCHNEIDER450 Partial differential equations arising from physics and geometry, M. BEN AYED et al (eds)451 Topological methods in group theory, N. BROADDUS, M. DAVIS, J.-F. LAFONT & I. ORTIZ (eds)452 Partial differential equations in fluid mechanics, C.L. FEFFERMAN, J.C. ROBINSON & J.L. ROORIGO (eds)453 Stochastic stability of differential equations in abstract spaces, K. LIU454 Beyond hyperbolicity, M. HAGEN, R. WEBB & H. WILTON (eds)455 Groups St Andrews 2017 in Birmingham, C.M. CAMPBELL et al (eds)

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London Mathematical Society Lecture Note Series: 450

Partial Differential Equations arising fromPhysics and Geometry

A Volume in Memory of Abbas Bahri

Edited by

MOHAMED BEN AYEDUniversite de Sfax, Tunisia

MOHAMED ALI JENDOUBIUniversite de Carthage, Tunisia

YOMNA REBAIUniversite de Carthage, Tunisia

HASNA RIAHIEcole Nationale d’Ingenieurs de Tunis, Tunisia

HATEM ZAAGUniversite de Paris XIII

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University Printing House, Cambridge CB2 8BS, United Kingdom

One Liberty Plaza, 20th Floor, New York, NY 10006, USA

477 Williamstown Road, Port Melbourne, VIC 3207, Australia

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Cambridge University Press is part of the University of Cambridge.

It furthers the University’s mission by disseminating knowledge in the pursuit ofeducation, learning, and research at the highest international levels of excellence.

www.cambridge.orgInformation on this title: www.cambridge.org/9781108431637

DOI: 10.1017/9781108367639

c© Cambridge University Press 2019

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2019

Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A.

A catalogue record for this publication is available from the British Library.

ISBN 978-1-108-43163-7 Paperback

Cambridge University Press has no responsibility for the persistence or accuracy ofURLs for external or third-party internet web sites referred to in this publication

and does not guarantee that any content on such websites is, or will remain,accurate or appropriate.

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Contents

Preface page viiMohamed Ben Ayed, Mohamed Ali Jendoubi, Yomna Rebaı,Hasna Riahi and Hatem Zaag

Abbas Bahri: A Dedicated Life ix

1 Blow-up Rate for a Semilinear Wave Equation withExponential Nonlinearity in One Space Dimension 1Asma Azaiez, Nader Masmoudi and Hatem Zaag

2 On the Role of Anisotropy in the Weak Stability of theNavier–Stokes System 33Hajer Bahouri, Jean-Yves Chemin and Isabelle Gallagher

3 The Motion Law of Fronts for Scalar Reaction-diffusionEquations with Multiple Wells: the Degenerate Case 88Fabrice Bethuel and Didier Smets

4 Finite-time Blowup for some Nonlinear ComplexGinzburg–Landau Equations 172Thierry Cazenave and Seifeddine Snoussi

5 Asymptotic Analysis for the Lane–Emden Problemin Dimension Two 215Francesca De Marchis, Isabella Ianni and Filomena Pacella

6 A Data Assimilation Algorithm: the Paradigm of the3D Leray-α Model of Turbulence 253Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

7 Critical Points at Infinity Methods in CR Geometry 274Najoua Gamara

v

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vi Contents

8 Some Simple Problems for the Next Generations 296Alain Haraux

9 Clustering Phenomena for Linear Perturbation of theYamabe Equation 311Angela Pistoia and Giusi Vaira

10 Towards Better Mathematical Models for Physics 332Luc Tartar

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Preface

From March 20 to 29, 2015, a conference bearing the book’s name took placein Hammamet, Tunisia.1

It was organized by MIMS2 and CIMPA,3 and it gave us the opportunityto celebrate the 60th birthday of Professor Abbas Bahri, Rutgers University.Shortly after, Professor Bahri passed away, on January 10, 2016, after a longstruggle against sickness. His death caused deep sadness among the academiccommunity and beyond, particularly in Tunisia, France and the United Statesof America, given the great influence he had in those countries. In Tunisia hecreated a new school of thought in PDEs, by supervising several students whocontinue to develop that innovative style. Several memorial tributes took placeafter his death and many obituaries were published. He will be missed a lot.In this book, we include a chapter presenting a short biography of ProfessorBahri, concentrating on his scientific achievements.

Following the Hammamet conference, and given the high quality of thepresentations, we felt we should record those contributions by publishing theproceedings of the conference as a book. The majority of the speakers agreedto participate, and we are very grateful to them for their participation in theconference and their commitment to this book.

After the death of Professor Bahri, the book, which was undergoing therefereeing process, suddenly acquired a deeper meaning for all of us, editorsand authors: it changed from the status of a simple conference proceedings tothat of a tribute to Professor Bahri, dedicated to his memory.

The book’s contents reflect, to some extent, the conference talks and courses,which present the state of the art in PDEs, in connection with Professor Bahri’s

1 http://archive.schools.cimpa.info/archivesecoles/20160922162631/2 http://www.mims.tn/3 https://www.cimpa.info/

vii

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viii Preface

contributions. Accordingly, the main speakers at the conference were amongthe best in their field, mainly from France, the USA, Italy and Tunisia.

MIMS is the Mediterranean Institute for the Mathematical Sciences,founded in Tunis in 2012 to promote mathematics education and researchin Tunisia and in the Mediterranean area. It was designed to be a bridgebetween countries from the North and the South promoting better cooperation.

CIMPA is the International Center for Pure and Applied Mathematics basedin Nice, France. It is funded by France, Spain, Switzerland and Norway,together with UNESCO. It promotes mathematical research in developingcountries by enhancing North–South cooperation.

Given that many of our contributors are leaders in their field, we expect thebook to attract readers from the community of researchers in PDEs interestedin interactions with geometry and physics.

We also aim to attract PhD students as readers, since some papers in thebook are lecture notes from the six-hour courses given during the conference.We would like to stress the fact that lecturers made the effort of makingtheir courses accessible to PhD students with a basic background in PDEs,as required by CIMPA.

Before closing this preface, we would like to warmly thank again the authorsfor their valuable contributions. Our thanks go also to Cambridge UniversityPress, for its support with this project, and for carefully considering oursubmission. We also thank all the production team for handling our LATEX fileswith a lot of care and patience. We would also like to acknowledge financialsupport we received from various institutions, which made the Hammametconference possible: the Commission for Developing Countries (CDC) ofthe International Mathematical Union (IMU), the French Embassy in Tunis,University of Carthage, University of Paris 13, University of Tunis El-Manar,University of Sfax, the Tunisian Mathematical Society (SMT) and the TunisianAssociation for Applied and Industrial Mathematics (ATMAI).

Paris, June 11, 2017The editors

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Abbas Bahri: A Dedicated Life

This volume is dedicated to the memory of Abbas Bahri. Most of the con-tributors to this book participated in the conference organized in Hammamet,Tunisia in March 2015, on the occasion of his 60th birthday. A short whilelater, Abbas passed away on January 10, 2016 after a long illness. In this note,we would like to pay tribute to him, stressing in particular his mathematicalachievements and influence.

Abbas Bahri was a leading figure in Nonlinear Analysis and ConformalGeometry. As a matter of fact, he played a fundamental role in our under-standing of the lack of compactness arising in some variational problems.For example, his book entitled Critical Points at Infinity in Some VariationalProblems [3] had a tremendous influence on researchers working in the field ofNonlinear Partial Differential Equations involving critical Sobolev exponents.In particular, he performed in that book the finite-dimensional reductionfor Yamabe type problems and the related shadow flow for an appropriatepseudogradient. He also gave the accurate expansion of the Euler–Lagrangefunctional and its gradient. All these techniques later became widely-used toolsin the field.

0.1 A short biography

Abbas Bahri was born on January 1, 1955 in Tunis, Tunisia. At the age of 16he moved to Paris, where he was admitted to the prestigious Ecole NormaleSuperieure, Rue d’Ulm at the age of 19. He later obtained his Agregationin mathematics, then defended a These d’Etat in 1981 at the age of 26 inUniversite Pierre et Marie Curie (Paris 6), under the direction of ProfessorHaim Brezis.

Starting his career as a Research Assistant in CNRS between 1979 and1981, he later obtained other positions in the University of Chicago, Ecole

ix

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x Abbas Bahri: A Dedicated Life

Polytechnique, Palaiseau, and Ecole Nationale d’Ingenieurs (ENIT), Tunis.In 1988 he was appointed Professor at Rutgers University. As director of theCenter for Nonlinear Analysis he organized many seminars and supervised anumber of PhD students. He also received many prestigious invitations all overthe world. His remarkable achievements have been widely recognized. He wasawarded the Langevin and Fermat prizes in 1989 for “introducing new tools inthe calculus of variation” and he received in 1990 the Board of Trustees Awardfor Excellence, Rutgers University’s highest honor for outstanding research.

Beyond his mathematical achivements, which will be discussed in the nextsection, we would like to pay tribute to Abbas Bahri for two other reasons.

The first reason, which is connected to research, concerns his total commit-ment to “transmission”, in particular in his homeland, Tunisia. The decisive actbegan in the early 1990s, when he started supervising about ten PhD students inTunisia, including two Mauritanians. He devoted much energy and time to this,dividing his holidays in Tunisia between his family and his students. He was infact establishing a new “mathematical tradition” in Tunisia, a tradition whichis proudly continued by his students who hold many outstanding positions inTunisia and abroad. More recently, despite his illness, he displayed tremendouscourage and went on a “math tour” in Tunisia in 2014–2015, giving lecturesat many universities, including Ecole Polytechnique de Tunisie, La Marsa, theUniversity of Kairouan, and the University of Sfax.

The second reason concerns his commitment to progress, democracy, andsocial justice in the world. He particularly believed in, and fought for, thedemocratization of his country of origin, where free rational thinking wouldprevail, and he was confident in the intellectual potential of the Tunisianpeople.

Besides being a gifted mathematician with an exceptional sense of origi-nality and depth, Abbas Bahri was also interested in – among other things– history, art, music, literature, philosophy, and politics. He believed inthe contribution of Arab and Muslim culture to the development of humanknowledge and intellect, and as a source of inspiration for progress. He alsoviewed this contribution as a way to transcend cultural differences. AbbasBahri valued diversity and nurtured friendships all over the world.

0.2 Mathematical contributions

Abbas Bahri’s mathematical interests were very broad, ranging from nonlinearPDEs arising from geometry and physics to systems of differential equationsof Celestial Mechanics. However, his research focused mainly on fundamentalproblems in Contact Forms and Conformal Geometry. Bahri’s contributions

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Abbas Bahri: A Dedicated Life xi

are various and he published many important results in collaboration with anumber of authors.

Bahri was fascinated by variational problems arising in Contact Geometry atthe beginning of his career, and he continued to work on this topic for the rest ofhis life. He was, in particular, motivated by the Weinstein conjecture about theexistence of periodic orbits of the Reeb vector field ξ of a given contact formα defined in M3, a three-dimensional closed and oriented manifold. Althoughthis problem features a variational structure, its corresponding variational formis defined on the loop space of M, H1(S1,M), by

J(x) :=∫ 1

0α(x)dt, x ∈H1(S1,M).

In fact, the critical points of J are the periodic orbits of ξ .J is a very bad variational problem on H1(S1,M) because the variational

flows do not seem to be Fredholm and the critical points of J have an infiniteMorse index.

It is in this framework that Bahri developed the concept of critical points atinfinity. In fact, he discovered that the ω-limit set of non-compact orbits of thegradient flow behave like a usual critical point, once a Morse reduction in theneighborhood of such geometric objects is performed. In particular, one canassociate with such asymptotes a Morse index as well as stable and unstablemanifolds.

To study the functional J, Bahri tried to restrict the variations of the curve.In order to do so, taking a non-vanishing vector field v in kerα and denotingβ(.) := dα(v, .), he defined the subspace Cβ := {x ∈ H1(S1,M) : β(x) =0 and α(x) = a}, where a is a positive constant (which may depend on x).Assuming that β is a contact form with the same orientation as α, he proved(in collaboration with D. Bennequin [1]) that:

J is a C2 function on Cβ whose critical points are the periodic orbits of ξ .Moreover, those orbits have a finite Morse index.

We notice that the curve in Cβ can be expressed in a simple way, that is, ifx ∈ Cβ then x = aξ + bv, where a is a positive constant (depending on x) andtherefore J(x)= a > 0.

It is easy to see that J does not satisfy the Palais–Smale (PS) condition sinceit just controls the value a of the curve but the b-component along v is free.Therefore, it can have any behavior along a PS sequence.

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xii Abbas Bahri: A Dedicated Life

Crucially, Bahri used the deformation of the level sets of the associatedfunctional J. For this purpose, in general, the used vector field is −∇J. Inthe case of the contact form α, taking w such that dα(v, w) = 1, if z :=λξ +μv + ηw belongs to TxCβ (eventually λ, μ and η have to satisfy someconditions (see (2.7) of [8])) then

∇J(x).z=−∫ 1

0bηdt.

In view of this formula, there is a “natural decreasing pseudo-gradient” thatcan be derived by taking η = b in the formula above (the other variables λ

and μ will be computed using (2.7) of [8]). This flow has several remarkablegeometric properties. One of them is that the linking of two curves under theJ-decreasing evolution through the flow of z (with η = b) never decreases.However, this flow has several “undesired” blow-ups and it is thereforedifficult to define a homology related to the periodic orbits of ξ with thispseudo-gradient.

Bahri’s main idea was to use a special (constructed) decreasingpseudo-gradient Z for (J,Cβ). This program was done in several of hisworks (in particular [11]) since he required many properties to be satisfied. Inparticular, the new vector-field Z blows up only along the stratified set ∪k�2k,where �2k := {curves made of k−pieces of ξ−orbits, alternating with k−pieces of ± v−orbits}. Furthermore, along its (semi)-flow-lines, the numberof zeros of b (the v-component of x) never increases and the L1-norm ofb is bounded. The two points are very important in the study of the PSsequences.

Bahri extended this pseudo-gradient on ∪�2k and he defined the functional

J∞(x) :=∞∑

k=1

ak, x ∈ ∪�2k,

where ak is the length of the kth piece of ξ . The critical points of J∞ are whatBahri called critical points at infinity. This precise pseudo-gradient allowedhim to understand the lack of compactness and to characterize the criticalpoints at infinity [11, 9]. These points are characterized as follows. A curve in∪�2k is a critical point at infinity if it satisfies one of the following assertions.

1. The v-jumps are between conjugate points (conjugate points are pointson the same v-orbit such that the form α is transported onto itself by thetransport map along v). These critical points are called true critical pointsat infinity.

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Abbas Bahri: A Dedicated Life xiii

2. The ξ -pieces have characteristic length and, in addition, the v-jumps sendkerα to itself (a ξ -piece [x0,x1] is characteristic if v completes an exactnumber n ∈ Z of half revolutions from x0 to x1).

Furthermore, in [13], Bahri proved that the linking property is conserved,that is: for any decreasing flow-line Cs, originating at a periodic orbit andending at another periodic orbit O′ (contractible in M) of ξ with a differenceof indices equal to 1, the linking number lk(Cs,O′) never decreases with s.

The properties required for the constructed pseudo-gradient Z allowed Bahrito define an intersection operator ∂ for the variational problem (J,Cβ), andhe was therefore able to define a kind of homology for the critical points (atinfinity) of J [9, 12]. However, he noticed that the critical points (at infinity)do not change the topology of the level sets of J (because J is not Fredholm)and this is a serious difficulty to overcome.

In his last paper [14] Bahri used these properties, combined with theFadel–Rabinowitz Morse index, to present a new beautiful proof of the Wein-stein conjecture on S3. This new proof combines the case of the tight contactstructure on S3 and the case of all the over-twisted ones and could thereforelead to a better understanding of the existence process for periodic orbits ofξ . It could also possibly lead to multiplicity results on all three-dimensionalclosed manifolds with finite fundamental group. Moreover, it can be extendedto closed manifolds M2n+1 with n≥ 1 satisfying some topological assumptions(with some technical difficulties).

When Bahri developed the theory of critical points at infinity he applied itto many problems. In collaboration with P. Rabinowitz he studied the 3-bodyproblem in Celestial Mechanics. This problem is modeled by the followingHamiltonian system: miqi+Vqi(q)= 0, i= 1,2,3, where mi > 0, qi ∈ R3, andV(q) =∑3

i=j,i,j=1 mimj/|qi − qj|α , with α > 0. This problem has a variationalstructure. Its T-periodic solutions correspond to critical points of the functionalI(q) := ∫ T

0 ( 12

∑mi|qi|2−V(q))dt defined on the class of T-periodic functions.

In [5] Bahri proved the existence of infinitely many T-periodic solutions ofthis problem (with α ≥ 2). The proof of this remarkable result is based on theunderstanding of the lack of compactness of I. In fact, sections 7 and 8 of [5]are devoted to the analysis of the critical points at infinity of I. This new objectallowed the authors to prove the result.

Moreover, Bahri applied his new theory to the Yamabe and scalar curvatureproblems. For this program, he collected in the monograph [3] many delicateand difficult estimates needed to understand the lack of compactness andto characterize the critical points at infinity of the associated variationalfunctional.

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xiv Abbas Bahri: A Dedicated Life

It is known that, in the region where the Palais–Smale condition fails, thefunctions have to be decomposed as the sum of some bubbles. In collaborationwith J.M. Coron, Bahri studied this lack of compactness of the scalar curvatureproblem on S3 [4] and he gave a criterion (depending on the scalar function tobe prescribed) for the existence of a solution for this problem. This criterionwas extended by various authors in other situations and equations with lack ofcompactness. Furthermore, Bahri used the theory of critical points at infinityto give another proof for the Yamabe conjecture for a locally conformally flatmanifold [7].

For a bounded domain , the Yamabe problem (�u+ u(n+2)/(n−2),u > 0 on; u= 0 in ∂) becomes more difficult. The associated variational functionalis defined by J(u) := 1/

∫|u|2n/(n−2) for u ∈ � = {u ∈ H1

0() : ‖u‖ = 1}.In collaboration with J.M. Coron, Bahri proved that if has a non-trivialtopology then this problem has at least one solution [2]. In fact, the proofis based on combining some analysis and algebraic topology arguments.The analysis part consists of (i) characterizing the levels where the lack ofcompactness occurs and (ii) proving that there is no difference of topologybetween the level sets Ja and Jb for a and b large, where Ja := {u ∈� : J(u) <a,u > 0}. Concerning the algebraic topology argument, since has no trivialtopology they were able to find a non-trivial class in the homology of thebottom level set. Furthermore, they proved an intrinsic argument which showsthat, for c1 < c2 < c3 three consecutive levels (where the lack of compactnessoccurs), starting from a non-trivial class in the homology of the pair (Jc2 ,Jc1),if there is no solution, they can find another non-trivial class in the homologyof the pair (Jc3 ,Jc2). Thus, by induction, they are able to find non-trivial classesin the homology of all the pairs (Jck ,Jck−1) where the cis are the levels wherethe lack of compactness occurs. This gives a contradiction with item (ii) of theanalysis part.

Concerning the subcritical case, in collaboration with P.L. Lions, takinga bounded and regular domain ⊂ Rn, n ≥ 2, Bahri studied the followingPDE: (P): −�u = f (x,u) in ; u = 0 on ∂ with |f (x,s)| ≤ C(1 + |s|p)(1 < p < (n + 2)/(n − 2) if n ≥ 3) and other assumptions on f . In [6],Bahri introduced another type of result. He proved that, taking a sequence ofsolutions (uk) of (P), the boundness of |uk|∞ is related to the Morse index of(uk). In fact, the authors proved that the sequence (|uk|∞) is bounded if andonly if the sequence of the Morse index of uk is bounded. The proof relieson some blow-up analysis. The limit problem becomes: −�u= |u|p−1u in Rn

(or −�u = |u|p−1u in : a half space with u = 0 on ∂ ). These problemswere studied in the positive case, but there is no result for the changing signsolution. Using a bootstrap argument, Bahri proved that these limit problems

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Abbas Bahri: A Dedicated Life xv

do not have non-trivial bounded solutions with bounded Morse index. This newcriterion (the notion of the Morse index) becomes very useful for classifyingthe solutions of other equations.

In his last book [10] Bahri studied, in the first part, the changing-signYamabe problem. He considered the case of R3 (or equivalently S3):

�u+ u5 = 0 in R3. (0.1)

In this case the solutions are known to exist, in fact in infinite number.Moreover, if we impose the positivity of the solutions then we see that theonly solutions are given by the family δ(a,λ) := c

√λ/√(1+λ2|.− a|2). As for

changing-sign solutions, we only know their asymptotic behavior at infinity.In this case Bahri studied the asymptotes generated by these solutions andtheir combinations under the action of the conformal group. As he said inhis monograph, “The Yamabe problem, without the positivity assumption, isa simpler model of less explicit non-compactness phenomena. The equation(0.1) is “un cas d’ecole””. In fact, this work provides a family of estimatesand techniques by which the problem of finding infinitely many solutions to thechanging-sign Yamabe-type problem on domains of Rn, n≥ 3, can be studied.Moreover, using the compactness result of Uhlenberg, the ideas introducedin this work could be useful in the study of the Yang–Mills equations. As amatter of fact, this was the topic of the course Bahri gave in February 2015 inthe Faculte des Sciences de Sfax.

An interesting idea used in [10] consists of deriving an a priori estimatefor the remainder term for a PS sequence. Let be a bounded domain andlet v be the unique solution of −�v = f (v) on , v = 0 in ∂. To prove that|v(x)| ≤ cϕ(x) for each x∈, where ϕ is a given function, Bahri introduces thefollowing PDE: −�v = f (min(|v|,ϕ)sign v) on , v = 0 in ∂. By studyingthe new function v, he proves that |v| is small with respect to ϕ and therefore itsatisfies−�v= f (v) on , v= 0 in ∂, which implies that v= v and therefore,v is small with respect to ϕ. The aim in introducing the new function v isto overcome some difficulties arising from some non-linear terms of f . Bahriintroduced this idea to get some a priori estimate on the remainder function ofa PS sequence for the Yamabe sign-changing problem.

As mentioned earlier, Abbas Bahri paved the way for future generations byintroducing a new “mathematical tradition” that is now being continued byhis students. His contributions go beyond mathematics and his influence hasreached many, in Tunisia and all over the world. He is missed not only by hisfamily and friends, but also by many people who met him and appreciated hishuman qualities and research achievements.

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xvi Abbas Bahri: A Dedicated Life

References

[1] Bahri A., Pseudo-orbits of Contact Forms, Pitman Research Notes in MathematicsSeries, 173. Longman Scientific & Technical, Harlow, 1988.

[2] Bahri A. and Coron J.M., On a nonlinear elliptic equation involving the criticalSobolev exponent: the effect of the topology of the domain, Comm. Pure Appl.Math. 41–3, 253–294, 1988.

[3] Bahri A., Critical points at infinity in some variational problems, Research Notesin Mathematics, 182, Longman-Pitman, London, 1989.

[4] Bahri A. and Coron J.M., The scalar-curvature problem on the standardthree-dimensional sphere, J. Funct. Anal. 95, no. 1, 106–172, 1991.

[5] Bahri, A. and Rabinowitz, P., Periodic orbits of hamiltonian systems of three bodytype. Ann. Inst. H. Poincare Anal. Non lineaire 8, 561–649, 1991.

[6] Bahri A and Lions P.L., Solutions of superlinear elliptic equations and theirMorse indices, Comm. Pure Appl. Math. 45, 1205–1215, 1992.

[7] Bahri A., Another proof of the Yamabe conjecture for locally conformally flatmanifolds, Nonlinear Anal. 20, no. 10, 1261–1278, 1993.

[8] Bahri A., Classical and quantic periodic motions of multiply polarized spinparticles, Pitman Research Notes in Mathematics Series, 378. Longman, Harlow,1998.

[9] Bahri A., Flow lines and algebraic invariants in contact form geometry, progressin nonlinear differential equations and their applications, 53. Birkhauser Boston,Inc., Boston, MA, 2003.

[10] Bahri A. and Y. Xu, Recent Progress in Conformal Geometry, ICP Advances Textsin Mathematics 1, London: Imperial College Press, 2007.

[11] Bahri A., Compactness, Adv. Nonlinear Studies, 8 (3), 465–568, 2008.[12] Bahri A., Homology computation, Adv. Nonlinear Studies, 8, 1–17, 2008.[13] Bahri A., Linking numbers in contact form geometry, with an application to the

computation of the intersection operator for the first contact form of J. Gonzaloand F. Varela, Arab J. Math, 3, 199–210, 2014.

[14] Bahri A., A Linking/S1-equivariant variational argument in the space of duallegendrian curves and the proof of the weinstein conjecture on S3 “in the large”,Adv. Nonlinear Studies, 15, 497–526, 2015.

Mohamed Ben Ayed, University of Sfax

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1

Blow-up Rate for a Semilinear WaveEquation with Exponential Nonlinearity

in One Space DimensionAsma Azaiez∗, Nader Masmoudi† and Hatem Zaag‡

We consider in this paper blow-up solutions of the semilinear wave equation in onespace dimension, with an exponential source term. Assuming that initial data are inH1

loc×L2loc or sometimes in W1,∞×L∞, we derive the blow-up rate near a

non-characteristic point in the smaller space, and give some bounds near otherpoints. Our results generalize those proved by Godin under high regularityassumptions on initial data.

1.1 Introduction

We consider the one dimensional semilinear wave equation:{∂2

t u= ∂2x u+ eu,

u(0)= u0 and ∂tu(0)= u1,(1.1)

where u(t) : x ∈ R→ u(x, t) ∈ R,u0 ∈ H1loc,u and u1 ∈ L2

loc,u. We may also addmore restrictions on initial data by assuming that (u0,u1) ∈ W1,∞ × L∞. TheCauchy problem for equation (1.1) in the space H1

loc,u × L2loc,u follows from

fixed point techniques (see Section 1.2).If the solution is not global in time, we show in this paper that it blows up

(see Theorems 1.1 and 1.2). For that reason, we call it a blow-up solution. Theexistence of blow-up solutions is guaranteed by ODE techniques and the finitespeed of propagation.

More blow-up results can be found in Kichenassamy and Littman [12], [13],where the authors introduce a systematic procedure for reducing nonlinearwave equations to characteristic problems of Fuchsian type and construct

∗ This author is supported by the ERC Advanced Grant no. 291214, BLOWDISOL.† This author is partially supported by NSF grant DMS- 1211806.‡ This author is supported by the ERC Advanced Grant no. 291214, BLOWDISOL and by ANR

project ANAE ref. ANR-13-BS01-0010-03.

1

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2 Asma Azaiez, Nader Masmoudi and Hatem Zaag

singular solutions of general semilinear equations which blow up on anon-characteristic surface, provided that the first term of an expansion of suchsolutions can be found.

The case of the power nonlinearity has been understood completely in aseries of papers, in the real case (in one space dimension) by Merle and Zaag[16], [17], [20] and [21] and in Cote and Zaag [6] (see also the note [18]), andin the complex case by Azaiez [3]. Some of those results have been extendedto higher dimensions for conformal or subconformal p:

1 < p≤ pc ≡ 1+ 4

N− 1, (1.2)

under radial symmetry outside the origin in [19]. For non-radial solutions, wewould like to mention [14] and [15] where the blow-up rate was obtained.We also mention the recent contribution of [23] and [22] where the blow-upbehavior is given, together with some stability results.

In [5] and [4], Caffarelli and Friedman considered semilinear waveequations with a nonlinearity of power type. If the space dimension N isat most 3, they showed in [5] the existence of solutions of Cauchy problemswhich blow up on a C1 spacelike hypersurface. If N = 1 and under suitableassumptions, they obtained in [4] a very general result which shows thatsolutions of Cauchy problems either are global or blow up on a C1 spacelikecurve. In [11] and [10], Godin shows that the solutions of Cauchy problemseither are global or blow up on a C1 spacelike curve for the following mixedproblem (γ = 1, |γ | ≥ 1):{

∂2t u= ∂2

x u+ eu, x > 0,∂xu+ γ ∂tu= 0 if x= 0.

(1.3)

In [11], Godin gives sharp upper and lower bounds on the blow-up rate forinitial data in C4 ×C3. It so happens that his proof can be extended for initialdata (u0,u1) ∈H1

loc,u×L2loc,u (see Proposition 1.15).

Let us consider u a blow-up solution of (1.1). Our aim in this paperis to derive upper and lower estimates on the blow-up rate of u(x, t). Inparticular, we first give general results (see Theorem 1.1), then, consideringonly non-characteristic points, we give better estimates in Theorem 1.2.

From Alinhac [1], we define a continuous curve � as the graph of a functionx �→ T(x) such that the domain of definition of u (or the maximal influencedomain of u) is

D= {(x, t)|0≤ t < T(x)}. (1.4)

From the finite speed of propagation, T is a 1-Lipschitz function. The graph �

is called the blow-up graph of u.

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Blow-up Rate for a Semilinear Wave Equation 3

Let us introduce the following non-degeneracy condition for �. If weintroduce for all x ∈R, t≤ T(x) and δ > 0, the cone

Cx,t,δ = {(ξ ,τ) = (x, t) |0≤ τ ≤ t− δ|ξ − x|}, (1.5)

then our non-degeneracy condition is the following: x0 is a non-characteristicpoint if

∃δ0 = δ0(x0) ∈ (0,1) such that u is defined on Cx0,T(x0),δ0 . (1.6)

If condition (1.6) is not true, then we call x0 a characteristic point. We denote byR⊂R (resp. S ⊂R) the set of non-characteristic (resp. characteristic) points.

We also introduce for each a ∈ R and T ≤ T(a) the following similarityvariables:

wa,T(y,s)= u(x, t)+ 2log(T− t), y= x− a

T− t, s=− log(T− t). (1.7)

If T = T(a), we write wa instead of wa,T(a).From equation (1.1), we see that wa,T (or w for simplicity) satisfies, for all

s≥− logT , and y ∈ (−1,1),

∂2s w− ∂y((1− y2)∂yw)− ew+ 2=−∂sw− 2y∂2

y,sw. (1.8)

In the new set of variables (y,s), deriving the behavior of u as t → T isequivalent to studying the behavior of w as s →+∞.

Our first result gives rough blow-up estimates. Introducing the following set:

DR ≡ {(x, t) ∈ (R,R+), |x|< R− t}, (1.9)

where R > 0, we have the following result.

Theorem 1.1 (Blow-up estimates near any point) We claim the following:

(i) (Upper bound) For all R > 0 and a ∈R such that (a,T(a)) ∈DR, it holdsthat:

∀|y|< 1, ∀s≥− logT(a), wa(y,s)≤−2log(1−|y|)+C(R),

∀t ∈ [0,T(a)), eu(a,t) ≤ C(R)

d((a, t),�)2≤ C(R)

(T(a)− t)2,

where d((x, t),�) is the (Euclidean) distance from (x, t) to �.(ii) (Lower bound) For all R > 0 and a∈R such that (a,T(a)) ∈DR, it holds

that

1

T(a)− t

∫I(a,t)

e−u(x,t)dx≤ C(R)√

d((a, t),�)≤ C(R)√

T(a)− t.

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4 Asma Azaiez, Nader Masmoudi and Hatem Zaag

If, in addition, (u0,u1) ∈W1,∞×L∞ then

∀t ∈ [0,T(a)), eu(a,t) ≥ C(R)

d((a, t),�)≥ C(R)

T(a)− t.

(iii) (Lower bound on the local energy “norm”) There exists ε0 > 0 suchthat for all a ∈R, and t ∈ [0,T(a)),

1

T(a)− t

∫I(a,t)

((ut(x, t))2+ (ux(x, t))2+ eu(x,t))dx≥ ε0

(T(a)− t)2, (1.10)

where I(a, t)= (a− (T(a)− t),a+ (T(a)− t)).

Remark The upper bound in item (i) was already proved by Godin [11], formore regular initial data. Here, we show that Godin’s strategy works even forless regular data. We refer to the integral in (1.10) as the local energy “norm”,since it is like the local energy as in Shatah and Struwe [24], though with the“+” sign in front of the nonlinear term. Note that the lower bound in item(iii) is given by the solution of the associated ODE u′′ = eu. However, thelower bound in (ii) doesn’t seem to be optimal, since it does not obey the ODEbehavior. Indeed, we expect the blow-up for equation (1.1) in the “ODE style”,in the sense that the solution is comparable to the solution of the ODE u′′ = eu

at blow-up. This is in fact the case with regular data, as shown by Godin [11].

If, in addition, a ∈R, we have optimal blow-up estimates.

Theorem 1.2 (An optimal bound on the blow-up rate near a non-charac-teristic point in a smaller space) Assume that (u0,u1) ∈ W1,∞ × L∞. Then,for all R > 0, for any a∈R such that (a,T(a))∈DR, we have the following:

(i) (Uniform bounds on w) For all s≥− logT(a)+ 1,

|wa(y,s)|+∫ 1

−1

((∂swa(y,s))2+ (∂ywa(y,s))2

)dy≤ C(R),

where wa is defined in (1.7).(ii) (Uniform bounds on u) For all t ∈ [0,T(a)),

|u(x, t)+ 2log(T(a)− t)|+ (T(a)− t)∫

I(∂xu(x, t))2+ (∂tu(x, t))2 dx≤ C(R).

In particular, we have

1

C(R)≤ eu(x,t)(T(a)− t)2 ≤ C(R).

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Blow-up Rate for a Semilinear Wave Equation 5

Remark This result implies that the solution indeed blows up on the curve �.

Remark Note that when a∈R, Theorem 1.1 already holds and directly followsfrom Theorem 1.2. Accordingly, Theorem 1.1 is completely meaningful whena ∈ S .

Following Antonini, Merle and Zaag in [2] and [15], we would like tomention the existence of a Lyapunov functional in similarity variables. Moreprecisely, let us define

E(w(s))=∫ 1

−1

(1

2(∂sw)2+ 1

2(1− y2)(∂yw)2− ew+ 2w

)dy. (1.11)

We claim that the functional E defined by (1.11) is a decreasing function oftime for solutions of (1.8) on (−1,1).

Proposition 1.3 (A Lyapunov functional for equation (1.1)) For all a ∈R, T ≤ T(a), s2 ≥ s1 ≥− logT, the following identities hold for w=wa,T :

E(w(s2))−E(w(s1))=−∫ s2

s1

(∂sw(−1,s))2+ (∂sw(1,s))2ds.

Remark The existence of such an energy in the context of the nonlinear heatequation has been introduced by Giga and Kohn in [7], [8] and [9].

Remark As for the semilinear wave equation with conformal power nonlin-earity, the dissipation of the energy E(w) degenerates to the boundary ±1.

This paper is organized as follows:In Section 1.2, we solve the local in time Cauchy problem.Section 1.3 is devoted to some energy estimates.In Section 1.4, we give and prove upper and lower bounds, following the

strategy of Godin [11].Finally, Section 1.5 is devoted to the proofs of Theorem 1.1, Theorem 1.2

and Proposition 1.3.

1.2 The Local Cauchy Problem

In this section, we solve the local Cauchy problem associated with (1.1) in thespace H1

loc,u×L2loc,u. In order to do so, we will proceed in three steps.

(1) In Step 1, we solve the problem in H1loc,u×L2

loc,u, for some uniform T > 0small enough.

(2) In Step 2, we consider x0 ∈ R, and use Step 1 and a truncation to find alocal solution defined in some cone Cx0,T(x0),1

for some T(x0) > 0. Then,

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6 Asma Azaiez, Nader Masmoudi and Hatem Zaag

by a covering argument, the maximal domain of definition is given byD=∪x0∈RCx0,T(x0),1

.(3) In Step 3, we consider some approximation of equation (1.1), and discuss

the convergence of the approximating sequence.

Step 1: The Cauchy problem in H1loc,u×L2

loc,u

In this step, we will solve the local Cauchy problem associated with (1.1) inthe space H = H1

loc,u × L2loc,u. In order to do so, we will apply a fixed point

technique. We first introduce the wave group in one space dimension:

S(t) : H →H,

(u0,u1) �→ S(t)(u0,u1)(x),

S(t)(u0,u1)(x)=⎛⎝ 1

2(u0(x+ t)+ u0(x− t))+ 1

2

∫ x+t

x−tu1dt

12 (u

′0(x+ t)− u′0(x− t))+ 1

2 (u1(x+ t)+ u1(x− t))

⎞⎠ .

Clearly, S(t) is well defined in H, for all t ∈R, and more precisely, there is auniversal constant C0 such that

||S(t)(u0,u1)||H ≤ C0(1+ t)||(u0,u1)||H . (1.12)

This is the aim of the step.

Lemma 1.4 (Cauchy problem in H1loc,u × L2

loc,u) For all (u0,u1) ∈ H, thereexists T > 0 such that there exists a unique solution of the problem (1.1) inC([0,T],H).

Proof Consider T > 0 (to be chosen later) small enough in terms of||(u0,u1)||H .

We first write the Duhamel formulation for our equation:

u(t)= S(t)(u0,u1)+∫ t

0S(t− τ)(0,eu(τ ))dτ . (1.13)

Introducing

R= 2C0(1+T)||(u0,u1)||H , (1.14)

we will work in the Banach space E = C([0,T],H) equipped with the norm||u||E = sup

0≤t≤T||u||H . Then, we introduce

� : E → E

V(t)=(v(t)v1(t)

)�→ S(t)(u0,u1)+

∫ t

0S(t− τ)(0,ev(t))dτ

and the ball BE(0,R).

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Blow-up Rate for a Semilinear Wave Equation 7

We will show that for T > 0 small enough, � has a unique fixed point inBE(0,R). To do so, we have to check two points:

1. � maps BE(0,R) to itself;2. � is k-Lipschitz with k < 1 for T small enough.

• Proof of 1: Let V =(v

v1

)∈ BE(0,R); this means that:

∀t ∈ [0,T], v(t) ∈H1loc,u(R)⊂ L∞(R)

and that

||v(t)||L∞(R) ≤ C∗R.

Therefore

||(0,ev)||E = sup0≤t≤T

||ev(t)||L2loc,u

≤ eC∗R√

2. (1.15)

This means that

∀τ ∈ [0,T] (0,ev(τ )) ∈H,

hence S(t − τ)(0,ev(τ )) is well defined from (1.12) and so is its integralbetween 0 and t. So � is well defined from E to E.

Let us compute ||�(v)||E.Using (1.12), (1.14) and (1.15) we write for all t ∈ [0,T],

||�(v)(t)||H ≤ ||S(t)(u0,u1)||H +∫ t

0||S(t− τ)(0,ev(τ ))||Hdτ

≤ R

2+∫ T

0C0(1+T)

√2eC∗Rdτ

≤ R

2+C0T(1+T)

√2eC∗R. (1.16)

Choosing T small enough so that

R

2+C0T(1+T)

√2eC∗R ≤ R

or

T(1+T)≤ Re−C∗R

2√

2C0

guarantees that � goes from BE(0,R) to BE(0,R).

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8 Asma Azaiez, Nader Masmoudi and Hatem Zaag

• Proof of 2: Let V , V ∈ BE(0,R). We have

�(V)−�(V)=∫ T

0S(t− τ)(0,ev(t)− ev(t))dτ .

Since ||v(t)||L∞(R) ≤ C∗R and the same for ||v(t)||L∞(R), we write

|ev(τ )− ev(τ )| ≤ eC∗R|v(τ )− v(τ )|,hence

||ev(τ )− ev(τ )||L2loc,u

≤ eC∗R||v(τ )− v(τ )||L2loc,u

≤ eC∗R||V− V||E. (1.17)

Applying S(t− τ) we write from (1.12), for all 0≤ τ ≤ t≤ T ,

||S(t− τ)(0,ev(τ )− ev(τ ))||H ≤ C0(1+T)||(0,ev(τ )− ev(τ ))||H≤ C0(1+T)||ev(τ )− ev(τ )||L2

loc,u

≤ C0(1+T)eC∗R||V− V||E. (1.18)

Integrating, we end up with

||�(V)−�(V)||E ≤ C0T(1+T)eC∗R||V− V||E. (1.19)

k= C0T(1+T)eC∗R can be made < 1 if T is small.

Conclusion From points 1 and 2, � has a unique fixed point u(t) in BE(0,R).This fixed point is the solution of the Duhamel formulation (1.13) and of ourequation (1.1). This concludes the proof of Lemma 1.4.

Step 2: The Cauchy problem in a larger regionLet (u0,u1) ∈ H1

loc,u × L2loc,u be initial data for the problem (1.1). Using the

finite speed of propagation, we will localize the problem and reduces it to thecase of initial data in H1

loc,u × L2loc,u already treated in Step 1. For (x0, t0) ∈

R× (0,+∞), we will check the existence of the solution in the cone Cx0,t0,1.In order to do so, we introduce χ , a C∞ function with compact support suchthat χ(x)= 1 if |x− x0|< t0; let also (u0, u1)= (u0χ ,u1χ) (note that u0 and u1

depend on (x0, t0) but we omit this dependence in the indices for simplicity).So, (u0, u1) ∈ H1

loc,u×L2loc,u. From Step 1, if u is the corresponding solution of

equation (1.1), then, by the finite speed of propagation, u= u in the intersectionof their domains of definition with the cone Cx0,t0,1. As u is defined for all (x, t)in R× [0,T) from Step 1 for some T = T(x0, t0), we get the existence of ulocally in Cx0,t0,1 ∩R× [0,T). Varying (x0, t0) and covering R× (0,+∞[ byan infinite number of cones, we prove the existence and the uniqueness of

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Blow-up Rate for a Semilinear Wave Equation 9

the solution in a union of backward light cones, which is either the wholehalf-space R× (0,+∞), or the subgraph of a 1-Lipschitz function x �→ T(x).We have just proved the following.

Lemma 1.5 (The Cauchy problem in a larger region) Consider (u0,u1) ∈H1

loc,u×L2loc,u. Then, there exists a unique solution defined in D, a subdomain of

R× [0,+∞), such that for any (x0, t0) ∈ D,(u,∂tu)(t0) ∈ H1loc × L2

loc(Dt0), withDt0 = {x ∈R|(x, t0) ∈D}. Moreover,

• either D=R×[0,+∞),• or D= {(x, t)|0≤ t < T(x)} for some 1-Lipschitz function x �→ T(x).

Step 3: Regular approximations for equation (1.1)Consider (u0,u1) ∈ H1

loc,u × L2loc,u, u its solution constructed in Step 2, and

assume that it is non-global, hence defined under the graph of a 1-Lipschitzfunction x �→ T(x). Consider for any n ∈N a regularized increasing truncationof F satisfying

Fn(u)={

eu if u≤ n,en if u≥ n+ 1

(1.20)

and Fn(u) ≤ min(eu, en+1). Consider also a sequence (u0,n,u1,n) ∈ (C∞(R))2

such that (u0,n,u1,n)→ (u0,u1) in H1×L2(−R,R) as n→∞, for any R > 0.Then, we consider the problem{

∂2t un = ∂2

x un+Fn(un),(un(0),∂tun(0))= (u0,n,u1,n) ∈H1

loc,u×L2loc,u.

(1.21)

Since Steps 1 and 2 clearly extend to locally Lipschitz nonlinearities, we get aunique solution un defined in the half-space R× (0,+∞), or in the subgraphof a 1-Lipschitz function. Since Fn(u)≤ en+1, for all u∈R, it is easy to see thatin fact un is defined for all (x, t) ∈R×[0,+∞). From the regularity of Fn, u0,n

and u1,n, it is clear that un is a strong solution in C2(R, [0,∞)). Introducing thefollowing sets:

K+(x, t)= {(y,s) ∈ (R,R+), |y− x|< s− t}, (1.22)

K−(x, t)= {(y,s) ∈ (R,R+), |y− x|< t− s},and

K±R (x, t)= K±(x, t)∩DR.

We claim the following.

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10 Asma Azaiez, Nader Masmoudi and Hatem Zaag

Lemma 1.6 (Uniform bounds on variations of un in cones) Consider R> 0;one can find C(R) > 0 such that if (x, t) ∈D∩DR, then ∀n ∈N:

un(y,s)≥ un(x, t)−C(R), ∀(y,s) ∈ K+R (x, t),

un(y,s)≤ un(x, t)+C(R), ∀(y,s) ∈ K−(x, t).

Remark Of course C depends also on initial data, but we omit that dependence,since we never change initial data in this setting. Note that since (x, t) ∈DR, itfollows that K−

R (x, t)= K−(x, t).

Proof We will prove the first inequality, the second one can be proved in thesame way. For more details see page 74 of [11].

Let R > 0, consider (x, t) fixed in D ∩ DR, and (y,s) in D ∩ K+R (x, t). We

introduce the following change of variables:

ξ = (y− x)− (s− t), η=−(y− x)− (s− t), un(ξ ,η)= un(y,s). (1.23)

From (1.21), we see that un satisfies:

∂ξηun(ξ ,η)= 1

4Fn(un)≥ 0. (1.24)

Let (ξ , η) be the new coordinates of (y,s) in the new set of variables. Note thatξ ≤ 0 and η ≤ 0. We note that there exists ξ0 ≥ 0 and η0 ≥ 0 such that thepoints (ξ0, η) and (ξ ,η0) lie on the horizontal line {s= 0} and have as originalcoordinates respectively (y∗,0) and (y,0) for some y∗ and y in [−R,R]. We notealso that in the new set of variables, we have:

un(y,s)− un(x, t)= un(ξ , η)−un(0,0)= un(ξ , η)−un(ξ ,0)+ un(ξ ,0)− un(0,0)

=−∫ 0

η

∂ηun(ξ ,η)dη−∫ 0

ξ

∂ξ un(ξ ,0)dξ . (1.25)

From (1.24), ∂ηun is monotonic in ξ . So, for example for η= η, as ξ ≤ 0≤ ξ0,we have:

∂ηun(ξ , η)≤ ∂ηun(0, η)≤ ∂ηun(ξ0, η).

Similarly, for any η ∈ (η,0), we can bound from above the function∂ηun(ξ ,η) by its value at the point (ξ ∗(η),η), which is the projection of (ξ ,η)on the axis {s= 0} in parallel to the axis ξ (as ξ ≤ 0≤ ξ ∗(η)).

In the same way, from (1.24), ∂ξ un is monotonic in η. As η≤ 0≤ η0, we canbound, for ξ ∈ (ξ ,0), ∂ξ un(ξ ,0) by its value at the point (ξ ,η∗(ξ)), which is theprojection of (ξ ,0) on the axis {s= 0} in parallel to the axis η (0 < η∗(ξ)). So

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Blow-up Rate for a Semilinear Wave Equation 11

it follows that:

∂ηun(ξ ,η)≤ ∂ηun(ξ∗(η),η), ∀η ∈ (η,0),

∂ξ un(ξ ,0)≤ ∂ξ un(ξ ,η∗(ξ)), ∀ξ ∈ (ξ ,0).(1.26)

By a straightforward geometrical construction, we see that the coordinates of(ξ ∗(η),η) and (ξ ,η∗(ξ)), in the original set of variables {y,s}, are respectively(x+ t−η

√2,0) and (x− t+η

√2,0). Both points are in [−R,R].

Furthermore, we have from (1.23):

∂ηun(ξ∗(η),η)= 1

2(−∂tun− ∂xun)(x+ t−η

√2,0)

= 1

2(−u1,n− ∂xu0,n)(x+ t−η

√2),

∂ξ un(ξ ,η∗(ξ))= 1

2(−∂tun+ ∂xun)(x− t+η

√2,0) (1.27)

= 1

2(−u1,n+ ∂xu0,n)(x− t+η

√2).

Using (1.27), the Cauchy–Schwarz inequality and the fact that u1,n and ∂xu0,n

are uniformly bounded in L2(−R,R) since they are convergent, we have:∫ 0η∂ηun(ξ

∗(η),η)dη≤ C(R),∫ 0ξ∂ξ un(ξ ,η∗(ξ))dξ ≤ C(R).

(1.28)

Using (1.25), (1.26) and (1.28), we reach the conclusion of Lemma 1.6.

Let us show the following.

Lemma 1.7 (Convergence of un as n→∞) Consider (x, t) ∈ R× [0,+∞).We have the following:

• if t > T(x), then un(x, t)→+∞,• if t < T(x), then un(x, t)→ u(x, t).

Proof We claim that it is enough to show the convergence for a subsequence.Indeed, this is clear from the fact that the limit is explicit and doesn’tdepend on the subsequence. Consider (x, t) ∈ R× [0,+∞); up to extractinga subsequence, there is an l(x, t) ∈R such that un(x, t)→ l(x, t) as n→∞.

Let us show that l = −∞. Since Fn(u)≥ 0, it follows that un(x, t)≥ un(x, t),where {

∂2t un = ∂2

x un,un(0)= u0,n and ∂tun(0)= u1,n.

(1.29)

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12 Asma Azaiez, Nader Masmoudi and Hatem Zaag

Since un ∈ L∞loc(R+,H1(−R,R)) ⊂ L∞loc(R

+,L∞(−R,R)), for any R > 0, fromthe fact that (u0,n,u1,n) is convergent in H1

loc × L2loc, it follows that l(x, t) ≥

limsupn→+∞ un(x, t) >−∞.Note from the fact that Fn(u)≤ eu that we have

∀x ∈R, t < T(x), un(x, t)≤ u(x, t). (1.30)

Introducing R = |x| + t+ 1, we see by definition (1.9) of DR that (x, t) ∈ DR.Let us handle two cases in the following.

Case 1: t < T(x)Let us introduce vn, the solution of{

∂2t vn = ∂2

x vn+ evn ,vn(0)= u0,n and ∂tvn(0)= u1,n ∈H1

loc,u×L2loc,u.

From the local Cauchy theory in H1loc,u×L2

loc,u and the Sobolev embedding, weknow that

vn → u uniformly as n→∞ in compact sets of D. (1.31)

Let us consider

K = K−(x,(t+T(x))/2)

and M =max(y,s)∈K |u(y,s)|<+∞, since K is a compact set in D.From (1.31), we may assume n large enough, so that

||u0,n− u0||L∞(K∩{t=0}) ≤ 1,

sup(y,s)∈K

|vn(y,s)| ≤ M+ 1 (1.32)

and

n≥ M+ 3. (1.33)

In particular,

||u0,n||L∞(K∩{t=0}) ≤ M+ 1. (1.34)

We claim that

∀(y,s) ∈ K, |un(y,s)| ≤ M+ 2. (1.35)

Indeed, arguing by contradiction, we may assume from (1.34) and continuityof un that

∀s ∈ [0, tn], ||un(s)||L∞(K∩{t=s}) ≤ M+ 2 (1.36)

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Blow-up Rate for a Semilinear Wave Equation 13

and

||un(tn)||L∞(K∩{t=tn}) = M+ 2, (1.37)

for some tn ∈ (0, t+T(x)2 ).

From (1.33), (1.36) and the definition (1.20) of Fn, we see that

∀(y,s) ∈ K with s≤ tn,Fn(un(y,s))= eun(y,s).

Therefore, un and vn satisfy the same equation with the same initial data onK ∩ {s ≤ tn}. From uniqueness of the solution to the Cauchy problem, we seethat

∀(y,s) ∈ K with s≤ tn,un(y,s)= vn(y,s).

A contradiction then follows from (1.32) and (1.37). Thus, (1.35) holds.Again, from the choice of n in (1.33), we see that

∀(y,s) ∈ K,Fn(un(y,s))= eun(y,s),

hence, from uniqueness,

∀(y,s) ∈ K,un(y,s)= vn(y,s).

From (1.31), and since (x, t) ∈ K, it follows that un(x, t)→ u(x, t) as n→∞.

Case 2: t > T(x)Assume by contradiction that l <+∞. From Lemma 1.6, it follows that

∀(y,s) ∈ K−(x, t), un(y,s)≤ un(x, t)+C(R).

For n≥ n0 large enough, this gives un(y,s)≤ l+ 1+C(R).If M = E(l+ 1+C(R))+ 1, then

∀n≥max(M,n0), ∀(y,s) ∈ K−(x, t),Fn(un(y,s))= eun(y,s),

and un satisfies (1.1) in K−(x, t) with initial data (u0,n,u1,n)→ (u0,u1) ∈ H1×L2(K−(x, t)∩ {t= 0}). From the finite speed of propagation and the continuityof solutions to the Cauchy problem with respect to the initial data, it followsthat un and u are both defined in K−(x, t) for n large enough, in particularu is defined at (x,s) with T(x) < s < t with u = un in K−(x, t). This gives acontradiction with the expression of the domain of definition (1.4) of u.

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14 Asma Azaiez, Nader Masmoudi and Hatem Zaag

1.3 Energy Estimates

In this section, we use some localized energy techniques from Shatah andStruwe [24] to derive a non-blow-up criterion which will give the lower boundin Theorem 1.1. More precisely, we give the following.

Proposition 1.8 (Non-blow-up criterion for a semilinear wave equation)∀c0 > 0, there exist M0(c0) > 0 and M(c0) > 0 such that, if

(H) :

{||∂xu0||2L2(−1,1)

+||u1||2L2(−1,1)≤ c2

0

∀|x|< 1, u0(x)≤M0,(1.38)

then equation (1.1) with initial data (u0,u1) has a unique solution (u,∂tu) ∈C([0,1),H1×L2(|x|< 1− t)) such that for all t ∈ [0,1) we have:

||∂xu(t)||2L2(|x|<1−t)+||∂tu(t)||2L2(|x|<1−t) ≤ 2c20 (1.39)

and

∀|x|< 1− t, u(x, t)≤M. (1.40)

Note that here we work in the space H1loc×L2

loc which is larger than the spaceH1

loc,u×L2loc,u which is adopted elsewhere for equation (1.1). Before giving the

proof of this result, let us first give the following corollary, which is a directconsequence of Proposition 1.8.

Corollary 1.9 There exists ε0 > 0 such that if∫ 1

−1(u1(x))

2+ (∂xu0(x))2+ eu0(x) dx≤ ε0, (1.41)

then the solution u of equation (1.1) with initial data (u0,u1) doesn’t blow upin the cone C0,1,1.

Let us first derive Corollary 1.9 from Proposition 1.8.

Proof of Corollary 1.9 assuming that Proposition 1.8 holdsFrom (1.41), if ε0 ≤ 1 we see that∫ 1

−1

((u1(x))

2+ (∂xu0(x))2)

dx≤ ε0 ≤ 1, (1.42)∫ 1

−1eu0(x)dx≤ ε0.

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Blow-up Rate for a Semilinear Wave Equation 15

Therefore, for some x0 ∈ (−1,1), we have 2eu0(x0) = ∫ 1−1 eu0(x)dx ≤ ε0, hence

u0(x0)≤ log ε02 . Using (1.42), we see that for all x ∈ (−1,1),

u0(x)= u0(x0)+∫ x

x0

∂xu0 ≤ u0(x0)+√

2

(∫ 1

−1(∂xu0(x))

2dx

) 12

≤ logε0

2+√

2ε0 ≤M0(1),

defined in Proposition 1.8, provided that ε0 is small enough. Therefore, thehypothesis (H) of Proposition 1.8 holds with c0= 1, and so does its conclusion.This concludes the proof of Corollary 1.9, assuming that Proposition 1.8 holds.

Now, we give the proof of Proposition 1.8.

Proof of Proposition 1.8 Consider c0 > 0 and introduce

M0 = log

(c2

0

16

)− c0

√2− c2

0

8and M(c0)= log

(c2

0

16

).

Then, we consider (u0,u1) satisfying hypothesis (H). From the solution of theCauchy problem in H1

loc×L2loc, which follows exactly by the same argument as

in the space H1loc,u×L2

loc,u presented in Section 1.2, there exists t∗ ∈ (0,1] suchthat equation (1.1) has a unique solution with (u,∂tu)∈C([0, t∗),H1×L2(|x|<1− t)). Our aim is to show that t∗ = 1 and that (1.39) and (1.40) hold for allt ∈ [0,1).

Clearly, from the solution of the Cauchy problem, it is enough to show that(1.39) and (1.40) hold for all t ∈ [0, t∗), so we only do that in the following.

Arguing by contradiction, we assume that there exists at least some timet∈ [0, t∗) such that either (1.39) or (1.40) doesn’t hold. If t is the lowest possiblet, then we have from continuity either

||∂xu(t)||2L2(|x|<1−t)+||∂tu(t)||2L2(|x|<1−t) = 2c0,

or

∃|x0|< 1− t, such that u(x0, t)=M.

Note that since (1.39) holds for all t ∈ [0, t), it follows that

∀t ∈ [0, t),∀|x|< 1− t,u(x, t)≤M = log

(c2

0

16

). (1.43)

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16 Asma Azaiez, Nader Masmoudi and Hatem Zaag

Following the alternative on t, two cases arise in the following.

Case 1: ||∂xu(t)||2L2(|x|<1−t)

+||∂tu(t)||2L2(|x|<1−t)= 2c2

0.Referring to Shatah and Struwe [24], we see that:∫

|x|<1−t( 1

2 (∂xu2+ ∂tu2)− eu) dx−

∫|x|<1

( 12 (∂xu2

0+ u21)− eu0) dx

=∫�

(eu− 12 |∂xu− x

|x|∂tu|2) dσ , (1.44)

where� = {(x, t) ∈R×R+, such that |x| = 1− t}∩ [0, t].

Using (1.43), it follows that∫|x|<1−t

eu(x,t)dx≤∫|x|<1−t

eM ≤ c20

8.

∫�

eudσ ≤∫ t

0(eu(1−t,t)+ eu(t−1,t))dt≤ c2

0

8.

Therefore, from (1.44) and (1.38), we write∫|x|<1−t

((∂xu)2+ (∂tu)2)dx≤

∫|x|<1

(∂xu0)2dx+ (u1)

2+∫|x|<1−t

eu(x,t)dx+∫�

eudσ

≤ c20+

3

8c2

0 < 2c20,

which is a contradiction.

Case 2: ∃x0 ∈ (−(1− t),1− t), u(x0, t)=M.Recall Duhamel’s formula:

∀|x|< 1− t,

u(x, t)= 1

2(u0(x− t)+ u0(x+ t)+ 1

2

∫ x+t

x−tu1(z)dz

+1

2

∫ t

0

∫ x+t−τ

x−t+τ

eu(z,τ) dzdτ . (1.45)

From (H), we write∫ x+t

x−tu1 dx≤

(∫ 1

−1u2

1

) 12

2√

2≤ c0

√2.

From (1.43), we write∫ t

0

∫ z+t−τ

z−t+τ

eu(z,τ)dzdτ ≤∫ t

0

∫ z+t−τ

z−t+τ

c20

16≤ c2

0

8.

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Blow-up Rate for a Semilinear Wave Equation 17

Since u0(x± t)≤M0 = log(c2

016 )− c0

√2− c2

08 , it follows from (1.45) that

u(x, t)≤M0+ c0

√2

2+ c2

0

16< log

(c2

0

16

)=M,

and a contradiction follows.This concludes the proof of Proposition 1.8. Since we have already derived

Corollary 1.9 from Proposition 1.8, this is also the conclusion of the proof ofCorollary 1.9.

1.4 ODE Type Estimates

In this section, we extend the work of Godin in [11]. In fact, we show that hisestimates hold for more general initial data. As in the introduction, we consideru(x, t) a non-global solution of equation (1.1) with initial data (u0,u1)∈H1

loc,u×L2

loc,u. This section is organized as follows.In the first subsection, we give some preliminary results and we show that

the solution goes to +∞ on the graph �.In the second subsection, we give and prove upper and lower bounds on the

blow-up rate.

1.4.1 Preliminaries

In this subsection, we first give some geometrical estimates on the blow-upcurve (see Lemmas 1.10, 1.11 and 1.12). Then, we use equation (1.1) to derivea kind of maximum principle in light cones (see Lemma 1.13), then a lowerbound on the blow-up rate (see Proposition 1.14).

We first give the following geometrical property concerning the distance to{t= T(x)}, the boundary of the domain of definition of u(x, t).

Lemma 1.10 (Estimate for the distance to the blow-up boundary) For all(x, t) ∈D, we have

1√2(T(x)− t)≤ d((x, t),�)≤ T(x)− t, (1.46)

where d((x, t),�) is the distance from (x, t) to �.

Proof Note first by definition that

d((x, t),�)≤ d((x, t),(x,T(x))= T(x)− t.

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18 Asma Azaiez, Nader Masmoudi and Hatem Zaag

Then, from the finite speed of propagation, � is above Cx,T(x),1, the backwardlight cone with vertex (x,T(x)). Since (x, t) ∈ Cx,T(x),1, it follows that

d((x, t),�)≥ d((x, t),Cx,T(x),1)=√

2

2(T(x)− t).

This concludes the proof of Lemma 1.10.

Now, we give a geometrical property concerning distances, specific fornon-characteristic points.

Lemma 1.11 (A geometrical property for non-characteristic points) Leta ∈R. There exists c := C(δ), where δ = δ(a) is given by (1.6), such that forall (x, t) ∈ Ca,T(a),1,

1

c≤ T(x)− t

T(a)− t≤ c.

Remark From Lemma 1.10, it follows that

1

c≤ d((x, t),�)

d((a, t),�)≤ c

whenever a ∈R and (x, t) ∈ Ca,T(a),1.

Proof Let a be a non-characteristic point. We recall from condition (1.6) that

∃δ = δ(a) ∈ (0,1) such that u is defined on Ca,T(a),δ .

Let (x, t) be in the light cone with vertex (a,T(a)). Using the fact that theblow-up graph is above the cone Ca,T(a),δ and the fact that (x, t) ∈ Ca,T(a),1, wesee that

T(x)− t≥ T(a)− δ|x− a|− t ≥ (T(a)− t)(1− δ). (1.47)

In addition, as � is a 1-Lipschitz graph, we have

T(x)≤ T(a)+|x− a|,so, for all (x, t) ∈ Ca,T(a),1,

T(x)− t≤ T(a)+|x− a|− t ≤ 2(T(a)− t). (1.48)

From (1.47) and (1.48), there exists c= c(δ) such that

1

c≤ T(x)− t

T(a)− t≤ c.

This concludes the proof of Lemma 1.11.

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Blow-up Rate for a Semilinear Wave Equation 19

M1

βP1

M2

ξ

τ

M

t

T(x)

τ = T (ξ)

slope δ

slope 1

M0N1

N0

x

α

Figure 1.1. Illustration for the proof of (1.49)

Finally, we give the following coercivity estimate on the distance to theblow-up curve, still specific for non-characteristic points.

Lemma 1.12 Let x∈R and t ∈ [0,T(x)). For all τ ∈ [0, t) and j= 1,2, we have

d((zj,wj),�)≥ 1

C(d((x, t),�)+|(x, t)− (zj,wj)|), (1.49)

where (z1,w1)= (x+ t− τ ,τ) and (z2,w2)= (x− t+ τ ,τ).

Remark Note that (zj,wj) for j = 1,2 lie on the backward light cone withvertex (x, t).

Proof Consider x ∈R, t ∈ [0,T(x)). By definition, there exists δ ∈ (0,1) suchthat Cx,T(x),δ ⊂ D. We will prove the estimate for j= 1 and τ ∈ [0, t), since thethe estimate for j= 2 follows by symmetry. In order to do so, we introduce thefollowing notations, as illustrated in Figure 1.1: M = (x,T(x)), M0 = (x, t) andM1 = (z1,w1) = (x+ t− τ ,τ), which is on the left boundary of the backwardlight cone Cx,t,1; N1 the orthogonal projection of M1 on the left boundary ofthe cone Cx,T(x),δ; P1 the orthogonal projection of M0 on [N1,M1]. Note thatthe quadrangle M0N0N1P1 is a rectangle. If α is such that tanα = δ and β =PM1M0, then we see from elementary considerations on angles that β = α+ π

4

and N0M0M = α.Therefore, using Lemma 1.10, and the angles on the triangle M0N0M, we

see that:

d((x, t),�)≤ T(x)− t=MM0 = M0N0

cosα= N1P1

cosα. (1.50)

Moreover, since the blow-up graph is above the cone Cx,T(x),δ , it follows that

d((z1,w1),�)≥M1N1.

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20 Asma Azaiez, Nader Masmoudi and Hatem Zaag

In particular,

M1N1 = N1P1+P1M1 = N1P1+ cos( π4 +α)M1M0. (1.51)

Since 0 < δ < 1, hence 0 < α < π4 , it follows that cos( π4 + α) > 0. Since

M1M0 = |(z1,w1)− (x, t)|, the result follows from (1.50) and (1.51).In the same way, we can prove this for the other point M2 = (z2,w2), which

gives (1.49).

Now, we give the following corollary from the approximation procedure inLemmas 1.6 and 1.7.

Lemma 1.13 (Uniform bounds on variations of u in cones) For any R > 0,there exists a constant C(R) > 0 such that if (x, t) ∈D∩DR then

u(y,s)≥ u(x, t)−C(R), ∀(y,s) ∈D∩K+R (x, t),

u(y,s)≤ u(x, t)+C(R), ∀(y,s) ∈ K−(x, t),

where the cones K± and K±R are defined in (1.22).

Remark The constant C(R) depends also on u0 and u1, but we omit thisdependence in the sequel.

In the following, we give a lower bound on the blow-up rate and we showthat u(x, t)→+∞ as t→ T(x).

Proposition 1.14 (A general lower bound on the blow-up rate)

(i) If (u0,u1) ∈W1,∞×L∞, then for all R > 0, there exists C(R) > 0 such thatfor all (x, t) ∈D∩DR,

d((x, t),�)eu(x,t) ≥ C.

In particular, for all (x, t) ∈D∩DR, u(x, t)→+∞ as d((x, t),�)→ 0.

(ii) If we only have (u0,u1) ∈ H1loc,u × L2

loc,u, then for all R > 0, there existsC(R) > 0 such that for all (x0, t) ∈D∩DR,

1

T(x0)− t

∫|x−x0|<T(x0)−t

e−u(x,t)dx≤ C(R)√

d(x0, t).

In particular, e−u converges to 0 on average over slices of the light cone, asd(x0, t)→ 0.

Remark Near non-characteristic points, we are able to derive the optimallower bound on the blow-up rate. See item (ii) of Proposition 1.15.

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Blow-up Rate for a Semilinear Wave Equation 21

Proof of Proposition 1.14

(i) Clearly, the last sentence in item (i) follows from the first, hence, we onlyprove the first.

Let R > 0 and (x, t)∈D∩DR. Using the approximation procedure defined in(1.21), we write un = un+ un with:

un(x, t)= 1

2

(u0,n(x− t)+ u0,n(x+ t)

)+ 1

2

∫ x+t

x−tu1,n(ξ)dξ ,

un(x, t)= 1

2

∫ t

0

∫ x+t−τ

x−t+τ

Fn(un(z,τ))dzdτ .

(Note that un was already defined in (1.29).)Since Fn ≥ 0 from (1.20), it follows that

un(x, t)≥ un(x, t)≥−C(R) for all (x, t) ∈D∩DR. (1.52)

Differentiating un, we see that

∂tun(x, t)= 1

2

(∂xu0,n(x+ t)− ∂xu0,n(x− t)

)+ 1

2

(u1,n(x+ t)+ u1,n(x− t)

)≤ ||∂xu0,n||L∞(−R,R)+||u1,n||L∞(−R,R) ≤ C(R). (1.53)

Differentiating un, we get

∂tun(x, t)= 1

2

∫ t

0(Fn(un(x− t+ τ ,τ),+Fn(un(x+ t− τ ,τ)))dτ

≤ 1

2

∫ t

0

(eun(x−t+τ ,τ)+ eun(x+t−τ ,τ)

)dτ

since Fn(u)≤ eu. Since un(x− t+ τ ,τ)≤ un(x, t)+C(R) and un(x+ t− τ ,τ)≤un(x, t)+C(R) from Lemma 1.6, it follows that

∂tun(x, t)≤ cteun(x,t) ≤ C(R)eun(x,t). (1.54)

Therefore, using (1.52) we see that

∂tun(x, t)= ∂tun(x, t)+ ∂tun(x, t)≤ C(R)+C(R)eun(x,t) ≤ C(R)eun(x,t),

hence

∂tun(x, t)e−un(x,t) ≤ C(R). (1.55)

Integrating (1.55) on any interval [t1, t2] with 0 ≤ t1 < T(x) < t2, we gete−un(x,t1) − e−un(x,t2) ≤ C(t2 − t1). Making n →∞ and using Lemma 1.7 wesee that e−u(x,t1) ≤ C(t2− t1).

Taking t1 = t and making t2 → T(x), we get e−u(x,t) ≤ C(T(x)− t). UsingLemma 1.10 concludes the proof of item (i) of Proposition 1.14.

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22 Asma Azaiez, Nader Masmoudi and Hatem Zaag

(ii) If (u0,u1)∈H1loc,u×L2

loc,u, then a small modification in the argument of item(i) gives the result. Indeed, if t0 ∈ [0,T(x0)),

a0 = x0− (T(x0)− t0), b0 = x0+ (T(x0)− t0),

and t≥ 0, we write from (1.52) and (1.53)∫ b0

a0

∂tun(x, t)e−un(x,t)dx≤ C(R)√

b0− a0.

Furthermore, from (1.54) we write∫ b0

a0

∂tun(x, t)e−un(x,t)dx≤ C(R)(b0− a0).

Therefore, it follows that

− d

dt

∫ b0

a0

e−un(x,t)dx=∫ b0

a0

∂tun(x, t)e−un(x,t)dx≤ C(R)√

b0− a0. (1.56)

Integrating (1.56) on an interval (t0, t′0), where

t′0 = 2T(x0)− t0, (1.57)

we get∫ b0

a0

e−un(x,t0)dx−∫ b0

a0

e−un(x,t′0)dx≤ C(R)√

b0− a0(t′0− t0)

= 2√

2C(R)(T(x0)− t0)32 . (1.58)

Since x �→ T(x) is 1-Lipschitz and T(x0) is the middle of [t0, t′0], we clearlysee that the segment [a0,b0] × {t′0} lies outside the domain of definition ofu(x, t); using Lemma 1.7, we see that∫ b0

a0

e−un(x,t′0)dx→ 0 as n→+∞,

on the one hand (we use the Lebesgue Lemma together with the bound (1.52)).On the other hand, similarly, we see that

∫ b0

a0

e−un(x,t0)dx→∫ b0

a0

e−u(x,t0)dx as n→+∞.

Thus, the conclusion follows from (1.58), together with Lemma 1.10, thisconcludes the proof of Proposition 1.14.

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Blow-up Rate for a Semilinear Wave Equation 23

1.4.2 The Blow-up Rate

This subsection is devoted to bounding the solution u. We have obtained thefollowing result.

Proposition 1.15 For any R > 0, there exists C(R) > 0, such that:

(i) (Upper bound on u) for all (x, t) ∈D∩DR we have

eud((x, t),�)2 ≤ C;

(ii) (Lower bound on u) if, in addition, (u0,u1) ∈ W1,∞ × L∞ and x is anon-characteristic point, then for all (x, t) ∈D∩DR,

eud((x, t),�)2 ≥ 1

C.

Remark In [11] Godin didn’t use the notion of characteristic point, butthe regularity of initial data was fundamental to achieve the result. In thiswork, our initial data are less regular, so we have focused on the case of anon-characteristic point in order to get his result.

Proof of Proposition 1.15(i) Consider R > 0. We will show the existence of some C(R) > 0 such that forany (x, t1) ∈D∩DR, we have

eud((x, t),�)2 ≤ C(R).

Consider then (x, t1) ∈ D∩DR. Since x �→ T(x) is 1-Lipschitz, we clearly seethat

(x,T(x)) ∈DR with R= 2R+T(a)+ 1. (1.59)

Consider now t2 ∈ (t1,T(x)), to be fixed later. We introduce the square domainwith vertices (x, t1),(x+ t2−t1

2 , t1+t22 ),(x, t2),(x− t2−t1

2 , t1+t22 ). Let

Tsup = {(ξ , t) | t1− t22

< t < t2, |x− ξ |< t2− t},

Tinf = {(ξ , t) | t1 < t <t1− t2

2, |x− ξ |< t− t1},

respectively the upper and lower half of the considered square. FromDuhamel’s formula, we write:

u(x, t2)= 1

2u

(x+ t2− t1

2,t2+ t1

2

)+ 1

2u

(x− t2− t1

2,t2+ t1

2

)+1

2

∫ x+ t2−t12

x− t2−t12

∂tu

(ξ ,

t2− t12

)dξ + 1

2

∫Tsup

eu(ξ ,t) dξdt

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24 Asma Azaiez, Nader Masmoudi and Hatem Zaag

and

u(x, t1)= 1

2u

(x+ t2− t1

2,t2+ t1

2

)+ 1

2u

(x− t2− t1

2,t2+ t1

2

)−1

2

∫ x+ t2−t12

x− t2−t12

∂tu

(ξ ,

t2− t12

)dξ + 1

2

∫Tinf

eu(ξ ,t) dξdt.

So,

u(x, t1)+ u(x, t2) (1.60)

= u

(x+ t2− t1

2,t2+ t1

2

)+ u

(x− t2− t1

2,t2+ t1

2

)+ 1

2

∫Tinf∪Tsup

eu(ξ ,t) dξdt.

Since the square Tinf ∪Tsup ⊂DR from (1.59), applying Lemma 1.13, we havefor all (x, t) ∈ Tinf ∪Tsup and for some C(R) > 0:

u(x, t)≥ u(x, t1)−C.

Applying this to (1.60), we get

u(x, t2)≥ u(x, t1)− 2C+ (t2− t1)2

4e(u(x,t1)−C).

Now, choosing t2= t1+σe−u(x,t1)/2, where σ = 2ec2√η+ 2C, we see that either

(x, t) /∈D or (x, t2) ∈D and u(x, t2)≥ u(x, t1)+ 1 by the above-given analysis.In the second case, we may proceed similarly and define for n≥ 3 a sequence

tn = tn−1+σe−u(x,tn−1)/2, (1.61)

as long as (x, tn−1) ∈ D. Clearly, the sequence (tn) is increasing whenever itexists. Repeating between tn and tn−1, for n ≥ 3, the argument we first wrotefor t1 and t2, we see that

u(x, tn)≥ u(x, tn−1)+ 1, (1.62)

as long as (x, tn) ∈D. Two cases arise then.

Case 1: The sequence (tn) can be defined for all n≥ 1, which means that

(x, tn) ∈D, ∀n ∈N∗. (1.63)

In particular, (1.61) and (1.62) hold for all n≥ 2.If t∞ = limn→∞ tn, then, from (1.63), we see that t∞ ≤ T(x).Since u(x, tn)→+∞ as n→∞ from (1.62), we need to have

t∞ = T(x),

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Blow-up Rate for a Semilinear Wave Equation 25

from the Cauchy theory. Therefore, using Lemma 1.10, (1.61) and (1.62), wesee that

d((x, t1),�)≤ T(x)− t1 =∞∑

n=1

(tn+1− tn)≤ σ

∞∑n=1

e−(u(x,t1)+(n−1))/2

≡ C(R)e−u(x,t1)/2,

which is the desired estimate.

Case 2: The sequence (tn) exists only for all n ∈ [1,k] for some k ≥ 2. Thismeans that (x, tk) /∈D, that is tk ≥ T(x).

Moreover, (1.61) holds for all n∈ [2,k], and (1.62) holds for all n∈ [2,k−1](in particular, it is never true if k = 2). As for Case 1, we use Lemma 1.10,(1.61) and (1.62) to write

d((x, t1),�)≤ T(x)− t1 ≤ tk− t1 =k−1∑n=1

(tn+1− tn)≤ σ

k−1∑n=1

e−(u(x,t1)+(n−1))/2

≤ σe−(u(x,t1))/2k−1∑n=1

e(n−1))/2 ≡ C(R)e−u(x,t1)/2,

which is the desired estimate. This concludes the proof of item (i) ofProposition 1.15.

(ii) Consider R > 0 and x a non-characteristic point such that (x, t) ∈ DR ∩D.We dissociate u into two parts u= u+ u with:

u(x, t)= 1

2(u0(x− t)+ u0(x+ t))+ 1

2

∫ x+t

x−tu1(ξ)dξ ,

u(x, t)= 1

2

∫ t

0

∫ x+t−τ

x−t+τ

eu(z,τ)dzdτ .

Differentiating u, we see that

∂tu(x, t)= 1

2(∂xu0(x+ t)− ∂xu0(x− t))+ 1

2(u1(x+ t)+ u1(x− t))

≤ ||∂xu0||L∞(−R,R)+||u1||L∞(−R,R) ≤ C(R),

since (x, t) ∈DR. Consider now an arbitrary a ∈ ( 12 ,1). Since u(x, t)→+∞ as

d((x, t),�)→ 0 (see Proposition 1.14), it follows that

e(a−1)u(x,t)∂tu(x, t)≤ C(R)d((x, t),�)−2a+1. (1.64)

Now, we will prove a similar inequality for u. Differentiating u, we see that

∂tu= 1

2

∫ t

0(eu(x−t+τ ,τ)+ eu(x+t−τ ,τ))dτ . (1.65)

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26 Asma Azaiez, Nader Masmoudi and Hatem Zaag

Using the upper bound in Proposition 1.15, part (i), which is already proved,and Lemma 1.13, we see that for all (y,s) ∈ K−(x, t),

u(y,s)= (1− a)u(y,s)+ au(y,s)≤ (1− a)(u(x, t)+C)

+ a(logC− 2logd((y,s),�)).

So,

eu(y,s) ≤ Ce(1−a)u(x,t)(d((y,s),�))−2a. (1.66)

Since x is a non-characteristic point, there exists δ0 ∈ (0,1) such that the coneCx,T(x),δ0 is below the blow-up graph �.

Applying (1.66) and Lemma 1.12 to (1.65), and using the fact that |(x, t)−(z1,w1)|2 = 2(τ − t)2, we write (recall that 1

2 < a < 1):

∂tu= 1

2

∫ t

0eu(z1,w1)+ eu(z2,w2)dτ

≤ 1

2

∫ t

0Ce(1−a)u(x,t)(d((z1,w1),�)

−2a+ d((z2,w2),�)−2a)dτ

≤ Ce(1−a)u(x,t)∫ t

0(d((x, t),�)+|(x, t)− (z1,w1)|)−2a

+ (d((x, t),�)+|(x, t)− (z1,w1)|)−2a dτ

≤ Ce(1−a)u(x,t)∫ t

0

(d((x, t),�)+√2(t− τ)

)−2adτ

≤ Ce(1−a)u(x,t) 1√2(2a− 1)

d((x, t),�)−2a+1,

which yields

e(a−1)u(x,t)∂tu(x, t)≤ Cd((x, t),�)−2a+1. (1.67)

In conclusion, we have from (1.64), (1.67) and Lemma 1.10:

e(a−1)u(x,t)∂tu(x, t)≤ Cd((x, t),�)−2a+1 ≤ C(T(x)− t)−2a+1. (1.68)

Since u(x, t) → +∞ as d((x, t),�) → 0 from Proposition 1.14, integrating(1.68) between t and T(x), we see that

e(a−1)u(x,t) ≤ C(T(x)− t)2−2a.

Using Lemma 1.10 again, we complete the proof of part (ii) of Proposi-tion 1.15.

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Blow-up Rate for a Semilinear Wave Equation 27

1.5 Blow-up Estimates for Equation (1.1)

In this section, we prove the three results of our paper: Theorem 1.1,Theorem 1.2 and Proposition 1.3. Each proof is given in a separate subsection.

1.5.1 Blow-up Estimates in the General Case

In this subsection, we use energy and ODE type estimates from previoussections and give the proof of Theorem 1.1.

Proof of Theorem 1.1 (i) Let R > 0, a ∈R such that (a,T(a)) ∈DR and (x, t) ∈Ca,T(a),1. Consider (ξ ,τ), the closest point of Ca,T(a),1 to (x, t). This means that

||(x, t)− (ξ ,τ)|| = infx′∈R

{||(x, t)− (x′, |x′| = 1− t)||} = d((x, t),Ca,T(a),1).

By a simple geometrical construction, we see that it satisfies the following:{τ = T(a)− (ξ − a),τ = t+ (ξ − x),

(1.69)

hence, T(a)− t− 2ξ + a+ x= 0, so

ξ − x= 1

2((T(a)− t)+ (a− x)) . (1.70)

Using the second equation of (1.69) and (1.70) we see that:

||(x, t)− (ξ ,τ)|| =√(ξ − x)2+ (τ − t)2 =√2|ξ − x|

=√

2

2|(T(a)− t)+ (a− x)|.

Thus,

d((x, t),�)≥ d((x, t),Ca,T(a),1)=√

2

2|(T(a)− t)+ (a− x)|. (1.71)

From Proposition 1.15, (1.71) and the similarity transformation (1.7) we have:

ewa(y,s) ≤ (T(a)− t)2eu(x,t) ≤ C(T(a)− t)2

d((x, t),�)2≤ C

(T(a)− t

T(a)− t−|x− a|)2

≤ C

(1−|y|)2,

which gives the first inequality of (i). The second one is given by Proposi-tion 1.15 and Lemma 1.10.

(ii) This is a direct consequence of Proposition 1.14 and Lemma 1.10.

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28 Asma Azaiez, Nader Masmoudi and Hatem Zaag

(iii) Now, we will use Section 1.3 to prove this. Arguing by contradiction,we assume that u is not global and that ∀ε0 > 0, ∃x0 ∈R,∃t0 ∈ [0,T(x0)), suchthat

1

T(x0)− t0

∫I((ut(x, t0))

2+ (ux(x, t0))2+ eu(x,t0))dx <

ε0

(T(x0)− t0)2,

where I = (x0− (T(x0)− t0),x0+ (T(x0)− t0)).

We introduce the following change of variables:

v(ξ ,τ)= u(x, t)+ log(T(x0)− t0), with x= x0+ ξ(T(x0)− t0),

t= t0+ τ(T(x0)− t0).

Note that v satisfies equation (1.1). For ε0 = ε0, v satisfies (1.41), so, byCorollary 1.9, v doesn’t blow-up in {(ξ ,τ)| |ξ | < 1 − τ ,τ ∈ [0,1)}, thus, udoesn’t blow-up in {(x, t)| |x − x0| < T(x0) − t, t ∈ [t0,T(x0))}, which is acontradiction. This concludes the proof of Theorem 1.1.

1.5.2 Blow-up Estimates in the Non-characteristic Case

In this subsection, we prove Theorem 1.2. We give first the following corollaryof Proposition 1.15.

Corollary 1.16 Assume that (u0,u1) ∈W1,∞×L∞. Then, for all R > 0, a ∈Rsuch that (a,T(a)) ∈DR, we have for all s≥− logT(a) and |y|< 1

|wa(y,s)| ≤ C(R).

Proof of Corollary 1.16 Assume that (u0,u1) ∈W1,∞ × L∞ and consider R >

0 and a ∈ R such that (a,T(a)) ∈ DR. On the one hand, we recall fromProposition 1.15 that

∀(x, t) ∈ Ca,T(a),1,1

C≤ eud((x, t),�)2 ≤ C.

Using (1.7), we see that

1

C≤(

d((x, t),�)

T(a)− t

)2

ewa(y,s) ≤ C, with y= x− a

T(a)− tand s=− log(T(a)− t).

Since

1

C≤ d((x, t),�)

T(x)− t≤ C,

from Lemmas 1.10 and 1.11, this yields the conclusion of Corollary 1.16.

Now, we give the proof of Theorem 1.2.

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Blow-up Rate for a Semilinear Wave Equation 29

Proof of Theorem 1.2 Consider R > 0 and a ∈R such that (a,T(a)) ∈DR. Wenote first that the fact that |wa(y,s)| ≤ C for all |y| < 1 and s ≥ − logT(a)follows from Corollary 1.16. It remains only to show that

∫ 1−1

((∂swa(y,s))2+

(∂ywa(y,s))2)

dy ≤ C. From Proposition 1.15 and Lemmas 1.10 and 1.11, wehave:

∀(x, t) ∈ Ca,T(a),1, eu(x,t) ≤ C(R)

(T(a)− t)2. (1.72)

We define Ea, the energy of equation (1.1), by

Ea(t)= 1

2

∫|x−a|<T(a)−t

((∂tu(x, t))2+ (∂xu(x, t))2)dx−∫|x−a|<T(a)−t

eu(x,t)dx.

(1.73)From Shatah and Struwe [24], we have

d

dtEa(t)≤ Ceu(a−(T(a)−t),t)+Ceu(a+(T(a)−t),t).

Integrating it over [0, t) and using (1.72), we see that

Ea(t)≤ Ea(0)+C∫ t

0eu(a−(T(a)−s),s)ds+C

∫ t

0eu(a+(T(a)−s),s)ds

≤ C(a, ||(u0,u1)||H1×L2(−R,R))+C∫ t

0

ds

(T(a)− s)2≤ C+ C

(T(a)− t).

Thus,

Ea(t)≤ C

(T(a)− t). (1.74)

Now, using (1.72) and (1.74) to bound the two first terms in the definition ofEa(t), (1.73), we get

1

2

∫|x−a|<T(a)−t

((∂tu(x, t))2+ (∂xu(x, t))2)dx≤∫|x−a|<T(a)−t

eudx+ C

(T(a)− t)

≤∫|x−a|<T(a)−t

C

(T(a)− t)2dx+ C

(T(a)− t)≤ C

(T(a)− t).

(1.75)

Writing inequality (1.75) in similarity variables, we get for all s ≥− logT(a), ∫ 1

−1(∂swa(y,s))2+

∫ 1

−1(∂ywa(y,s))2 ≤ C(R).

This yields the conclusion of Theorem 1.2.

Now, we give the proof of Proposition 1.3.

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30 Asma Azaiez, Nader Masmoudi and Hatem Zaag

Proof of Proposition 1.3 Multiplying (1.8) by ∂sw, and integrating over(−1,1), we see that∫ 1

−1∂sw∂2

s wdy−∫ 1

−1∂sw∂y((1− y2)∂yw)dy−

∫ 1

−1∂swew dy+ 2

∫ 1

−1∂swdy

=−∫ 1

−1(∂sw)2 dy+ 2

∫ 1

−1y∂sw∂2

y,swdy.

Thus,

d

ds

(∫ 1

−1

1

2(∂sw)2+ 2w− ew dy

)+ I1 =−

∫ 1

−1(∂sw)2 dy+ I2,

where

I1 =−∫ 1

−1∂y((1− y2)∂yw

)∂swdy=

∫ 1

−1

((1− y2)∂yw

)∂2

yswdy

= 1

2

d

ds

(∫ 1

−1(1− y2)(∂sw)2 dy

)and

I2 =−2∫ 1

−1y∂sw∂2

y,swdy=−∫ 1

−1y∂y(∂sw)2 dy

=∫ 1

−1(∂sw)2 dy− (∂sw(−1,s))2− (∂sw(1,s))2.

Thus,

d

ds

(∫ 1

−1

[1

2(∂sw)2+ 1

2(1− y2)(∂yw)2− ew+ 2w

]dy

)=−(∂sw(−1,s))2− (∂sw(1,s))2,

which yields the conclusion of Proposition 1.3 by integration in time.

References

[1] S. Alinhac. Blowup for nonlinear hyperbolic equations. Progress in NonlinearDifferential Equations and their Applications, 17. Birkhauser Boston Inc., Boston,MA, 1995.

[2] C. Antonini and F. Merle. Optimal bounds on positive blow-up solutions for asemilinear wave equation. Internat. Math. Res. Notices, (21):1141–1167, 2001.

[3] A. Azaiez. Blow-up profile for the complex-valued semilinear wave equation.Trans. Amer. Math. Soc., 367(8):5891–5933, 2015.

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Blow-up Rate for a Semilinear Wave Equation 31

[4] L. A. Caffarelli and A. Friedman. Differentiability of the blow-up curvefor one-dimensional nonlinear wave equations. Arch. Rational Mech. Anal.,91(1):83–98, 1985.

[5] L. A. Caffarelli and A. Friedman. The blow-up boundary for nonlinear waveequations. Trans. Amer. Math. Soc., 297(1):223–241, 1986.

[6] R. Cote and H. Zaag. Construction of a multisoliton blowup solution to thesemilinear wave equation in one space dimension. Comm. Pure Appl. Math.,66(10):1541–1581, 2013.

[7] Y. Giga and R. V. Kohn. Asymptotically self-similar blow-up of semilinear heatequations. Comm. Pure Appl. Math., 38(3):297–319, 1985.

[8] Y. Giga and R. V. Kohn. Characterizing blowup using similarity variables. IndianaUniv. Math. J., 36(1):1–40, 1987.

[9] Y. Giga and R. V. Kohn. Nondegeneracy of blowup for semilinear heat equations.Comm. Pure Appl. Math., 42(6):845–884, 1989.

[10] P. Godin. The blow-up curve of solutions of mixed problems for semilinear waveequations with exponential nonlinearities in one space dimension. II. Ann. Inst.H. Poincare Anal. Non Lineaire, 17(6):779–815, 2000.

[11] P. Godin. The blow-up curve of solutions of mixed problems for semilinear waveequations with exponential nonlinearities in one space dimension. I. Calc. Var.Partial Differential Equations, 13(1):69–95, 2001.

[12] S. Kichenassamy and W. Littman. Blow-up surfaces for nonlinear wave equations.I. Comm. Partial Differential Equations, 18(3–4):431–452, 1993.

[13] S. Kichenassamy and W. Littman. Blow-up surfaces for nonlinear wave equations.II. Comm. Partial Differential Equations, 18(11):1869–1899, 1993.

[14] F. Merle and H. Zaag. Determination of the blow-up rate for the semilinear waveequation. Amer. J. Math., 125(5):1147–1164, 2003.

[15] F. Merle and H. Zaag. Determination of the blow-up rate for a critical semilinearwave equation. Math. Ann., 331(2):395–416, 2005.

[16] F. Merle and H. Zaag. Existence and universality of the blow-up profile for thesemilinear wave equation in one space dimension. J. Funct. Anal., 253(1):43–121,2007.

[17] F. Merle and H. Zaag. Openness of the set of non-characteristic points andregularity of the blow-up curve for the 1 D semilinear wave equation. Comm.Math. Phys., 282(1):55–86, 2008.

[18] F. Merle and H. Zaag. Points caracteristiques a l’explosion pour une equationsemilineaire des ondes. In “Seminaire X-EDP”. Ecole Polytech., Palaiseau, 2010.

[19] F. Merle and H. Zaag. Blow-up behavior outside the origin for a semilinear waveequation in the radial case. Bull. Sci. Math., 135(4):353–373, 2011.

[20] F. Merle and H. Zaag. Existence and classification of characteristic points atblow-up for a semilinear wave equation in one space dimension. Amer. J. Math.,134(3):581–648, 2012.

[21] F. Merle and H. Zaag. Isolatedness of characteristic points at blowup for a1-dimensional semilinear wave equation. Duke Math. J, 161(15):2837–2908,2012.

[22] F. Merle and H. Zaag. On the stability of the notion of non-characteristicpoint and blow-up profile for semilinear wave equations. Comm. Math. Phys.,333(3):1529–1562, 2015.

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32 Asma Azaiez, Nader Masmoudi and Hatem Zaag

[23] F. Merle and H. Zaag. Dynamics near explicit stationary solutions in similarityvariables for solutions of a semilinear wave equation in higher dimensions. Trans.Amer. Math. Soc., 368(1):27–87, 2016.

[24] J. Shatah and M. Struwe. Geometric wave equations, volume 2 of Courant LectureNotes in Mathematics. New York University Courant Institute of MathematicalSciences, New York, 1998.

∗ Universite de Cergy-Pontoise, Laboratoire Analyse Geometrie Modelisation,CNRS-UMR 8088, 2 avenue Adolphe Chauvin 95302, Cergy-Pontoise, [email protected]

† Courant Institute, NYU, 251 Mercer Street, NY 10012, New [email protected]

‡ Universite Paris 13, Institut Galilee, Laboratoire Analyse Geometrie et Applications,CNRS-UMR 7539, 99 avenue J.B. Clement 93430, Villetaneuse, [email protected]

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2

On the Role of Anisotropy in the Weak Stabilityof the Navier–Stokes System

Hajer Bahouri∗, Jean-Yves Chemin† and Isabelle Gallagher‡

In this article, we investigate the weak stability of the three-dimensionalincompressible Navier–Stokes system. Because of the invariances of this system,a positive answer in general to this question would imply global regularity for anydata. Thus some restrictions have to be imposed if we hope to prove such a weakopenness result. The result we prove in this paper solves this issue under ananisotropy assumption. To achieve our goal, we write a new kind of profiledecomposition and establish global existence results for the Navier–Stokes systemassociated with new classes of arbitrarily large initial data, generalizing theexamples dealt with in [13, 14, 15].

Key words and phrases. Navier–Stokes equations; anisotropy; Besov spaces; profiledecomposition; weak stability

2.1 Introduction and Statement of Results

2.1.1 Setting of the Problem

We are interested in the Cauchy problem for the three-dimensional, incom-pressible Navier–Stokes system

(NS) :

⎧⎪⎪⎨⎪⎪⎩∂tu+ u · ∇u−�u=−∇p in R+ ×R3,

div u= 0,

u|t=0 = u0 ,

where u(t,x) and p(t,x) are respectively the velocity and the pressure of thefluid at time t≥ 0 and position x ∈R3.

An important point in the study of (NS) is its scale invariance, which readsas follows: defining the scaling operators, for any positive real number λ and

33

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34 H. Bahouri, J.-Y. Chemin and I. Gallagher

any point x0 of R3,

�λ,x0φ(t,x)def= 1

λφ( t

λ2,x− x0

λ

)and �λφ(t,x)

def= 1

λφ( t

λ2,

x

λ

), (2.1)

if u solves (NS) with data u0, then �λ,x0u solves (NS) with data �λ,x0 u0. Notein particular that in two space dimensions, L∞(R+;L2(R2)) is scale invariant,while in three space dimensions that is the case for L∞(R+;L3(R3)) or the

family of spaces L∞(R+;B−1+ 3

pp,q (R3)),1 with 1≤ p <∞ and 0 < q≤∞.

Let us also emphasize that the (NS) system formally conserves the energy,in the sense that smooth enough solutions satisfy the following equality for alltimes t≥ 0:

1

2‖u(t)‖2

L2(R3)+∫ t

0‖∇u(t′)‖2

L2(R3)dt′ = 1

2‖u0‖2

L2(R3). (2.2)

The energy equality (2.2) can easily be derived by observing that, thanks tothe divergence-free condition, the nonlinear term is skew-symmetric in L2: onehas indeed if u and p are smooth enough and decaying at infinity that(

u(t) · ∇u(t)+∇p(t)|u(t))L2 = 0.

The mathematical study of the Navier–Stokes system has a long historybeginning with the founding paper [35] of J. Leray in 1933. In this article,J. Leray proved that any finite energy initial data (meaning square-integrabledata) generates a (possibly non-unique) global in time weak solution; andin any dimension d ≥ 2. He moreover proved in [36] the uniqueness ofthe solution in two space dimensions, but in dimension three and more, thequestion of the uniqueness of Leray’s solutions is still an open problem.Actually the difference between dimension two and higher dimensions islinked to the fact that ‖u(t)‖L2(R2) is both scale invariant and bounded globallyin time thanks to the energy estimate, while it is not the case in dimensiond ≥ 3 (since L2(R3) is not scale invariant).

Recall that u ∈ L2loc([0,T] × R3) is said to be a weak solution of (NS)

associated with the data u0 if for any compactly supported, divergence-freevector field φ in C∞([0,T]×R3) the following holds for all t≤ T:∫R3

u·φ(t,x)dx=∫R3

u0(x)·φ(0,x)dx+∫ t

0

∫R3(u·�φ+u⊗u :∇φ+u·∂tφ)dxdt′,

1 Here B−1+ 3

pp,q (R3) denotes the usual homogeneous Besov space (see [2], [9] or [46] for a

precise definition).

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Anisotropy in the Weak Stability of the Navier–Stokes System 35

withu⊗ u :∇φ

def=∑

1≤j,k≤3

ujuk∂kφj .

Weak solutions satisfying the energy inequality

1

2‖u(t)‖2

L2(R3)+∫ t

0‖∇u(t′)‖2

L2(R3)dt′ ≤ 1

2‖u0‖2

L2(R3)(2.3)

are said to be turbulent solutions, following the terminology of J. Leray [35].In what follows, we say that a family (XT)T>0 of spaces of distributions

over [0,T] ×R3 is scaling invariant if for all T > 0 one has, under Notation(2.1):

∀λ> 0,∀x0 ∈R3 , u∈XT ⇐⇒�λ,x0 u∈Xλ−2T with ‖u‖XT =‖�λ,x0 u‖Xλ−2T

.

Similarly, a space X0 of distributions defined on R3 will be said to be scalinginvariant if

∀λ > 0,∀x0 ∈R3 , u0 ∈ X0 ⇐⇒�λ,x0u0 ∈ X0 with ‖u0‖X0 = ‖�λ,x0 u0‖X0 .

This leads to the definition of a scaled solution, which will be the notion ofsolution we consider throughout this article.

Definition 2.1 A vector field u is said to be a scaled solution to (NS) associatedwith the data u0 if it is a weak solution, belonging to a family of scalinginvariant spaces.

After Leray’s results, the question of the global wellposedness of theNavier–Stokes system in dimension d ≥ 3 was raised, and has been openever since, although several partial answers to the construction of a globalunique solution were established since (we refer for instance to [2] or [34] andthe references therein for recent surveys on the subject). Let us simply recallthe best result known to this day on the uniqueness of solutions to (NS), whichis due to H. Koch and D. Tataru in [33]: if

‖u0‖BMO−1(R3)

def= ‖u0‖B−1∞,∞(R3)+ sup

x∈R3

R>0

1

R32

(∫[0,R2]×B(x,R)

|(et�u0)(t,y)|2 dydt) 1

2

is small enough, then there is a global, unique solution to (NS), lyingin BMO−1∩X for all times, with X a scale invariant space to be specified – weshall not be using that space in the sequel. Note that the space BMO−1 isinvariant by the scaling operator �λ,x0 and that the norm in B−1∞,∞(R3) denotes aBesov norm. Actually, the Besov space B−1∞,∞(R3) is the largest space in whichany scale and translation invariant Banach space of tempered distributions

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36 H. Bahouri, J.-Y. Chemin and I. Gallagher

embeds (see [39]). However, it was proved in [10] and [22] that (NS) isill-posed for initial data in B−1∞,∞(R3).

Our goal in this paper is to investigate the stability of global solutions.Let us recall that strong stability results have been achieved. Specifically, itwas proved in [1] (see [19] for the Besov setting) that the set of initial datagenerating a global solution is open in BMO−1. More precisely, denotingby VMO−1 the closure of smooth functions in BMO−1, it was establishedin [1] that if u0 belongs to VMO−1 and generates a global, smooth solutionto (NS), then any sequence (u0,n)n∈N converging to u0 in the BMO−1 normalso generates a global smooth solution as soon as n is large enough.

In this paper we would like to address the question of weak stability.

If (u0,n)n∈N, bounded in some scale invariant space X0, converges to u0 in thesense of distributions, with u0 giving rise to a global smooth solution, is it thecase for u0,n when n is large enough ?

Because of the invariances of the (NS) system, a positive answer in general tothis question would imply global regularity for any data and so would solvethe question of the possible blow-up in finite time of solutions to (NS), whichis actually one of the Millennium Prize Problems in Mathematics. Indeed,consider for instance the sequence

u0,n = λn�0(λn ·)=�λn �0 with limn→∞

(λn+ 1

λn

)=∞ , (2.4)

with �0 any smooth divergence-free vector field. If the weak stability resultwere true, then since the weak limit of (u0,n)n∈N is zero (which gives rise to theunique, global solution which is identically zero) then for n large enough u0,n

would give rise to a unique, global solution. By scale invariance then sowould �0, for any �0, so that would solve the global regularity problem for(NS). Another natural example is the sequence

u0,n =�0(·− xn)=�1,xn�0 , (2.5)

with (xn)n∈N a sequence of R3 whose norm goes to infinity. Thus sequencesbuilt by rescaling fixed divergence-free vector fields according to the invari-ances of the equations have to be excluded from our analysis, since solving the(NS) system for any smooth initial data seems out of reach.

Thus clearly some restrictions have to be imposed if we hope to prove such aweak openness result. Let us note that a first step in that direction was achievedin [4], under two additional assumptions on the weak convergence. The firstone is an assumption on the asymptotic separation of the horizontal and verticalspectral supports of the sequence (u0,n)n∈N, while the second one requires that

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Anisotropy in the Weak Stability of the Navier–Stokes System 37

some of the profiles involved in the profile decomposition of (u0,n)n∈N vanishat zero. In this paper, we remove the second assumption and give a positiveanswer to the question of weak stability, provided that the convergence of thesequence (u0,n)n∈N towards u0 holds “anisotropically” in frequency space (seeDefinition 2.4). The main ingredient which enables us to eliminate the secondassumption required in [4] is a novel form of anisotropic profile decomposition.This new profile decomposition enables us to decompose the sequence ofinitial data u0,n, up to a small remainder term, into a finite sum of orthogonalsequences of divergence-free vector fields; these sequences are obtained fromthe classical anisotropic profile decompositions by grouping together all theprofiles having the same horizontal scale. The price to pay is that the profilesare no longer fixed functions as in the classical case, but bounded sequences.To carry out the strategy of proof developed in [4] in this framework, weare led to establishing global existence results for (NS) associated with newclasses of arbitrarily large initial data, generalizing the examples dealt with in[13, 14, 15], and where regularity is sharply estimated.

2.1.2 Statement of the Main Result

We prove in this article a weak stability result for the (NS) system underan anisotropy assumption. This leads us naturally to introducing anisotropicBesov spaces. These spaces generalize the more usual isotropic Besov spaces,which are studied for instance in [2, 9, 46].

Definition 2.2 Let χ (the Fourier transform of χ ) be a radial function in D(R)

such that χ (t) = 1 for |t| ≤ 1 and χ(t) = 0 for |t| > 2. For (j,k) ∈ Z2, thehorizontal truncations are defined by

Shk f (ξ)

def= χ(2−k|(ξ1,ξ2)|

)f (ξ) and �h

kdef= Sh

k+1− Shk ,

and the vertical truncations by

Svj f

def= χ (2−j|ξ3|)f (ξ) and �vj

def= Svj+1− Sv

j .

For all p in [1,∞] and q in ]0,∞], and all (s,s′) in R2, with s < 2/p,s′ <1/p (or s ≤ 2/p and s′ ≤ 1/p if q = 1), the anisotropic homogeneous Besovspace Bs,s′

p,q is defined as the space of tempered distributions f such that

‖f‖Bs,s′

p,q

def=∥∥∥2ks+js′ ‖�h

k�vj f‖Lp

∥∥∥�q<∞ .

In all other cases of indices s and s′, the Besov space is defined similarly, up totaking the quotient with polynomials.

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38 H. Bahouri, J.-Y. Chemin and I. Gallagher

Remark 2.3 The Besov spaces Bs,s′p,q (for s = s′) are anisotropic in essence,

which, as pointed out above, will be an important feature of our analysis.These spaces have properties which look very much like the ones of classicalBesov spaces. We refer for instance to [2], [17], [23] and [41] forall necessary details. By construction, these spaces are defined using ananisotropic Littlewood–Paley decomposition. It is useful to point out thatthe horizontal and vertical truncations Sh

k , �hk , Sv

j and �vj , introduced in

Definition 2.2, map Lp into Lp with norms independent of k, j and p. For ourpurpose, it is crucial to recall the following inequalities, known as Bernsteininequalities: if 1≤ p1 ≤ p2 ≤∞, then for any α ∈N2 and m ∈N

‖∂α(x1,x2)

�hkf‖Lp2 (R2;Lr(R)) � 2k(|α|+2(1/p1−1/p2))‖�h

kf‖Lp1 (R2;Lr(R)) (2.6)

and ‖∂mx3�v

j f‖Lr(R2;Lp2 (R)) � 2j(m+1/p1−1/p2)‖�vj f‖Lr(R2;Lp1 (R)) , (2.7)

as well as the action of the heat flow on frequency localized distributions in ananisotropic context, namely for any p in [1,∞]

‖et��hk�

vj f‖Lp � e−ct(22k+22j)‖�h

k�vj f‖Lp . (2.8)

Notation For clarity, in what follows we denote by Bs,s′ the space Bs,s′2,1 , by Bs

the space Bs, 12 and by Bp,q the space B

−1+ 2p , 1

pp,q . In particular B2,1 = B0.

Let us point out that the scaling operators (2.1) enjoy the followinginvariances:

‖�λ,x0ϕ‖Bp,q = ‖ϕ‖Bp,q

and ∀r ∈ [1,∞] , ‖�λ,x0�‖Lr(R+;B−1+2/p+2/r,1/pp,q )

= ‖�‖Lr(R+;B−1+2/p+2/r,1/p

p,q ),

and also the following scaling property:

∀r ∈ [1,∞] , ∀σ ∈R ,

‖�λ,x0�‖Lr(R+;B−1+2/p+2/r−σ ,1/pp,q )

∼ λσ‖�‖Lr(R+;B−1+2/p+2/r−σ ,1/p

p,q ). (2.9)

The Navier–Stokes system in anisotropic spaces has been studied in a numberof frameworks. We refer, for instance, to [4], [17], [23], [25] and [41]. Inparticular, in [4] it is proved that if u0 belongs to B0, then there is a uniquesolution (global in time if the data are small enough) in L2([0,T];B1). Thatnorm controls the equation, in the sense that as soon as the solution belongsto L2([0,T];B1), then it lies in fact in Lr([0,T];B 2

r ) for all 1 ≤ r ≤ ∞. Thespace B1 is included in L∞ and since the seminal work [35] of J. Leray,it is known that the L2([0,T];L∞(R3)) norm controls the propagation ofregularity and also ensures weak uniqueness among turbulent solutions. Thusthe space B0 is natural in this context.

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Anisotropy in the Weak Stability of the Navier–Stokes System 39

As mentioned above, the result we establish in this paper involves ananisotropy assumption on the sequence (u0,n)n∈N of initial data. Let usintroduce this assumption that we call the notion of anisotropically oscillatingsequences, and which is a natural adaptation to our setting of the vocabularyof P. Gerard in [21].

Definition 2.4 Let 0 < q ≤ ∞ be given. We say that a sequence (fn)n∈N,bounded in B1,q, is anisotropically oscillating if the following property holds.There exists p≥ 2 such that for all sequences (kn, jn) in ZN×ZN,

liminfn→∞ 2kn(−1+ 2

p )+ jnp ‖�h

kn�v

jn fn‖Lp(R3) = C > 0 �⇒ limn→∞ |jn− kn| =∞ .

(2.10)

Remark 2.5 In view of the Bernstein inequalities (2.6) and (2.7), it is easy tosee that any function f in B1,q belongs also to Bp,∞ for any p≥ 1, hence

f ∈ B1,q �⇒ sup(k,j)∈Z2

2k(−1+ 2p )+ j

p ‖�hk�

vj f‖Lp <∞ .

The left-hand side of (2.10) indicates which ranges of frequencies are pre-

dominant in the sequence (fn): if liminfn→∞ 2kn(−1+ 2

p )+ jnp ‖�h

kn�v

jn fn‖Lp is zero for

a couple of frequencies (2kn ,2jn), then the sequence (fn)n∈N is “unrelated” tothose frequencies, with the vocabulary of P. Gerard in [21]. The right-hand sideof (2.10) is then an anisotropy property. Indeed one sees easily that a sequencesuch as (u0,n)n∈N defined in (2.4) is precisely not anisotropically oscillating: forthe left-hand side of (2.10) to hold for this example one would need jn∼ kn∼ n,which is precisely not the condition required on the right-hand side of (2.10).A typical sequence satisfying Assumption (2.10) is rather (for a ∈R3)

fn(x) := 2αnf(2αn(x1− a1),2

αn(x2− a2),2βn(x3− a3)

), (α,β) ∈R2, α = β

with f smooth.

Our main result is stated as follows.

Theorem 2.6 Let q be given in ]0,1[ and let u0 in B1,q generate a unique globalsolution to (NS) in L2(R+;B1). Let (u0,n)n∈N be a sequence of divergence freevector fields converging towards u0 in the sense of distributions, and suchthat (u0,n− u0)n∈N is anisotropically oscillating. Then for n large enough, u0,n

generates a unique, global solution to the (NS) system in the space L2(R+;B1).

Remark 2.7 One can see from the proof of Theorem 2.6 that the solution un(t)associated with u0,n converges for all times, in the sense of distributions to the

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40 H. Bahouri, J.-Y. Chemin and I. Gallagher

solution associated with u0. In this sense the Navier–Stokes system is stable byweak convergence.

The proof of Theorem 2.6 enables us to infer easily the following result, whichgeneralizes the statement of Theorem 2.6 to the case when the solution to the(NS) system generated by u0 is assumed to blow up in finite time (for a strategyof proof, one can consult [4]).

Corollary 2.8 Let (u0,n)n∈N be a sequence of divergence-free vector fieldsbounded in the space B1,q for some 0 < q < 1, converging towards some u0 inB1,q in the sense of distributions, with u0−(u0,n)n∈N anisotropically oscillating.Let u be the solution to the Navier–Stokes system associated with u0 andassume that the life span of u is T∗ <∞. Then for all positive times T < T∗,there is a subsequence such that the life span of the solution associatedwith u0,n is at least T.

Remark 2.9 As explained above, the natural space in our context wouldbe B0. For technical reasons, we assume in our result more smoothness onthe sequence of initial data, since obviously by Bernstein inequalities (2.6) and(2.7), we have B1,q ↪→ B0.

2.1.3 Layout

The proof of Theorem 2.6 is addressed in Section 2.2. In Subsection 2.2.2,we provide a new kind of “anisotropic profile decomposition” of the sequenceof initial data, whose proof can be found in Section 2.3. This enables us toreplace the sequence of Cauchy data, up to an arbitrarily small remainder term,by a finite (but large) sum of orthogonal sequences of divergence-free vectorfields. In Subsection 2.2.3, we state that each individual element involvedin the decomposition derived in Subsection 2.2.2 gives rise to a uniqueglobal solution to the (NS) system (the proof is postponed to Section 2.4).Subsection 2.2.4 is devoted to the proof of the fact that the sum of eachindividual profile does provide an approximate solution to the Navier–Stokessystem, thanks to an orthogonality argument, which completes the proof ofTheorem 2.6.

For all points x = (x1,x2,x3) in R3 and all vector fields u = (u1,u2,u3), wedenote by

xhdef= (x1,x2) and uh def= (u1,u2)

their horizontal parts. We also define horizontal differentiation opera-

tors ∇h def= (∂1,∂2) and divhdef= ∇h·, as well as �h

def= ∂21 + ∂2

2 .

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Anisotropy in the Weak Stability of the Navier–Stokes System 41

We also use the following shorthand notation: XhYv := X(R2;Y(R)),where X is a function space defined on R2 and Y is defined on R.

As we shall be considering functions which have different types of variationsin the x3 variable and the xh variable, the following notation will be used:[

f]β(x)

def= f (xh,βx3) . (2.11)

Clearly, for any function f , we have the following identity which will be ofconstant use throughout this paper:∥∥[f ]β∥∥B

s1,s2p,1

∼ βs2− 1

p ‖f‖Bs1,s2p,1

. (2.12)

Finally, we denote by C a constant which does not depend on the variousparameters appearing in this paper, and which may change from line to line.We also denote sometimes x≤ Cy by x � y.

2.2 Proof of the Main Theorem

2.2.1 General Scheme of the Proof

The main arguments leading to Theorem 2.6 are the following: by a profiledecomposition argument, the sequence of initial data is decomposed into theweak limit u0 and the sum of sequences of divergence-free vector fields, upto a small remainder term. Then to prove that each individual element of thedecomposition generates a unique global solution to (NS), it is necessary toestimate sharply the regularity in scaling invariant (anisotropic) norms. Themutual orthogonality of each term in the decomposition of the initial dataimplies finally that the sum of the solutions associated with each element isitself an approximate solution to (NS), globally in time, which concludes theproof of the result.

2.2.2 Anisotropic Profile Decomposition

The study of the lack of compactness in critical Sobolev embeddings hasattracted a lot of attention in the past decades, both for its interesting geometricfeatures and for its applications to nonlinear partial differential equations. Thisstudy originates in the works of P.-L. Lions (see [37] and [38]) by means ofdefect measures, and earlier decompositions of bounded sequences into a sumof “profiles” can be found in the studies by H. Brezis and J.-M. Coron in [11]and M. Struwe in [45]. Our source of inspiration here is the work [21] of P.Gerard in which the defect of compactness of the critical Sobolev embeddings

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42 H. Bahouri, J.-Y. Chemin and I. Gallagher

(for L2-based Sobolev spaces) in Lebesgue spaces is described by means ofan asymptotic, orthogonal decomposition in terms of rescaled and translatedprofiles. This was generalized to Lp-based Sobolev spaces by S. Jaffard in [26],to Besov spaces by G. Koch [32], and finally to general critical embeddings byH. Bahouri, A. Cohen and G. Koch in [3] (see also [6, 7, 8] for the limiting caseof Sobolev embeddings in Orlicz spaces and [44] for an abstract, functionalanalytic presentation of the concept in various settings).

In the pioneering works [5] (for the critical 3D wave equation) and [40](for the critical 2D Schrodinger equation), it was highlighted that this typeof decomposition provides applications to the study of nonlinear partialdifferential equations. The ideas of [5] were revisited in [31] and [18] in thecontext of the Schrodinger equations and Navier–Stokes system, respectively,with an aim of describing the structure of bounded sequences of solutions tothose equations. These profile decomposition techniques have since been usedsuccessfully to study the possible blow-up of solutions to nonlinear partialdifferential equations, in various contexts; we refer for instance to [20], [24],[27], [28], [29], [30], [42], [43].

The first step in the proof of Theorem 2.6 consists of writing down ananisotropic profile decomposition of the sequence of initial data (u0,n)n∈N (seeTheorem 2.12). To state our result in a clear way, let us start by introducingsome definitions and notations.

Definition 2.10 We say that two sequences of positive real numbers (λ1n)n∈N

and (λ2n)n∈N are orthogonal if

λ1n

λ2n

+ λ2n

λ1n

→∞ , n→∞ .

A family of sequences((λ

jn)n∈N

)j is said to be a family of scales if λ0

n ≡ 1 and

if (λjn)n∈N and (λk

n)n∈N are orthogonal when j = k.

Definition 2.11 Let μ be a positive real number less than 1/2, fixed fromnow on.

We define Dμ

def= [−2+μ,1−μ]×[1/2,7/2] and Dμ

def= [−1+μ,1−μ]×[1/2,3/2]. We denote by Sμ the space of functions a belonging to

⋂(s,s′)∈Dμ

Bs,s′

such that

‖a‖Sμdef= sup

(s,s′)∈Dμ

‖a‖Bs,s′ <∞ .

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Anisotropy in the Weak Stability of the Navier–Stokes System 43

Notation In all that follows, θ is a given function in D(BR3(0,1)) which has

value 1 near BR3(0,1/2). For any positive real number η, we denote

θη(x)def= θ(ηx) and θh,η(xh)

def= θη(xh,0) . (2.13)

In order to make notations as clear as possible, the letter v (possibly withindices) will always denote a two-component divergence free vector field,which may depend on the vertical variable x3.

The following result, the proof of which is postponed to Section 2.3, is inthe spirit of the profile decomposition theorem of P. Gerard in [21] concerningthe critical Sobolev embedding in Lebesgue spaces.

Theorem 2.12 Under the assumptions of Theorem 2.6 and up to the extractionof a subsequence, the following holds. There is a family of scales

((λ

jn)n∈N

)j∈N

and for all L ≥ 1 there is a family of sequences((hj

n)n∈N)

j∈N going to zerosuch that for any real number α in ]0,1[, there are families of sequences ofdivergence-free vector fields (for j ranging from 1 to L), (vj

n,α,L)n∈N, (wjn,α,L)n∈N,

(v0,∞n,α,L)n∈N, (w0,∞

0,n,α,L)n∈N, (v0,loc0,n,α,L)n∈N and (w0,loc

0,n,α,L)n∈N, all belonging to Sμ, anda smooth, compactly supported function u0,α such that the sequence (u0,n)n∈Ncan be written in the form

u0,n ≡ u0,α+[(v0,loc

0,n,α,L+h0nw

0,loc,h0,n,α,L,w0,loc,3

0,n,α,L

)]h0

n+[(v0,∞

0,n,α,L+h0nw

0,∞,h0,n,α,L,w0,∞,3

0,n,α,L)]

h0n

+L∑

j=1

�λ

jn

[(v

jn,α,L+ hj

nwj,hn,α,L,wj,3

n,α,L)]

hjn+ρn,α,L,

where u0,α approximates u0 in the sense that

limα→0

‖u0,α − u0‖B1,q = 0, (2.14)

where the remainder term satisfies

limL→∞ lim

α→0limsup

n→∞‖et�ρn,α,L‖L2(R+;B1) = 0, (2.15)

while the following uniform bounds hold:

M def= supL≥1

supα∈]0,1[

supn∈N

(∥∥(v0,∞0,n,α,L,w0,∞,3

0,n,α,L)∥∥B0 +

∥∥(v0,loc0,n,α,L,w0,loc,3

0,n,α,L)∥∥B0

+‖u0,α‖B0 +L∑

j=1

∥∥(vjn,α,L,wj,3

n,α,L)∥∥B0

)<∞

(2.16)

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44 H. Bahouri, J.-Y. Chemin and I. Gallagher

and for all α in ]0,1[,

def= supL≥1

sup1≤j≤Ln∈N

(∥∥(v0,∞0,n,α,L,w0,∞,3

0,n,α,L)∥∥

Sμ+∥∥(v0,loc

0,n,α,L,w0,loc,30,n,α,L)

∥∥Sμ

+‖u0,α‖Sμ +∥∥(vj

n,α,L,wj,3n,α,L)

∥∥Sμ

) (2.17)

is finite. Finally, we have

limL→∞ lim

α→0limsup

n→∞

∥∥(v0,loc0,n,α,L,w0,loc,3

0,n,α,L

)(·,0)∥∥B0

2,1(R2)= 0,

(2.18)

∀(α,L) ,∃η(α,L)/ ∀η≤ η(α,L) ,∀n ∈N , (1− θh,η)(v0,loc0,n,α,L,w0,loc,3

0,n,α,L)= 0 and

(2.19)

∀(α,L,η) , ∃n(α,L,η)/ ∀n≥ n(α,L,η) , θh,η(v0,∞0,n,α,L,w0,∞,3

0,n,α,L)= 0. (2.20)

Theorem 2.12 states that the sequence u0,n is equal, up to a small remainderterm, to a finite sum of orthogonal sequences of divergence-free vector fields.These sequences are obtained from the profile decomposition derived in [4](see Proposition 2.4 in [4]) by grouping together all the profiles having thesame horizontal scale λn, and the form they take depends on whether the scaleλn is identically equal to one or not.

Note that in contrast with classical profile decompositions (see for instance[21]), cores of concentration do not appear in the profile decomposition givenin Theorem 2.12 since all the profiles with the same horizontal scale aregrouped together, and thus the decomposition is written in terms of scalesonly. The price to pay is that the profiles are no longer fixed functions,but bounded sequences. To carry out the strategy of proof developed in [4]in this framework, we have to establish that each element involved in thedecomposition of Theorem 2.12 generates a global solution to the (NS) systemas soon as n is large enough. Since we deal with bounded sequences, it isnecessary to sharply estimate the regularity.

Let us emphasize that in the case when λn goes to 0 or infinity, thesesequences are of the type

�λn

[(vh

0,n+ hnwh0,n,w3

0,n)]

hn, (2.21)

where we used Notation (2.11), and with hn a sequence going to zero. It isessential (to establish our result) that the profiles that must be considered inthat case are only profiles of type (2.21) with hn tending to zero. Actually thedivergence-free assumption on u0,n allows us to include the terms of type (2.21)with hn tending to infinity into the remainder term; and the anisotropically

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Anisotropy in the Weak Stability of the Navier–Stokes System 45

oscillating assumption for (u0 − u0,n)n∈N allows us to exclude in the profiledecomposition of u0,n sequences of type (2.21) with hn ≡ 1.

In the case when λn is identically equal to one, we deal with three typesof orthogonal sequence: the first one consists in u0,α an approximation of theweak limit u0, the second one is of type (2.21) with λn ≡ 1 and hn tending tozero, and is uniformly localized in the horizontal variable and vanishes at x3 =0, while the third one is also of type (2.21) with λn ≡ 1 and hn convergingto zero, and its support in the horizontal variable goes to infinity. Note that,contrary to the case when the horizontal scale λn tends to 0 or infinity, allthe profiles involved in the anisotropic decomposition of the sequence (u0 −u0,n)n∈N having the same horizontal scale λn ≡ 1 are not grouped together:the sum of these profiles is divided into two parts depending on whether thehorizontal cores of concentration escape to infinity or not. This splitting playsa key role in establishing our result under the only assumption of anisotropicoscillation, by removing the second assumption required in [4].

2.2.3 Propagation of Profiles

The second step of the proof of Theorem 2.6 consists of proving that eachindividual profile involved in the decomposition of Theorem 2.12 generates aglobal solution to (NS) as soon as n is large enough. This is mainly based onthe following results concerning respectively profiles of the type

�λ

jn

[(v

jn,α,L+ hj

nwj,hn,α,L,wj,3

n,α,L)]

hjn

with λjn going to 0 or infinity and hj

n converging to zero, and the profiles ofhorizontal scale one, see respectively Theorems 2.14 and 2.15.

In order to state these theorems, let us begin by defining the function spaceswe shall be working with.

Definition 2.13 We define the space As,s′ = L∞(R+;Bs,s′) ∩ L2(R+;Bs+1,s′)equipped with the norm

‖a‖As,s′def= ‖a‖L∞(R+;Bs,s′ )+‖a‖L2(R+;Bs+1,s′ ) ,

and we denote As =As, 12 .

We denote by F s,s′ any function space such that

‖L0f‖L2(R+;Bs+1,s′ ) � ‖f‖F s,s′ ,

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46 H. Bahouri, J.-Y. Chemin and I. Gallagher

where, for any non-negative real number τ , Lτ f denotes the solution of the heatequation {

∂tLτ f −�Lτ f = f ,Lτ f|t=τ = 0.

We denote F s =F s, 12 .

Examples Using the smoothing effect of the heat flow, it is easy to provethat the spaces L2(R+;Bs−1,s′), L2(R+;Bs,s′−1) are F s,s′ spaces, as well asthe spaces L1(R+;Bs,s′) and L1(R+;Bs+1,s′−1). Actually, recalling that L0f =∫ t

0e(t−t′)�f (t′)dt and taking advantage of (2.8), we get for any function

in L2(R+;Bs−1,s′)

‖�hk�

vj L0f‖L2 �

∫ t

0e−ct′(22k+22j)‖�h

k�vj f (t′)‖L2 dt′,

where we make use of notations of Definition 2.2. We deduce that there is asequence dj,k(t′) in the sphere of �1(Z×Z;L2(R+)) such that

‖�hk�

vj L0f‖L2 � ‖f‖L2(R+;Bs−1,s′ )2

−k(s−1)2−js′∫ t

0e−ct′(22k+22j)dj,k(t

′)dt′ .

Young’s inequality in time therefore gives

‖�hk�

vj L0f‖L2(R+;L2) � ‖f‖L2(R+;Bs−1,s′ )2

−k(s−1)−js′dj,k ,

where dj,k is a generic sequence in the sphere of �1(Z× Z), which ends theproof of the result in the case when f belongs to L2(R+;Bs−1,s′). The argumentis similar in the other cases.

Notation In the following we designate by T0(A,B) a generic constant depend-ing only on the quantities A and B. We denote by T1 a generic non-decreasingfunction from R+ into R+ such that

limsupr→0

T1(r)

r<∞ , (2.22)

and by T2 a generic locally bounded function from R+ into R+. All thosefunctions may vary from line to line. Let us notice that for any positivesequence (an)n∈N belonging to �1, we have∑

n

T1(an)≤ T2

(∑n

an

). (2.23)

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Anisotropy in the Weak Stability of the Navier–Stokes System 47

As in the isotropic case, the following space-time (quasi)-norms, first intro-duced by J.-Y. Chemin and N. Lerner in [16]:

‖f‖Lr([0,T];Bs,s′

p,q )

def= ∥∥2ks+js′ ‖�hk�

vj f‖Lr([0,T];Lp)

∥∥�q , (2.24)

are very useful in the context of the Navier–Stokes system, and will be ofconstant use all along this paper. Notice that of course Lr([0,T];Bs,s′

p,r ) =Lr([0,T];Bs,s′

p,r ), and by Minkowski’s inequality, we have the embedding

Lr([0,T];Bs,s′p,q)⊂ Lr([0,T];Bs,s′

p,q) if r ≥ q.Our first theorem of global existence for the Navier–Stokes system, which

concerns profiles with horizontal scales going to 0 or infinity, generalizes theexample considered in [13].

Theorem 2.14 A locally bounded function ε1 from R+ into R+ exists, whichsatisfies the following. For any (v0,w3

0) in Sμ (see Definition 2.11), for anypositive real number β such that β ≤ ε1(‖(v0,w3

0)‖Sμ), the divergence-freevector field

�0def= [

(v0−β∇h�−1h ∂3w

30,w3

0)]β

generates a global solution �β to (NS), which satisfies

‖�β‖A0 ≤ T1(‖(v0,w30)‖B0)+β T2(‖(v0,w3

0)‖Sμ) . (2.25)

Moreover, for any (s,s′) in [−1+μ,1− μ] × [1/2,7/2], we have, for any rin [1,∞],

‖�β‖Lr(R+;Bs+ 2

r )+ 1

βs′− 12

‖�β‖Lr(R+;B 2

r ,s′ )≤ T2(‖(v0,w3

0)‖Sμ) . (2.26)

The proof of Theorem 2.14 is provided in Subsection 2.4.1.The existence of a global regular solution for the set of profiles associated

with the horizontal scale 1 is ensured by the following theorem, which can beviewed as a generalization of Theorem 3 of [14] and of Theorem 2 of [15].

Theorem 2.15 With the notation of Theorem 2.12, let us consider the initialdata:

�00,n,α,L

def= u0,α +[(v0,∞

0,n,α,L+ h0nw

0,∞,h0,n,α,L,w0,∞,3

0,n,α,L

)]h0

n

+ [(v0,loc

0,n,α,L+ h0nw

0,loc,h0,n,α,L,w0,loc,3

0,n,α,L)]

h0n

.

There is a constant ε0, depending only on u0 and on Mα , such that if h0n ≤ ε0,

then the initial data �00,n,α,L generates a global smooth solution �0

n,α,L which

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48 H. Bahouri, J.-Y. Chemin and I. Gallagher

satisfies for all s in [−1+μ,1−μ] and all r in [1,∞],‖�0

n,α,L‖Lr(R+;Bs+ 2r )≤ T0(u0,Mα) . (2.27)

The proof of Theorem 2.15 is provided in Subsection 2.4.2.

2.2.4 End of the Proof of the Main Theorem

To end the proof of Theorem 2.6, we need to check that the sum of thepropagation of the remainder term through the transport-diffusion equationand the solutions to (NS) associated with each individual profile (providedby Theorems 2.14 and 2.15) is an approximate solution to the Navier–Stokessystem. This can be achieved by proving that the nonlinear interactions of allthe solutions are negligible, thanks to the orthogonality between the scales. Forthat purpose, let us look at the profile decomposition given by Theorem 2.12.For a given positive and small ε, Assertion (2.15) allows us to choose α, Land N0 (depending of course on ε) such that

∀n≥ N0 , ‖et�ρn,α,L‖L2(R+;B1) ≤ ε . (2.28)

The parameters α and L are fixed so that (2.28) holds. Let us consider thetwo functions ε1, T1 and T2 (resp. ε0 and T0) which appear in the statement ofTheorem 2.14 (resp. Theorem 2.15). Since each sequence (hj

n)n∈N, for 0≤ j≤L, goes to zero as n goes to infinity, one can choose an integer N1 greater thanor equal to N0 such that

∀n≥ N1 , ∀j ∈ {0, . . . ,L} , hjn ≤min

{ε1(Mα),ε0,

ε

LT2(Mα)

}· (2.29)

Now for 1 ≤ j ≤ L (resp. j = 0), let us denote by �jn,ε (resp. �0

n,ε) the globalsolution of (NS) associated with the initial data:[

(vjn,α,L+ hj

nwj,hn,α,L,wj,3

n,α,L)]

hjn(

resp. u0,α +[(v0,∞

0,n,α,L+ h0nw

0,∞,h0,n,α,L,w0,∞,3

0,n,α,L

)]h0

n

+ [(v0,loc

0,n,α,L+ h0nw

0,loc,h0,n,α,L,w0,loc,3

0,n,α,L)]

h0n

)given by Theorem 2.14 (resp. Theorem 2.15). We look for the global solutionassociated with u0,n in the form

un = uappn,ε +Rn,ε with uapp

n,εdef=

L∑j=0

�λ

jn�j

n,ε+ et�ρn,α,L . (2.30)

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Anisotropy in the Weak Stability of the Navier–Stokes System 49

In view of the scaling invariance of the Navier–Stokes system, �λ

jn�

jn,ε solves

(NS) with the initial data �λ

jn

[(v

jn,α,L+ hj

nwj,hn,α,L,wj,3

n,α,L)]

hjn. This gives the

following equation on Rn,ε:

∂tRn,ε − �Rn,ε+ div(Rn,ε⊗Rn,ε+Rn,ε⊗ uapp

n,ε + uappn,ε ⊗Rn,ε

)+∇pn,ε

= Fn,εdef= F1

n,ε+F2n,ε+F3

n,ε

with F1n,ε

def= −div(et�ρn,α,L⊗ et�ρn,α,L

),

F2n,ε

def= −L∑

j=0

div(�

λjn�j

n,ε⊗ et�ρn,α,L+ et�ρn,α,L⊗�λ

jn�j

n,ε

)and F3

n,εdef= −

∑0≤j,k≤L

j=k

div(�

λjn�j

n,ε⊗�λkn�k

n,ε

), (2.31)

and where(div(u⊗ v)

)j =3∑

k=1

∂k(ujvk).

In order to establish that the function un defined by (2.30) provides a globalsolution to the (NS) system, it suffices to prove that there exist some space F0

as in Definition 2.13 and an integer N ≥ N1 such that

∀n≥ N , ‖Fn,ε‖F0 ≤ Cε , (2.32)

where C depends only on L and Mα . In the next estimates we omit thedependence of all constants on α and L, which are fixed. Indeed, if (2.32)holds, then Rn,ε exists globally thanks to strong stability in B0 (see [4] for thesetting of B1,1).

Let us start with the estimate of F1n,ε. Using the fact that B1 is an algebra, we

have ∥∥et�ρhn,α,L⊗ et�ρn,α,L

∥∥L1(R+;B1)

� ‖et�ρn,α,L

∥∥2L2(R+;B1)

,

so

‖divh(et�ρh

n,α,L⊗ et�ρn,α,L)‖L1(R+;B0) � ‖et�ρn,α,L

∥∥2L2(R+;B1)

and

‖∂3(et�ρ3

n,α,Let�ρn,α,L)‖

L1(R+;B1,− 12 )� ‖et�ρn,α,L

∥∥2L2(R+;B1)

.

According to Inequality (2.28), this gives rise to

∀n≥ N1 , ‖F1n,ε‖F0 � ε2. (2.33)

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50 H. Bahouri, J.-Y. Chemin and I. Gallagher

Now let us consider F2n,ε. By the scaling invariance of the operators �

λjn

in L2(R+;B1) and again the fact that B1 is an algebra, we get∥∥�λ

jn�j

n,ε⊗ et�ρn,α,L+ et�ρn,α,L⊗�λ

jn�j

n,ε

∥∥L1(R+;B1)

� ‖�jn,ε‖L2(R+;B1)‖et�ρn,α,L‖L2(R+;B1) .

(2.34)

Making use of Estimates (2.25) and (2.27), we infer that

L∑j=0

∥∥�jn,ε

∥∥L2(R+;B1)

≤ T0(u0,Mα)+T2(M)+L∑

j=1

hjnT2(Mα) ,

which in view of Condition (2.29) on the sequences (hjn)n∈N implies that∥∥∥ L∑

j=0

�jn,ε

∥∥∥L2(R+;B1)

≤ T0(u0,Mα)+T2(M)+ ε .

It follows (of course up to a change of T2) that for small enough ε∥∥∥ L∑j=0

�jn,ε

∥∥∥L2(R+;B1)

≤ T0(u0,Mα)+T2(M) . (2.35)

Thanks to (2.28) and (2.34), this gives rise to

∀n≥ N1 , ‖F2n,ε‖F0 ≤ ε

(T0(u0,Mα)+T2(M)

). (2.36)

Finally let us consider F3n,ε. Using the fact that B1 is an algebra along with the

Holder inequality, we infer that for a small enough γ in ]0,1[,∥∥�λ

jn�j

n,ε⊗�λkn�k

n,ε

∥∥L1(R+;B1)

≤‖�λ

jn�j

n,ε‖L

21+γ (R+;B1)

‖�λkn�k

n,ε‖L

21−γ (R+;B1)

.

The scaling invariance (2.9) gives

‖�λ

jn�j

n,ε‖L

21+γ (R+;B1)

∼ (λjn)

γ ‖�jn,ε‖

L2

1+γ (R+;B1)and

‖�λkn�k

n,ε‖L

21−γ (R+;B1)

∼ 1

(λkn)

γ‖�k

n,ε‖L

21−γ (R+;B1)

.

For small enough γ , Theorems 2.14 and 2.15 imply that

∥∥�λ

jn�j

n,ε⊗�λkn�k

n,ε

∥∥L1(R+;B1)

�(λj

n

λkn

)γ ·We deduce that

‖F3n,ε‖F0 �

∑0≤j,k≤L

j=k

min{λj

n

λkn

,λkn

λjn

}γ.

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Anisotropy in the Weak Stability of the Navier–Stokes System 51

As the sequences (λjn)n∈N and (λk

n)n∈N are orthogonal (see Definition 2.10), wehave for any j and k such that j = k that

limn→∞min

{λjn

λkn

,λkn

λjn

}= 0.

Thus an integer N2 greater than or equal to N1 exists such that

∀n≥ N2 , ‖F3n,ε‖F0 � ε .

Together with (2.33) and (2.36), this implies that

n≥ N2 �⇒‖Fn,ε‖F0 � ε ,

which proves (2.32) and thus concludes the proof of Theorem 2.6.

2.3 Profile Decomposition of the Sequence of Initial Data:Proof of Theorem 2.12

The proof of Theorem 2.12 is structured as follows. First, in Section 2.3.1 wewrite down the profile decomposition of any bounded sequence of anisotrop-ically oscillating divergence-free vector fields, following the results of [4].Next we reorganize the profile decomposition by grouping together all profileshaving the same horizontal scale and we check that all the conclusions ofTheorem 2.12 hold: this is performed in Section 2.3.2.

2.3.1 Profile Decomposition of Anisotropically Oscillating,Divergence-free Vector Fields

In this section we start by recalling the result of [4], where an anisotropic pro-file decomposition of sequences of B1,q anisotropically oscillating is achieved.Let us first define anisotropic scaling operators, similar to the operators definedin (2.1): for any two sequences of positive real numbers (εn)n∈N and (γn)n∈N,and for any sequence (xn)n∈N of points in R3, we denote

�εn,γn,xnφ(x)def= 1

εnφ

(xh− xn,h

εn

,x3− xn,3

γn

Let us also introduce the definition of orthogonal triplets of sequences,analogous to Definition 2.10.

Definition 2.16 We say that two triplets of sequences (ε�n,γ �n ,x�n)n∈N with �

belonging to {1,2}, where (ε�n,γ �n )n∈N are two sequences of positive real

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52 H. Bahouri, J.-Y. Chemin and I. Gallagher

numbers and x�n are sequences in R3, are orthogonal if, when n tends to infinity,

eitherε1

n

ε2n

+ ε2n

ε1n

+ γ 1n

γ 2n

+ γ 2n

γ 1n

→∞

or (ε1n ,γ 1

n )≡ (ε2n ,γ 2

n ) and |(x1n)

ε1n ,γ 1

n − (x2n)

ε1n ,γ 1

n |→∞ ,

where we have denoted (x�n)εk

n,γ kn

def=(x�n,h

εkn

,x�n,3

γ kn

)· A family of sequences(

(εjn,γ j

n,xjn)n∈N

)j≥0 is said to be a family of scales and cores if ε0

n ≡ γ 0n ≡ 1,

x0n ≡ 0, and if (ε�n,γ �

n ,x�n)n∈N and (εkn,γ k

n ,xkn)n∈N are orthogonal when � = k.

Now, let us recall without proof the following result.

Proposition 2.17 ([4]) Under the assumptions of Theorem 2.6, the followingholds. For all integers �≥ 0 there is a triplet of scales and cores in the sense ofDefinition 2.16, denoted by (ε�n,γ �

n ,x�n)n∈N, and for all α in ]0,1[ there are arbi-trarily smooth divergence-free vector fields (φh,�

α ,0) and (−∇h�−1h ∂3φ

�α ,φ�

α)

with φh,�α and φ�

α compactly supported, and such that, up to extracting asubsequence, one can write the sequence (u0,n)n∈N in the following form, foreach L≥ 1:

u0,n = u0+L∑

�=1

�ε�n,γ �n ,x�n

h,�α + r h,�

α − ε�n

γ �n

∇h�−1h ∂3(φ

�α+ r�α),φ

�α+ r�α

)+ (

ψh,Ln −∇h�−1

h ∂3ψLn ,ψL

n

),

(2.37)where ψh,L

n and ψLn are independent of α and satisfy

limsupn→∞

(‖ψh,L

n ‖B0 +‖ψLn ‖B0

)→ 0, L→∞ , (2.38)

while r h,�α and r�α are independent of n and L and satisfy for each � ∈N

‖r h,�α ‖B1,q +‖r�α‖B1,q ≤ α . (2.39)

Moreover, the following properties hold:

∀�≥ 1, limn→∞ (γ �

n )−1ε�n ∈ {0,∞} ,

and limn→∞ (γ �

n )−1ε�n =∞ �⇒ φ�

α ≡ r�α ≡ 0,(2.40)

as well as the following stability result, which is uniform in α:∑�≥1

(‖φ h,�α ‖B1,q +‖r h,�

α ‖B1,q +‖φ�α‖B1,q +‖r�α‖B1,q

)� sup

n‖u0,n‖B1,q +‖u0‖B1,q .

(2.41)

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Anisotropy in the Weak Stability of the Navier–Stokes System 53

Remark 2.18 As pointed out in [4, Section 2], if two scales appearing in theabove decomposition are not orthogonal, then they can be chosen to be equal.We shall therefore assume from now on that this is the case: two sequences ofscales are either orthogonal or equal.

2.3.2 Regrouping of Profiles According to Horizontal Scales

In order to proceed with the re-organization of the profile decompositionprovided in Proposition 2.17, we introduce some more definitions, keepingthe notation of Proposition 2.17. For a given L ≥ 1 we define recursively anincreasing (finite) sequence of indices �k ∈ {1, . . . ,L} by

�0def= 0, �k+1

def= min{� ∈ {�k+ 1, . . . ,L}/ ε�n

γ �n

→ 0 and � /∈k⋃

k′=0

�L(ε�k′n )

},

(2.42)where for 0≤ �≤ L, we define (recalling that by Remark 2.18 if two scales arenot orthogonal, then they are equal),

�L(ε�n)def=

{�′ ∈ {1, . . . ,L}/ε�′n ≡ ε�n and ε�

′n (γ

�′n )−1→ 0, n→∞

}. (2.43)

We call L(L) the largest index of the sequence (�k) and we may then introducethe following partition:

{� ∈ {1, . . . ,L}/ε�n(γ �

n )−1 → 0

}= L(L)⋃k=0

�L(ε�kn ) . (2.44)

We shall now regroup profiles in the decomposition (2.37) of u0,n according tothe value of their horizontal scale. We fix from now on an integer L≥ 1.

Construction of the Profiles for � = 0Before going into the technical details of the construction, let us discuss anexample explaining the computations of this paragraph. Consider the particularcase when u0,n is given by

u0,n(x)= u0(x)+(v0

0(xh,2−nx3)+w0,h0 (xh,2−2nx3),0

)+ (

v00(x1+ 2n,x2,2−nx3),0

),

with v00 and w0,h

0 smooth (say in Bs,s′1,q for all s,s′ in R) and compactly supported.

Let us assume that u0,n converges towards u0 in the sense of distributions, andthat (u0,n− u0)n∈N is anisotropically oscillating. Then we can write

u0,n(x)= u0(x)+(v0,loc

0,n (xh,2−nx3),0)+ (

v0,∞0,n (xh,2−nx3),0

),

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54 H. Bahouri, J.-Y. Chemin and I. Gallagher

with v0,loc0,n (y) := v0

0(y)+w0,h0 (yh,2−ny3) and v0,∞

0,n (y)= v00(y1+ 2n,y2,y3). Now

since u0,n ⇀ u0 as n goes to infinity, we have that v00(xh,0)+wh

0(xh,0) ≡ 0,hence v0,loc

0,n (xh,0)= 0. The initial data u0,n has therefore been re-written as

u0,n(x)= u0(x)+(v0,loc

0,n (xh,2−nx3),0)+ (

v0,∞0,n (xh,2−nx3),0

)with v0,loc

0,n (xh,0)= 0

and where the support in xh of v0,loc0,n (xh,2−nx3) is in a fixed compact set whereas

the support in xh of v0,∞0,n (xh,2−nx3) escapes to infinity. This is of the same form

as in the statement of Theorem 2.12.When considering all the profiles having the same horizontal scale (1 here),

the point is therefore to choose the smallest vertical scale (2n here) and towrite the decomposition in terms of that scale only. Of course this impliesthat, contrary to usual profile decompositions, the profiles are no longer fixedfunctions in B1,q, but sequences of functions, bounded in B1,q.

In view of the above example, let �−0 be an integer such that γ�−0n is the

smallest vertical scale going to infinity, associated with profiles for 1≤ �≤ L,having 1 for horizontal scale. More precisely, we ask that

γ�−0n = min

�∈�L(1)γ �

n ,

where according to (2.43),

�L(1)={�′ ∈ {1, . . . ,L}/ε�′n ≡ 1 and γ �′

n →∞ , n→∞}

.

Notice that the minimum of the sequences γ �n is well defined in our context,

thanks to the fact that due to Remark 2.18, either two sequences are orthogonalin the sense of Definition 2.16, or they are equal. Observe also that �−0 is byno means unique, as several profiles may have the same horizontal scale aswell as the same vertical scale (in which case the concentration cores must beorthogonal).

Now we denote

h0n

def= (γ�−0n )−1 , (2.45)

and we notice that h0n goes to zero as n goes to infinity for each L. Note also

that h0n depends on L through the choice of �−0 , since if L increases then �−0 may

also increase; this dependence is omitted in the notation for simplicity. Let usdefine (up to a subsequence extraction)

a� def= limn→∞

(x�n,h,

x�n,3

γ �n

)· (2.46)

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Anisotropy in the Weak Stability of the Navier–Stokes System 55

We then define the divergence-free vector fields

v0,loc0,n,α,L(y)

def=∑

�∈�L(1)a�h∈R2

φh,�α

(yh− x�n,h ,

y3

h0nγ

�n

− x�n,3

γ �n

)(2.47)

and

w0,loc0,n,α,L(y)

def=∑

�∈�L(1)a�h∈R2

(− 1

h0nγ

�n

∇h�−1h ∂3φ

�α ,φ�

α

)(yh− x�n,h ,

y3

h0nγ

�n

− x�n,3

γ �n

).

(2.48)By construction we have

w0,loc,h0,n,α,L =−∇h�−1

h ∂3w0,loc,30,n,α,L .

Similarly, we define

v0,∞0,n,α,L(y)

def=∑

�∈�L(1)|a�h|=∞

φh,�α

(yh− x�n,h ,

y3

h0nγ

�n

− x�n,3

γ �n

)(2.49)

and

w0,∞0,n,α,L(y)

def=∑

�∈�L(1)|a�h|=∞

(− 1

h0nγ

�n

∇h�−1h ∂3φ

�α ,φ�

α

)(yh− x�n,h ,

y3

h0nγ

�n

− x�n,3

γ �n

).

(2.50)By construction we have again

w0,∞,h0,n,α,L =−∇h�−1

h ∂3w0,∞,30,n,α,L .

Moreover, recalling the notation

[f ]h0n(x)

def= f (xh,h0nx3)

and

�εn,γn,xnφ(x)def= 1

εnφ

(xh− xn,h

εn,x3− xn,3

γn

),

one can compute that∑�∈�L(1)a�h∈R2

�1,γ �n ,x�n

(φh,�α − 1

γ �n

∇h�−1h ∂3φ

�α ,φ�

α

)= [

(v0,loc0,n,α,L+h0

nw0,loc,h0,n,α,L,w0,loc,3

0,n,α,L)]

h0n

(2.51)

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56 H. Bahouri, J.-Y. Chemin and I. Gallagher

and∑�∈�L(1)|a�h|=∞

�1,γ �n ,x�n

(φh,�α − 1

γ �n

∇h�−1h ∂3φ

�α ,φ�

α

)= [

(v0,∞0,n,α,L+h0

nw0,∞,h0,n,α,L,w0,∞,3

0,n,α,L)]

h0n

.

(2.52)Let us now check that v0,loc

0,n,α,L, w0,loc0,n,α,L, v0,∞

0,n,α,L and w0,∞0,n,α,L satisfy the bounds

given in the statement of Theorem 2.12. We shall study only v0,loc0,n,α,L and w0,loc

0,n,α,L

as the other study is very similar. By translation and scale invariance of B0 andusing definitions (2.47) and (2.48), we get

‖v0,loc,h0,n,α,L‖B0 ≤

∑�≥1

‖φh,�α ‖B0 and ‖w0,loc,3

0,n,α,L‖B0 ≤∑�≥1

‖φ�α‖B0 . (2.53)

According to (2.41) and the Sobolev embedding B1,q ↪→ B0, this gives rise to

‖v00,n,α,L‖B0 +‖w0,loc,3

0,n,α,L‖B0 ≤ C uniformly in α ,L ,n . (2.54)

Moreover, for each given α, the profiles are as smooth as needed, and since

in the above sums by construction γ�−0n,L ≤ γ �

n , one gets also, after an easycomputation

∀s∈R ,∀s′ ≥ 1/2, ‖v0,loc0,n,α,L‖Bs,s′ + ‖w0,loc,3

0,n,α,L‖Bs,s′ ≤C(α) uniformly in n ,L .(2.55)

Estimates (2.54) and (2.55) easily give (2.16) and (2.17).Finally, let us estimate v0,loc,h

0,n,α,L(·,0) and w0,loc,30,n,α,L(·,0) in B0

2,1(R2) and

prove (2.18). On the one hand, by assumption we know that u0,n ⇀ u0 inthe sense of distributions. On the other hand, we can take weak limits in thedecomposition of u0,n provided by Proposition 2.17. We recall that by (2.40),if ε�n/γ

�n →∞ then φ�

α ≡ r�α ≡ 0. Then we notice that clearly

ε�n → 0 or ε�n →∞ �⇒ �ε�n,γ �n ,x�n

f ⇀ 0

for any value of the sequences γ �n ,x�n and any function f . Moreover,

γ �n → 0 �⇒ �1,γ �

n ,x�nf ⇀ 0

for any sequence of cores x�n and any function f , so we are left with the studyof profiles such that ε�n ≡ 1 and γ �

n →∞. Then we also notice that if γ �n →∞,

then with Notation (2.46),

|a�h| =∞ �⇒ �1,γ �

n ,x�nf ⇀ 0. (2.56)

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Anisotropy in the Weak Stability of the Navier–Stokes System 57

In that case, in view of (2.38) and (2.41),

L∑�=1

�ε�n,γ �n ,x�n

ε�n

γ �n

∇h�−1h ∂3(φ

�α+ r�α)+∇h�−1

h ∂3ψLn ⇀ 0.

Consequently, for each L ≥ 1 and each α in ]0,1[, we have in view of (2.37),as n goes to infinity

−ψLn −

∑�∈�L(1)

r�α(·− x�n,h,·− x�n,3

γ �n

)⇀∑

�∈�L(1)s.t.a�h∈R2

φ�α(·− a�

h,0)

−ψh,Ln −

∑�∈�L(1)

r h,�α (·− x�n,h,

·− x�n,3

γ �n

)⇀∑

�∈�L(1)s.t.a�h∈R2

φh,�α (·− a�

h,0) .

(2.57)

Now let η > 0 be given. Then, thanks to (2.38) and (2.39), there is an L0 ≥ 1such that for all L ≥ L0 there is an α0 ≤ 1 (depending on L) such that forall L≥ L0 and α ≤ α0, uniformly in n≥ n(L0,η),∥∥∥(ψ h,L

n ,ψLn

)∥∥∥B0+∥∥∥ ∑�∈�L(1)

(rh,�α ,r�α)(·− x�n,h,

·− x�n,3

γ �n

)

∥∥∥B0≤ η .

Using the fact that B0 is embedded in L∞(R;B02,1(R

2)), we infer from (2.57)that for L≥ L0 and α ≤ α0∥∥∥ ∑

�∈�L(1)s.t.a�h∈R2

φh,�α (·− a�

h,0)∥∥∥

B02,1(R

2)≤ η (2.58)

and ∥∥∥ ∑�∈�L(1)

s.t.a�h∈R2

φ�α(·− a�

h,0)∥∥∥

B02,1(R

2)≤ η . (2.59)

But by (2.47), we have

v0,loc,h0,n,α,L(·,0)=

∑�∈�L(1)a�h∈R2

φh,�α

(·−x�n,h,−x�n,3

γ �n

)

and by (2.48) we have also

w0,loc,30,n,α,L(·,0)=

∑�∈�L(1)a�h∈R2

φ�α

(·−x�n,h,−x�n,3

γ �n

).

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58 H. Bahouri, J.-Y. Chemin and I. Gallagher

It follows that we can write for all L≥ L0 and α ≤ α0,

limsupn→∞

‖v0,loc,h0,n,α,L(·,0)‖B0

2,1(R2) ≤

∥∥ ∑�∈�L(1)a�h∈R2

φh,�α (·− a�

h,0)∥∥

B02,1(R

2)

≤ η

thanks to (2.58). A similar estimate for w0,loc,30,n,α,L(·,0) using (2.59) gives finally

limL→∞ lim

α→0limsup

n→∞

(‖v0,loc,h

0,n,α,L(·,0)‖B02,1(R

2)+‖w0,loc,30,n,α,L(·,0)‖B0

2,1(R2)

)= 0. (2.60)

The results (2.19) and (2.20) involving the cut-off function θ are simply due tothe fact that the profiles are compactly supported.

Construction of the Profiles for � ≥ 1The construction is very similar to the previous one. We start by considering afixed integer j ∈ {1, . . . ,L(L)}.

Then we define an integer �−j so that, up to a sequence extraction,

γ�−jn = min

�∈�L(ε�jn )

γ �n ,

whereas in (2.43)

�L(ε�n)def=

{�′ ∈ {1, . . . ,L}/ε�′n ≡ ε�n and ε�

′n (γ

�′n )−1 → 0, n→∞

}.

Notice that necessarily ε�−j ≡ 1. Finally, we define

hjn

def= ε�jn (γ

�−jn )−1 .

By construction we have that hjn → 0 as n →∞ (recall that ε

�jn ≡ ε

�−jn ). Then

we define for j≤L(L)

vj,hn,α,L(y)

def=∑

�∈�L(ε�jn )

φh,�α

(yh−

x�n,h

ε�jn

,ε�jn

hjnγ �

n

y3−x�n,3

γ �n

)(2.61)

and

wjn,α,L(y)

def=∑

�∈�L(ε�jn )

(− ε

�jn

hjnγ �

n

∇h�−1h ∂3φ

�α ,φ�

α

)(yh−

x�n,h

ε�jn

,ε�jn

hjnγ �

n

y3−x�n,3

γ �n

)and we choose

L(L) < j≤ L ⇒ vj,hn,α,L ≡ 0 and w

jn,α,L ≡ 0. (2.62)

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Anisotropy in the Weak Stability of the Navier–Stokes System 59

We notice thatw

j,hn,α,L =−∇h�−1

h ∂3wj,3n,α,L .

Defining

λjn

def= ε�jn ,

a computation, similar to that giving (2.51), implies directly that∑�∈�L(ε

�jn )

�ε�jn ,γ �

n ,x�n

(φh,�α − λ

jn

γ �n

∇h�−1h ∂3φ

�α ,φ�

α

)=�

λjn

[(v

j,hn,α,L+ hj

nwj,hn,α,L,wj,3

n,α,L)]

hjn

.

(2.63)

Notice that since ε�jn ≡ 1 as recalled above, we have that λ

jn → 0 or ∞

as n → ∞.The a priori bounds for the profiles (v

j,hn,α,L,wj,3

n,α,L)1≤j≤L are obtained exactlyas in the previous paragraph: let us prove that∑

j≥1

(‖vj,hn,α,L‖B0 +‖wj,3

n,α,L‖B0

)≤ C , and

∀s ∈R , ∀s′ ≥ 1/2,∑j≥1

(‖vj,hn,α,L‖Bs,s′ + ‖wj,3

n,α,L‖Bs,s′)≤ C(α) .

(2.64)

We shall detail the argument for the first inequality only, and in the caseof vj,h

n,α,L, as the study of wj,3n,α,L is similar. We write, using the definition of vj,h

n,α,L

in (2.61),

L∑j=1

‖vj,hn,α,L‖B0 =

L(L)∑j=1

∥∥∥ ∑�∈�L(ε

�jn )

φh,�α

(yh−

x�n,h

ε�jn

,ε�jn

hjnγ �

n

y3−x�n,3

γ �n

)∥∥∥B0

,

so by definition of the partition (2.44) and by scale and translation invarianceof B0 we find, thanks to (2.41), that there is a constant C independent of L suchthat

L∑j=1

‖vj,hn,α,L‖B0 ≤

L∑�=1

‖φh,�α ‖B0 ≤ C .

The result is proved.

Construction of the Remainder TermWith the notation of Proposition 2.17, let us first define the remainder terms

ρ(1),hn,α,L

def= −L∑

�=1

ε�n

γ �n

�ε�n,γ �n ,x�n∇h�−1

h ∂3r�α−∇h�−1h ∂3ψ

Ln (2.65)

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60 H. Bahouri, J.-Y. Chemin and I. Gallagher

and

ρ(2)n,α,L

def=L∑

�=1

�ε�n,γ �n ,x�n

(rh,�α ,0

)+ L∑�=1

�ε�n,γ �n ,x�n

(0,r�α)+(ψ

h,Ln ,ψL

n

). (2.66)

Observe that by construction, thanks to (2.38) and (2.39) and to the fact thatif r�α ≡ 0, then ε�n/γ

�n goes to zero as n goes to infinity, we have

limL→∞ lim

α→0limsup

n→∞‖ρ(1),h

α,n,L‖B1,− 12= 0,

and limL→∞ lim

α→0limsup

n→∞‖ρ(2)

α,n,L‖B0 = 0.(2.67)

Then we notice that for each � ∈ N and each α ∈]0,1[, we have by a directcomputation ∥∥∥�ε�n,γ �

n ,x�n(φh,�

α ,0)∥∥∥B1,− 1

2∼ γ �

n

ε�n

∥∥φh,�α

∥∥B1,− 1

2.

We deduce that if ε�n/γ�n →∞, then �ε�n,γ �

n ,x�n(φh,�

α ,0) goes to zero in B1,− 12 as n

goes to infinity, hence so does the sum over � ∈ {1, . . . ,L}. It follows that foreach given α in ]0,1[ and L≥ 1 we may define

ρ(1)n,α,L

def= ρ(1),hn,α,L+

L∑�=1

ε�n/γ�n→∞

�ε�n,γ �n ,x�n

(φh,�α ,0)

and we havelim

L→∞ limα→0

limsupn→∞

‖ρ(1)n,α,L‖B1,− 1

2= 0. (2.68)

Finally, as D(R3) is dense in B1,q, let us choose a family (u0,α)α of functionsin D(R3) such that ‖u0− u0,α‖B1,q ≤ α and let us define

ρn,α,Ldef= ρ

(1)α,n,L+ρ

(2)n,α,L+ u0− u0,α . (2.69)

Inequalities (2.67) and (2.68) give

limL→∞ lim

α→0limsup

n→∞‖et�ρn,α,L‖L2(R+;B1) = 0. (2.70)

End of the Proof of Theorem 2.12Let us return to the decomposition given in Proposition 2.17, and use defini-tions (2.65), (2.66) and (2.69), which imply that

u0,n = u0,α+L∑

�=1ε�n/γ

�n→0

�ε�n,γ �n ,x�n

(φh,�α − ε�n

γ �n

∇h�−1h ∂3φ

�α ,φ�

α

)+ρn,α,L .

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Anisotropy in the Weak Stability of the Navier–Stokes System 61

We recall that for all � in N, we have limn→∞ (γ �n )−1ε�n ∈ {0,∞} and in the case

where the ratio ε�n/γ�n goes to infinity then φ�

α ≡ 0. Next we separate the casewhen the horizontal scale is one from the others: with the notation (2.43) wewrite

u0,n = u0,α +∑

�∈�L(1)

�1,γ �n ,x�n

(φh,�α − 1

γ �n

∇h�−1h ∂3φ

�α ,φ�

α

)

+L∑

�=1ε�n ≡1

ε�n/γ�n→0

�ε�n,γ �n ,x�n

(φh,�α − ε�n

γ �n

∇h�−1h ∂3φ

�α ,φ�

α

)+ρn,α,L .

With (2.51) this can be written

u0,n = u0,α+[(v0,loc,h

0,n,α,L+h0nw

0,loc,h0,n,α,L,w0,loc,3

0,n,α,L)]

h0n+[(v0,∞,h

0,n,α,L+ h0nw

0,∞,h0,n,α,L,w0,∞,3

0,n,α,L)]

h0n

+∑�=1ε�n ≡1

ε�n/γ�n→0

�ε�n,γ �n ,x�n

(φh,�α − ε�n

γ �n

∇h�−1h ∂3φ

�α ,φ�

α

)+ρn,α,L .

Next we use the partition (2.44), so that with notation (2.42) and (2.43),

u0,n = u0,α+[(v0,loc,h

0,n,α,L+h0nw

0,loc,h0,n,α,L,w0,loc,3

0,n,α,L)]

h0n+[(v0,∞,h

0,n,α,L+ h0nw

0,∞,h0,n,α,L,w0,∞,3

0,n,α,L)]

h0n

+L(L)∑j=1

∑�∈�L(ε

�jn )

ε�jn ≡1

�ε�jn ,γ �

n ,x�n

(φh,�α − ε

�jn

γ �n

∇h�−1h ∂3φ

�α ,φ�

α

)+ρn,α,L .

Then we finally use the identity (2.63) which gives

u0,n = u0,α+[(v0,loc,h

0,n,α,L+h0nw

0,loc,h0,n,α,L,w0,loc,3

0,n,α,L)]

h0n+[(v0,∞,h

0,n,α,L+ h0nw

0,∞,h0,n,α,L,w0,∞,3

0,n,α,L)]

h0n

+L∑

j=1

�λ

jn[(vj,h

n,α,L+ hjnw

j,hn,α,L,wj,3

n,α,L)]hjn+ρn,α,L .

The end of the proof follows from the estimates (2.54), (2.55), (2.60), (2.64),along with (2.70). Theorem 2.12 is proved.

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62 H. Bahouri, J.-Y. Chemin and I. Gallagher

2.4 Proof of Theorems 2.14 and 2.15

2.4.1 Proof of Theorem 2.14

In order to prove that the initial data defined by

�0def= [

(v0−β∇h�−1h ∂3w

30,w3

0)]β

,

with (v0,w30) satisfying the assumptions of Theorem 2.14, gives rise to a global

smooth solution for small enough β, we look for the solution in the form

�β =�app+ψ with �app def= [(v+βwh,w3)

]β, (2.71)

where v solves the two-dimensional Navier–Stokes equations

(NS2D)x3:

⎧⎨⎩∂tv+ v · ∇hv−�hv =−∇hp in R+ ×R2,divhv = 0,v|t=0 = v0(·,x3) ,

while w3 solves the transport-diffusion equation

(Tβ):

{∂tw

3+ v · ∇hw3−�hw3−β2∂2

3w3 = 0 in R+ ×R3,

w3|t=0 =w3

0

and wh is determined by the divergence-free condition on w.In Subsection 2.4.1 (resp. 2.4.1), we establish a priori estimates on v (resp.

w), and in Subsection 2.4.1, we achieve the proof of Theorem 2.14 by studyingthe perturbed Navier–Stokes equation satisfied by ψ .

Two-Dimensional Flows with ParameterThe goal of this section is to prove the following proposition on v, the solutionof (NS2D)x3

. It is a general result on the regularity of the solution of (NS2D)

when the initial data depends on a real parameter x3, measured in terms ofBesov spaces with respect to the variable x3.

Proposition 2.19 Let v0 be a two-component divergence-free vector fielddepending on the vertical variable x3, and belonging to Sμ. Then the unique,global solution v to (NS2D)x3

belongs to A0 and satisfies the followingestimate:

‖v‖A0 ≤ T1(‖v0‖B0) . (2.72)

Moreover, for all (s,s′) in Dμ, we have

∀r ∈ [1,∞] , ‖v‖Lr(R+;Bs+ 2

r ,s′ )≤ T2(‖v0‖Sμ). (2.73)

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Anisotropy in the Weak Stability of the Navier–Stokes System 63

Proof The proof of Proposition 2.19 is done in three steps. First, we deducefrom the classical energy estimate for the two-dimensional Navier–Stokessystem a stability result in the spaces Lr(R+;Hs+ 2

r (R2))2 with r in [2,∞]and s in ] − 1,1[. This is the purpose of Lemma 2.20, the proof of whichuses essentially energy estimates together with paraproduct laws. Then wehave to translate the stability result of Lemma 2.20 in terms of Besov spaceswith respect to the third variable, seen before simply as a parameter. Thisis the object of Lemma 2.21, the proof of which relies on the equivalenceof two definitions of Besov spaces with regularity index in ]0,1[: the firstone involving the dyadic decomposition of the frequency space, and theother one consisting of estimating integrals in physical space. Finally, invok-ing the Gronwall lemma and product laws we conclude the proof of theproposition. �

Step 1: 2D-stability result Let us start by proving the following lemma.

Lemma 2.20 For any compact set I included in ] − 1,1[, a constant C existssuch that, for any r in [2,∞] and any s in I, we have for any two solutions v1

and v2 of the two-dimensional Navier–Stokes equations

‖v1− v2‖Lr(R+;Hs+ 2

r (R2))� ‖v1(0)− v2(0)‖Hs(R2) E12(0) , (2.74)

where we define

E12(0)def= expC

(‖v1(0)‖2L2 +‖v2(0)‖2

L2

).

Proof Defining v12(t)def= v1(t)− v2(t), we find that

∂tv12+ v2 · ∇hv12−�hv12 =−v12 · ∇hv1−∇hp . (2.75)

Thus, taking the Hs scalar product with v12, we get, thanks to thedivergence-free condition,

1

2

d

dt‖v12(t)‖2

Hs +‖∇hv12(t)‖2Hs =−(v2(t) · ∇hv12(t)|v12(t)

)Hs

− (v12(t) · ∇hv1(t)|v12(t)

)Hs ,

2 Here Hs+ 2r (R2) denotes the usual homogeneous Sobolev space.

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64 H. Bahouri, J.-Y. Chemin and I. Gallagher

whence, by time integration we get

‖v12(t)‖2Hs + 2

∫ t

0‖∇hv12(t

′)‖2Hsdt′

= ‖v12(0)‖2Hs − 2

∫ t

0

(v2(t

′) · ∇hv12(t′)|v12(t

′))

Hs dt′

− 2∫ t

0

(v12(t

′) · ∇hv1(t′)|v12(t

′))

Hs dt′ .

Now making use of the following estimate proved in [12, Lemma 1.1]:(v · ∇ha|a)Hs � ‖∇hv‖L2‖a‖Hs‖∇ha‖Hs , (2.76)

available uniformly with respect to s in any compact set of ]−2,1[, we deducethat there is a positive constant C such that for any s in I, we have

2∣∣∣∫ t

0

(v2(t

′) · ∇hv12(t′)|v12(t

′))

Hs dt′∣∣∣

≤ 1

2

∫ t

0‖∇hv12(t

′)‖2Hs dt′ + C2

2

∫ t

0‖v12(t

′)‖2Hs‖∇hv2(t

′)‖2L2 dt′ .

(2.77)Noticing that∫ t

0

(v12(t

′)·∇hv1(t′)|v12(t

′))

Hsdt′ ≤∫ t

0‖∇hv12(t

′)‖Hs‖v12(t′)·∇hv1(t

′)‖Hs−1 dt′ ,

we deduce by the Cauchy–Schwarz inequality and product laws in Sobolevspaces on R2 that for s in I,

2∣∣∣∫ t

0

(v12(t

′) · ∇hv1(t′)|v12(t

′))

Hs dt′∣∣∣

≤ 1

2

∫ t

0‖∇hv12(t

′)‖2Hs dt′ + C2

2

∫ t

0‖v12(t

′)‖2Hs‖∇hv1(t

′)‖2L2 dt′ .

(2.78)

Combining (2.77) and (2.78), we get for s in I,

‖v12(t)‖2Hs +

∫ t

0‖∇hv12(t

′)‖2Hsdt′

� ‖v12(0)‖2Hs +

∫ t

0‖v12(t

′)‖2Hs

(‖∇hv1(t′)‖2

L2 +‖∇hv2(t′)‖2

L2

)dt′ .

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Anisotropy in the Weak Stability of the Navier–Stokes System 65

Gronwall’s lemma implies that there exists a positive constant C such that

‖v12(t)‖2Hs +

∫ t

0‖∇hv12(t

′)‖2Hs dt′

� ‖v12(0)‖2Hs expC

∫ t

0

(‖∇hv1(t′)‖2

L2 +‖∇hv2(t′)‖2

L2

)dt′ .

But for any i in {1,2}, we have by the L2 energy estimate (2.2)

∫ t

0‖∇hvi(t

′)‖2L2 dt′ ≤ 1

2‖vi(0)‖2

L2 . (2.79)

Consequently, for s in I,

‖v12(t)‖2Hs +

∫ t

0‖∇hv12(t

′)‖2Hsdt′ � ‖v12(0)‖2

Hs E12(0) ,

which leads to the result by interpolation.

Step 2: propagation of vertical regularity Thanks to Lemma 2.20, we canpropagate vertical regularity as stated in the following result.

Lemma 2.21 For any compact set I included in ] − 1,1[, a constant C existssuch that, for any r in [2,∞] and any s in I, we have for any solution v

to (NS2D)x3,

‖v‖Lr(R+;L∞v (H

s+ 2r

h ))� ‖v0‖BsE(0) with E(0)

def= exp(C‖v(0)‖2

L∞v L2h

).

Proof As mentioned above, the proof of Lemma 2.21 uses crucially thecharacterization of Besov spaces via differences in physical space, namely thatfor any Banach space X of distributions one has (see for instance Theorem 2.36of [2]) ∥∥(2 j

2 ‖�vj u‖L2

v(X)

)j

∥∥�1(Z)

∼∫R

‖u− (τ−zu)‖L2v(X)

|z| 12

dz

|z| , (2.80)

where the translation operator τ−z is defined by

(τ−zu)(t,xh,x3)def= u(t,xh,x3+ z) .

Lemma 2.20 asserts that, for any r in [2,∞], any s in I and any couple (x3,z)in R2, the solution v to (NS2D)x3

satisfies

‖v− τ−zv‖Ysr � ‖v0− τ−zv0‖Hs

hE(0) with Ys

rdef= Lr(R+;H

s+ 2r

h ) .

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66 H. Bahouri, J.-Y. Chemin and I. Gallagher

Taking the L2 norm of the above inequality with respect to the x3 variable and

then the L1 norm with respect to the measure |z|− 32 dz gives∫

R

‖v− τ−zv‖L2v(Y

sr )

|z| 12

dz

|z| �∫R

‖v0− τ−zv0‖L2v(H

sh)

|z| 12

dz

|z| E(0) . (2.81)

Now, making use of the characterization (2.80) with X = Ysr , we find that∫

R

‖v− τ−zv‖L2v(Y

sr )

|z| 12

dz

|z| ∼∑j∈Z

2j2

∥∥∥∥∥(2k(s+ 2r )�v

j �hkv(t, ·,z)

)k

∥∥Lr(R+;�2(Z;L2

h))

∥∥∥L2

v.

Similarly, we have∫R

‖v0− τ−zv0‖L2v(H

sh)

|z| 12

dz

|z| ∼∑j∈Z

2j2∥∥(2ks‖�v

j �hkv0‖L2

h

)k

∥∥�2(Z;L2

v).

Thus, by the embedding from �1(Z) to �2(Z), we get∫R

‖v0− τ−zv0‖L2v(H

sh)

|z| 12

dz

|z| �∑

(j,k)∈Z2

2j2 2ks‖�v

j �hkv0‖L2(R3) .

This implies that Estimate (2.81) can also be written as∑j∈Z

2j2

∥∥∥∥∥(2k(s+ 2r )�v

j �hkv(t, ·,z)

)k

∥∥Lr(R+;�2(Z;L2

h))

∥∥∥L2

v� ‖v0‖Bs E(0) .

As r ≥ 2, Minkowski’s inequality implies that∑j∈Z

2j2

∥∥∥∥∥(2k(s+ 2r )�v

j �hkv(t, ·)

)k

∥∥�2(Z;L2(R3))

∥∥∥Lr(R+)

� ‖v0‖Bs E(0) .

Bernstein inequalities (2.6) and (2.7) ensure that

‖�vj �

hkv(t, ·)‖L∞v (L2

h)� 2

j2 ‖�h

kv(t, ·)‖L2(R3) ,

which gives rise to∥∥∥∥∥(2k(s+ 2r )‖�h

kv‖L∞v (L2h)

)k

∥∥�2(Z)

∥∥∥Lr(R+)

� ‖v0‖Bs E(0) .

Permuting the �2 norm and the L∞v norm, thanks to Minkowski’s inequalityagain, achieves the proof of the lemma.

Step 3: end of the proof of Proposition 2.19 Our aim is to establish (2.73)for all (s,s′) in Dμ. Let us start by proving the following inequality: for any v

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Anisotropy in the Weak Stability of the Navier–Stokes System 67

solving (NS2D)x3, for any r in [4,∞], any s in ]− 1

2 , 12 [ and any positive s′,

‖v‖Lr(R+;Bs+ 2

r ,s′ )� ‖v0‖Bs,s′ exp

(∫ ∞

0C(‖v(t)‖4

L∞v (L4h))+‖v(t)‖2

L∞v (H1h )

)dt)

.

(2.82)For that purpose, let us introduce, for any non-negative λ, the followingnotation: for any function F we define

Fλ(t)def= F(t)exp

(−λ

∫ t

0φ(t′)dt′

)with φ(t)

def= ‖v(t)‖4L∞v (L4

h)+‖v(t)‖2

L∞v (H1h )

.

Combining Lemma 2.21 with the Sobolev embedding of H12 (R2) into L4(R2),

we find that ∫ t

0φ(t′)dt′ � E(0)(‖v0‖2

B0 +‖v0‖4B0) . (2.83)

Now making use of the Duhamel formula and the action of the heat flow (seefor instance Proposition B.2 in [4]), we infer that

‖�vj �

hkvλ(t)‖L2 ≤ Ce−c22kt‖�v

j �hkv0‖L2

+C2k∫ t

0exp

(−c(t− t′)22k−λ

∫ t

t′φ(t′′)dt′′

)‖�v

j �hk(v⊗ v)λ(t

′)‖L2 dt′ .

(2.84)

Recall that (v⊗v)λ = v⊗vλ. Now to estimate the term ‖�vj �

hk(v⊗v)λ(t′)‖L2 ,

we make use of the anisotropic version of Bony’s paraproduct decompo-sition (one can consult [2] and [41] for an introduction to anisotropicLittlewood–Paley theory), writing

ab=4∑

�=1

T�(a,b) with

T1(a,b)=∑

j,k

Svj Sh

ka�vj �

hkb , (2.85)

T2(a,b)=∑

j,k

Svj �

hka�v

j Shk+1b ,

T3(a,b)=∑

j,k

�vj Sh

kaSvj+1�

hkb ,

T4(a,b)=∑

j,k

�vj �

hkaSv

j+1Shk+1b .

In light of the Bernstein inequality (2.6), we have

‖�vj �

hkT1(v(t),vλ(t))‖L2 � 2

k2 ‖�v

j �hkT1(v(t),vλ(t))‖L2

v(L4/3h )

,

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68 H. Bahouri, J.-Y. Chemin and I. Gallagher

which, in view of (2.85), Holder’s inequalities and the action of the horizontaland vertical truncations on Lebesgue spaces, ensures the existence of somefixed nonzero integer N0 such that

‖�vj �

hkT1(v(t),vλ(t))‖L2 � 2

k2

∑j′≥j−N0k′≥k−N0

‖Svj′S

hk′v(t)‖L∞v (L4

h)‖�v

j′�hk′vλ(t)‖L2

� 2k2 ‖v(t)‖L∞v (L4

h)

∑j′≥j−N0k′≥k−N0

‖�vj′�

hk′vλ(t)‖L2 .

According to the definition of L4(R+;Bs+ 12 ,s′), we get

2js′2ks‖�vj �

hkT1(v(t),vλ(t))‖L2

� ‖vλ‖L4(R+;Bs+ 1

2 ,s′)‖v(t)‖L∞v (L4

h)

∑j′≥j−N0k′≥k−N0

2−(j′−j)s′2−(k′−k)(s+ 12 )fj′,k′(t) ,

where fj′,k′(t), defined by

fj′,k′(t)def= ‖vλ‖−1

L4(R+;Bs+ 12 ,s′

)

2k′(s+ 12 )2j′s′ ‖�v

j′�hk′vλ(t)‖L2 ,

is on the sphere of �1(Z2;L4(R+)).Since s >−1/2 and s′ > 0, it follows by Young’s inequality on series that

2js′2ks‖�vj �

hkT1(v(t),vλ(t))‖L2 � ‖vλ‖

L4(R+;Bs+ 12 ,s′

)‖v(t)‖L∞v (L4

h)fj,k(t),

where fj,k(t) is on the sphere of �1(Z2;L4(R+)).As by definition φ(t) is greater than ‖v(t)‖4

L∞v (L4h)

, we infer that

T 1j,k,λ(t)

def= 2k2js′2ks∫ t

0exp

(−c(t− t′)22k−λ

∫ t

t′φ(t′′)dt′′

)×‖�v

j �hkT1(v(t′),vλ(t′))‖L2 dt′

� ‖vλ‖L4(R+;Bs+1/2,s′ )

× 2k∫ t

0exp

(−c(t− t′)22k−λ

∫ t

t′φ(t′′)dt′′

14 (t′)fj,k(t′)dt′ .

(2.86)

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Anisotropy in the Weak Stability of the Navier–Stokes System 69

By Holder’s inequality, this leads to

T 1j,k,λ(t)� ‖vλ‖

L4(R+;Bs+ 12 ,s′

)

(∫ t

0e−c(t−t′)22k

f 4j,k(t

′)dt′) 1

4

× 2k

(∫ t

0exp

(−c(t− t′)22k− 4

∫ t

t′φ(t′′)dt′′

)φ(t′)

13 dt′

) 34

.

Finally, applying Holder’s inequality in the last term of the above inequality,we get

T 1j,k,λ(t)�

1

λ14

(∫ t

0e−c(t−t′)22k

f 4j,k(t

′)dt′) 1

4 ‖vλ‖L4(R+;Bs+1/2,s′ ) . (2.87)

Now let us study the term with T2. Using again that the support of the Fouriertransform of the product of two functions is included in the sum of the twosupports, let us write that

‖�vj �

hkT2(v(t),vλ(t))‖L2 �

∑j′≥j−N0k′≥k−N0

‖Svj′�

hk′v(t)‖L∞v (L2

h)‖�v

j′Shk′+1vλ(t)‖L2

v(L∞h ) .

Combining the Bernstein inequality (2.6) with the definition of the function φ,we get

‖Svj′�

hk′v(t)‖L∞v (L2

h)� 2−k′ ‖v(t)‖L∞v (H1

h )� 2−k′φ

12 (t) . (2.88)

Now let us observe that, using the Bernstein inequality again, we have

‖�vj′S

hk′+1vλ(t)‖L2

v(L∞h ) �

∑k′′≤k′

‖�vj′�

hk′′vλ(t)‖L2

v(L∞h )

�∑k′′≤k′

2k′′ ‖�vj′�

hk′′vλ(t)‖L2 .

By definition of the L4(R+;Bs+ 12 ,s′) norm, we have

2j′s′2k′(s− 12 ) ‖�v

j′Shk′+1vλ(t)‖L2

v(L∞h ) � ‖vλ‖L4(R+;Bs+ 1

2 ,s′)

∑k′′≤k′

2(k′−k′′)(s− 12 )f

j′,k′′(t),

where fj′,k′′(t), on the sphere of �1(Z2;L4(R+)), is defined by

fj′,k′′(t)

def= ‖vλ‖−1L4(R+;Bs+1/2,s′ )2

j′s′2k′′(s+1/2)‖�vj′�

hk′′vλ(t)‖L2 .

Since s < 12 , this ensures by Young’s inequality that

‖�vj′S

hk′+1vλ(t)‖L2

v(L∞h ) � 2−j′s′2−k′(s− 1

2 ) ‖vλ‖L4(R+;Bs+1/2,s′ ) fj′,k′(t),

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70 H. Bahouri, J.-Y. Chemin and I. Gallagher

where fj′,k′(t) is on the sphere of �1(Z2;L4(R+)). Together with Inequal-ity (2.88), this gives

2js′2k(s+ 12 ) ‖�v

j �hkT2(v(t),vλ(t))‖L2 � φ(t)

12 ‖vλ‖L4(R+;Bs+1/2,s′ ) fj,k(t) ,

where fj,k(t) is on the sphere of �1(Z2;L4(R+)). We deduce that

T 2j,k,λ(t)

def= 2k2js′2ks∫ t

0exp

(−c(t− t′)22k−λ

∫ t

t′φ(t′′)dt′′

)×‖�v

j �hkT2(v(t′),vλ(t′))‖L2 dt′

� ‖vλ‖L4(R+;Bs+1/2,s′ )

× 2k2

∫ t

0exp

(−c(t− t′)22k−λ

∫ t

t′φ(t′′)dt′′

)φ(t′)

12 fj,k(t

′)dt′ .

(2.89)Using Holder’s inequality twice, we get

T 2j,k,λ(t)� ‖vλ‖L4(R+;Bs+1/2,s′ )

(∫ t

0e−c(t−t′)22k

f 4j,k(t

′)dt′) 1

4

× 2k2

(∫ t

0exp

(−c(t− t′)22k−λ

∫ t

t′φ(t′′)dt′′

)φ(t′)

23 dt′

) 34

� 1

λ12

‖vλ‖L4(R+;Bs+1/2,s′ )

(∫ t

0e−c(t−t′)22k

f 4j,k(t

′)dt′) 1

4

. (2.90)

As T3 is estimated like T1 and T4 is estimated like T2, this implies finally that

2js′2ks‖�vj �

hkvλ(t)‖L2 � 2js′2kse−c22kt‖�v

j �hkv0‖L2

+(∫ t

0e−c(t−t′)22k

f 4j,k(t

′)dt′) 1

4( 1

λ14

+ 1

λ12

)‖vλ‖L4(R+;Bs+1/2,s′ ) .

As we have(∫ ∞

0

(∫ t

0e−c(t−t′)22k

f 4j,k(t

′)dt′) 1

4×4dt

) 14

= c−1dj,k2−k2

and supt∈R+

(∫ t

0e−c(t−t′)22k

f 4j,k(t

′)dt′) 1

4 = dj,k , withdj,k ∈ �1(Z2) ,

we infer that

2js′2ks(‖�v

j �hkvλ‖L∞(R+;L2)+ 2

k2 ‖�v

j �hkvλ‖L4(R+;L2)

)� 2js′2ks‖�v

j �hkv0‖L2 + dj,k

( 1

λ14

+ 1

λ12

)‖vλ‖L4(R+;Bs+1/2,s′ ) .

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Anisotropy in the Weak Stability of the Navier–Stokes System 71

This ends the proof of (2.82) by taking the sum over j and k and choosing λ

large enough.Now to show that Estimate (2.82) remains available for r = 2, we start

from Formula (2.84) with λ = 0. Applying again anisotropic paraproductdecomposition, we find by arguments similar to those conducted above

2js′2k(s+1)‖�vj �

hkv(t)‖L2 � 2js′2k(s+1)e−c22kt‖�v

j �hkv0‖L2

+ 22k ‖v‖L4(R+;Bs+ 1

2 ,s′)

∫ t

0e−c(t−t′)22k(

(gj,k(t′)+ 2−

k2 hj,k(t

′))dt′ ,

where gj,k (resp. hj,k) are in �1(Z2;L2(R+)) (resp. �1(Z2;L43 (R+))), with∑

(j,k)∈Z2

‖gj,k‖L2(R+) � ‖φ‖14L1 and

∑(j,k)∈Z2

‖hj,k‖L

43 (R+)

� ‖φ‖12L1 .

Laws of convolution in the time variable, summation over j and k and (2.82)imply that

‖v‖L2(R+;Bs+1,s′ ) � ‖v0‖Bs,s′ exp(

C∫ ∞

0φ(t)dt

).

This implies by interpolation in view of (2.82) that for all r in [2,∞], all s in]− 1

2 , 12 [ and all positive s′

‖v‖Lr(R+;Bs+2/r,s′ ) � ‖v0‖Bs,s′ exp(

C∫ ∞

0φ(t)dt

), (2.91)

which in view of (2.83) ensures Inequality (2.72) and achieves the proof ofEstimate (2.73) in the case when s belongs to ]− 1

2 , 12 [·

To conclude the proof of the proposition, it remains to complete the rangeof indices. Let us first double the interval on the index s, by proving that forany s in ]− 1,1[, any s′ ≥ 1/2 and any r in [2,∞] we have

‖v‖Lr(R+;Bs+2/r,s′ ) � ‖v0‖Bs,s′ + ‖v0‖Bs/2,s′ ‖v0‖Bs/2 exp(C‖v0‖B0 E0) . (2.92)

Anisotropic product laws (see for instance Appendix B in [4]) ensure that forany s in ]− 1,1[ and any s′ ≥ 1/2, we have

‖v(t)⊗ v(t)‖Bs,s′ � ‖v(t)‖B s+12‖v(t)‖

Bs+1

2 ,s′ .

According to Formula (2.84) and the smoothing effect of the horizontal heatflow, we find that, for any s belonging to ] − 1,1[, any s′ ≥ 1/2 and any rin [2,∞],

‖v‖Lr(R+;Bs+2/r,s′ ) � ‖v0‖Bs,s′ + ‖v⊗ v‖L2(R+;Bs,s′ )� ‖v0‖Bs,s′ + ‖v‖

L4(R+;Bs+1

2 )‖v‖

L4(R+;Bs+1

2 ,s′).

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72 H. Bahouri, J.-Y. Chemin and I. Gallagher

Finally, Inequality (2.82) ensures that for any s in ] − 1,1[, any s′ ≥ 1/2 andany r in [2,∞],‖v‖Lr(R+;Bs+2/r,s′ ) � ‖v0‖Bs,s′ + ‖v0‖Bs/2‖v0‖Bs/2,s′ exp(C‖v0‖B0 E(0)) , (2.93)

which ends the proof of Inequality (2.92), and thus for (2.73) when r isin [2,∞] and s is in ]− 1,1[.Let us now treat the case when s belongs to ] − 2,0] and s′ ≥ 1/2. Again byanisotropic product laws, we have

‖v(t)⊗ v(t)‖Bs+1,s′ � ‖v(t)‖Bs/2+1‖v(t)‖Bs/2+1,s′ ,

which implies that

‖v⊗ v‖L1(R+;Bs+1,s′ ) � ‖v‖L2(R+;Bs/2+1)‖v‖L2(R+;Bs/2+1,s′ ) .

The smoothing effect of the heat flow gives then, for any r in [1,∞], any sin ]− 2,0] and any s′ ≥ 1/2,

‖v‖Lr(R+;Bs+2/r,s′ ) � ‖v0‖Bs,s′ + ‖v‖L2(R+;Bs/2+1)‖v‖L2(R+;Bs/2+1,s′ ) .

Inequality (2.93) implies that, for any r in [1,∞], any s in ] − 2,0] and anys′ ≥ 1/2,

‖v‖Lr(R+;Bs+2/r,s′ ) � ‖v0‖Bs,s′ + ‖v0‖Bs/2

(‖v0‖Bs/2,s′ + ‖v0‖Bs/4‖v0‖Bs/4,s′

exp(C‖v0‖B0 E0))

+‖v0‖2Bs/4

(‖v0‖Bs/2,s′ + ‖v0‖Bs/4‖v0‖Bs/4,s′

exp(C‖v0‖B0E0))

.

This concludes the proof of Estimate (2.73), and thus achieves the proof ofProposition 2.19.

Propagation of Regularity by a Transport-Diffusion EquationNow let us estimate the norm of the function w3 defined as the solution of (Tβ)

defined in Subsection 2.4.1. This is described in the following proposition.

Proposition 2.22 Let v0 and v be as in Proposition 2.19. For any non-negativereal number β, let us consider w3, the solution of

(Tβ)

{∂tw

3+ v · ∇hw3−�hw3−β2∂2

3w3 = 0 in R+ ×R3,

w3|t=0 =w3

0 .

Then w3 satisfies the following estimates, where all the constants are indepen-dent of β:

‖w3‖A0 � ‖w30‖B0 exp

(T1(‖v0‖B0)

), (2.94)

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Anisotropy in the Weak Stability of the Navier–Stokes System 73

and for any s in [−2+μ,0] and any s′ ≥ 1/2, we have

‖w3‖As,s′ �(‖w3

0‖Bs,s′ + ‖w30‖B0T2(‖v0‖Sμ)

)exp

(T1(‖v0‖B0)

). (2.95)

Proof Proposition 2.22 follows easily from the following lemma whichis a general result about the propagation of anisotropic regularity by atransport-diffusion equation.

Lemma 2.23 Let us consider (s,s′), a couple of real numbers, and Q, abilinear operator which continuously maps B1×Bs+1,s′ into Bs,s′ . A constant Cexists such that for any two-component vector field v in L2(R+;B1), any f

in L1(R+;Bs,s′), any a0 in Bs,s′ and for any non-negative β, if �β

def= �h+β2∂2z

and a is the solution of

∂ta−�βa+Q(v,a)= f and a|t=0 = a0 ,

then a satisfies

∀r ∈ [1,∞] , ‖a‖Lr(R+;Bs+2/r,s′ ) ≤ C(‖a0‖Bs,s′ + ‖f‖L1(R+;Bs,s′ )

)exp

(C∫ ∞

0‖v(t)‖2

B1dt)

.

The fact that the third index of the Besov spaces is one induces sometechnical difficulties which lead us to work first on subintervals I of R+ onwhich ‖v‖L2(I;B1) is small.

Let us then start by considering any subinterval I = [τ0,τ1] of R+. TheDuhamel formula and the smoothing effect of the heat flow imply that

‖�hk�

vj a(t)‖L2 ≤ e−c22k(t−τ0)‖�h

k�vj a(τ0)‖L2

+C∫ t

τ0

e−c22k(t−t′)∥∥�hk�

vj

(Q(v(t′),a(t′))+ f (t′)

)∥∥L2 dt′ .

After multiplication by 2ks+js′ and using Young’s inequality in the time integral,we deduce that

2ks+js′(‖�hk�

vj a‖L∞(I;L2)+ 22k‖�h

k�vj a‖L1(I;L2)

)≤ C2ks+js′ ‖�hk�

vj a(τ0)‖L2

+C∫

Idk,j(t

′)(‖v(t′)‖B1‖a(t′)‖Bs+1,s′ + ‖f (t′)‖Bs,s′

)dt′ ,

where for any t, dk,j(t) is an element of the sphere of �1(Z2). By summationover (k, j) and using the Cauchy–Schwarz inequality, we infer that

‖a‖L∞(I;Bs,s′ )+‖a‖L1(I;Bs+2,s′ ) ≤ C‖a(τ0)‖Bs,s′ +C‖f‖L1(I;Bs,s′ )

+C‖v‖L2(I;B1)‖a‖L2(I;Bs+1,s′ ) .(2.96)

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74 H. Bahouri, J.-Y. Chemin and I. Gallagher

Let us define an increasing sequence (Tm)0≤m≤M+1 by induction such that T0 =0, TM+1 =∞ and

∀m < M ,∫ Tm+1

Tm

‖v(t)‖2B1dt= c0 and

∫ ∞

TM

‖v(t)‖2B1dt≤ c0 ,

for some given c0 which will be chosen later on. Obviously, we have∫ ∞

0‖v(t)‖2

B1dt≥∫ TM

0‖v(t)‖2

B1dt=Mc0 . (2.97)

Thus the number M of T ′ms such that Tm is finite is less than c−10 ‖v‖2

L2(R+;B1).

Applying Estimate (2.96) to the interval [Tm,Tm+1], we get

‖a‖L∞([Tm,Tm+1];Bs,s′ )+‖a‖L1([Tm,Tm+1];Bs+2,s′ )

≤ ‖a‖L2([Tm,Tm+1];Bs+1,s′ )+C(‖a(Tm)‖Bs,s′ +C‖f‖L1([Tm,Tm+1];Bs,s′ )

),

if c0 is chosen such that C√

c0 ≤ 1.

Since

‖a‖L2([Tm,Tm+1];Bs+1,s′ ) ≤ ‖a‖12

L∞([Tm,Tm+1];Bs,s′ )‖a‖12

L1([Tm,Tm+1];Bs+2,s′ ) ,

we infer that

‖a‖L∞([Tm,Tm+1];Bs,s′ )+‖a‖L1([Tm,Tm+1];Bs+2,s′ )

≤ 2C(‖a(Tm)‖Bs,s′ + ‖f‖L1([Tm,Tm+1];Bs,s′ )

).

(2.98)

Now let us us prove by induction that

‖a‖L∞([0,Tm];Bs,s′ ) ≤ (2C)m(‖a0‖Bs,s′ + ‖f‖L1([0,Tm],Bs,s′ )

).

Using (2.98) and the induction hypothesis we get

‖a‖L∞([Tm,Tm+1];Bs,s′ ) ≤ 2C(‖a‖L∞([0,Tm];Bs,s′ )+‖f‖L1([Tm,Tm+1];Bs,s′ )

)≤ (2C)m+1(‖a0‖Bs,s′ + ‖f‖L1([0,Tm+1],Bs,s′ )

),

provided that 2C ≥ 1, which ensures in view of (2.97) that

‖a‖L∞(R+;Bs,s′ ) ≤ C(‖a0‖Bs,s′ + ‖f‖L1(R+;Bs,s′ )

)exp

(C∫ ∞

0‖v(t)‖2

B1dt)

.

We deduce from (2.98) that

‖a‖L1([Tm,Tm+1];Bs+2,s′ ) ≤ C(‖a0‖Bs,s′ + ‖f‖L1(R+;Bs,s′ )

)exp

(C∫ ∞

0‖v(t)‖2

B1dt)

+C‖f‖L1([Tm,Tm+1];Bs,s′ ) .

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Anisotropy in the Weak Stability of the Navier–Stokes System 75

Once we have noticed that xeCx2 ≤ eC′x2, the result comes by summation over m

and the fact that the total number of ms is less than or equal to c−10 ‖v‖2

L2(R+;B1).

This ends the proof of the lemma and thus of Proposition 2.22.

As wh is defined by wh =−∇h�−1h ∂3w

3, we deduce from Proposition 2.22 thefollowing corollary.

Corollary 2.24 For any s in [−2+μ,0] and any s′ ≥ 1/2,

‖wh‖As+1,s′−1 �(‖w3

0‖Bs,s′ + ‖w30‖B0T2(‖v0‖Sμ)

)exp

(T1(‖v0‖B0)

).

Conclusion of the Proof of Theorem 2.14Using the definition of the approximate solution �app given in (2.71), we inferfrom Propositions 2.19 and 2.22 and Corollary 2.24 that

‖�app‖L2(R+;B1) ≤ T1(‖(v0,w30)‖B0)+βT2(‖(v0,w3

0)‖Sμ) . (2.99)

Moreover, the error term ψ satisfies the following modified Navier–Stokessystem, with null Cauchy data:

∂tψ + div(ψ ⊗ψ +�app⊗ψ +ψ ⊗�app)−�ψ =−∇qβ +

4∑�=1

E�β with

E1β

def= ∂23 [(v,0)]β +β(0, [∂3p]β) ,

E2β

def= β[(

w3∂3(v,w3)+ (∇h�−1h divh∂3(vw

3),0))]

β,

E3β

def= β[(

wh · ∇h(v,w3)+ v · ∇h(wh,0))]

βand

E4β

def= β2[(

wh · ∇h(wh,0)+w3∂3(wh,0)

)]β

.

(2.100)

If we prove that ∥∥∥ 4∑�=1

E�β

∥∥∥F0≤ βT2

(‖(v0,w30)‖Sμ

), (2.101)

then according to the fact that ψ|t=0 = 0, ψ exists globally and satisfies

‖ψ‖L2(R+;B1) � β T2(‖(v0,w3

0)‖Sμ

). (2.102)

This in turn implies that �0 generates a global regular solution �β

in L2(R+;B1) which satisfies

‖�β‖L2(R+;B1) ≤ T1(‖(v0,w3

0)‖B0

)+β T2(‖(v0,w3

0)‖Sμ

). (2.103)

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76 H. Bahouri, J.-Y. Chemin and I. Gallagher

Once this bound in L2(R+;B1) is obtained, the bound in A0 follows by heatflow estimates, and in As,s′ by propagation of regularity for the Navier–Stokessystem.

So all we need to do is to prove Inequality (2.101). Let us first estimate theterm ∂2

3 [(v,0)]β . This requires the use of some L2(R+;Bs,s′) norms. Clearly,we have

‖∂23 [v]β‖

L2(R+;B0,− 12 )� ‖[v]β‖

L2(R+;B0, 32 )

,

which implies in view of the vertical scaling property (2.12) of the space B0, 32

‖∂23 [v]β‖

L2(R+;B0,− 12 )� β ‖v‖

L2(R+;B0, 32 )

.

Therefore Proposition 2.19 ensures that

‖∂23 [v]β‖

L2(R+;B0,− 12 )≤ β T2(‖v0‖Sμ) . (2.104)

Now let us study the pressure term. By applying the horizontal divergence tothe equation satisfied by v we get, thanks to the fact that divhv = 0,

∂3p=−∂3�−1h

2∑�,m=1

∂�∂m(v�vm) .

Since � and m belong to {1,2}, the operator �−1h ∂�∂m is a zero-order horizontal

Fourier multiplier, which implies that∥∥[∂3p]β∥∥

L1(R+;B0)= ‖∂3p‖L1(R+;B0)

� ‖v∂3v‖L1(R+;B0).

According to laws of product in anisotropic Besov spaces, we obtain

‖v(t)∂3v(t)‖B0 � ‖v(t)‖B1‖∂3v(t)‖B0 ,

which gives rise to∥∥[∂3p]β∥∥

L1(R+;B0)� ‖v‖L2(R+;B1)‖∂3v‖L2(R+;B0)

� ‖v‖L2(R+;B1)‖v‖L2(R+;B0,3/2) . (2.105)

Combining (2.104) and (2.105), we get by virtue of Proposition 2.19

‖E1β‖F0 ≤ β T2

(‖v0‖Sμ

). (2.106)

In the same way, we treat the terms E2β , E3

β and E4β , achieving the proof of

Estimate (2.101). This ends the proof of the fact that the solution �β of (NS)

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Anisotropy in the Weak Stability of the Navier–Stokes System 77

with initial data

�0 =[(v0−β∇h�−1

h ∂3w30,w3

0)]β

is global and belongs to L2(R+;B1).The proof of the whole of Theorem 2.14 is then achieved.

2.4.2 Proof of Theorem 2.15

The proof of Theorem 2.15 is done in three steps. First we define anapproximate solution, using results proved in the previous section, and then weprove useful localization results on the different parts entering in the definitionof the approximate solution. In the last step, we conclude the proof of thetheorem, using those localization results.

The Approximate SolutionWith the notation of Theorem 2.12, let us consider the divergence-free vectorfield:

�00,n,α,L

def= u0,α+[(v0,∞

0,n,α,L+ h0nw

0,∞,h0,n,α,L,w0,∞,3

0,n,α,L

)]h0

n

+ [(v0,loc

0,n,α,L+ h0nw

0,loc,h0,n,α,L,w0,loc,3

0,n,α,L)]

h0n

.

Our purpose is to establish that for h0n small enough, depending only on the

weak limit u0 and on∥∥(v0,∞

0,n,α,L,w0,∞,30,n,α,L)

∥∥Sμ

as well as∥∥(v0,loc

0,n,α,L,w0,loc,30,n,α,L)

∥∥Sμ

,

there is a unique, global smooth solution to (NS) with data �00,n,α,L.

Let us start by solving (NS) globally with the data u0,α . By using the globalstrong stability of (NS) in B1,1 (see [4], Corollary 3) and the convergenceresult (2.14), we deduce that, for α small enough, u0,α generates a unique,

global solution uα to the (NS) system belonging to L2(R+;B2, 1

21,1 ). Actually, in

view of the Sobolev embedding of B2, 1

21,1 into B1, uα ∈ L2(R+;B1).

Next let us define

�0,∞0,n,α,L

def= [(v0,∞

0,n,α,L+ h0nw

0,∞,h0,n,α,L,w0,∞,3

0,n,α,L

)]h0

n.

Thanks to Theorem 2.14, we know that for h0n smaller than

ε1(∥∥(v0,∞

0,n,α,L,w0,∞,30,n,α,L)

∥∥Sμ

)there is a unique global smooth solution �

0,∞n,α,L

associated with �0,∞0,n,α,L, which belongs to A0, and using the notation and

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78 H. Bahouri, J.-Y. Chemin and I. Gallagher

results of Subsection 2.2.3, in particular (2.71) and (2.102), we can write

�0,∞n,α,L

def= �0,∞,appn,α,L +ψ

0,∞n,α,L with

�0,∞,appn,α,L

def= [v0,∞

n,α,L+ h0nw

0,∞,hn,α,L ,w0,∞,3

n,α,L

]h0

nand

‖ψ0,∞n,α,L‖Lr(R+;B 2

r� h0

nT2(∥∥(v0,∞

0,n,α,L,w0,∞,30,n,α,L)

∥∥Sμ

),

(2.107)

for all r in [2,∞],lim

L→∞ limα→0

limsupn→∞

‖�0,locn,α,L(·,0)‖

Lr(R+;B2r2,1(R

2))= 0, (2.108)

where v0,∞n,α,L solves (NS2D)x3

with data v0,∞0,n,α,L, w0,∞,3

n,α,L solves the transport-

diffusion equation (Th0n) defined in Subsection 2.4.1 with data w0,∞,3

0,n,α,L andwhere

w0,∞,hn,α,L =−∇h�−1

h ∂3w0,∞,3n,α,L .

Similarly, defining

�0,loc0,n,α,L

def= [(v0,loc

0,n,α,L+ h0nw

0,loc,h0,n,α,L,w0,loc,3

0,n,α,L

)]h0

n,

then for h0n smaller than ε1

(∥∥(v0,loc0,n,α,L,w0,loc,3

0,n,α,L)∥∥

)there is a unique global

smooth solution �0,locn,α,L associated with �

0,loc0,n,α,L, which belongs to A0, and

�0,locn,α,L

def= �0,loc,appn,α,L +ψ

0,locn,α,L with

�0,loc,appn,α,L

def= [v0,loc

n,α,L+ h0nw

0,loc,hn,α,L ,w0,loc,3

n,α,L

]h0

nand for all r in [2,∞]

‖ψ0,locn,α,L‖Lr(R+;B 2

r� h0

nT2(∥∥(v0,loc

0,n,α,L,w0,loc,30,n,α,L)

∥∥Sμ

),

(2.109)where v0,loc

n,α,L solves (NS2D)x3with data v0,loc

0,n,α,L and w0,loc,3n,α,L solves (Th0

n) with

data w0,loc,30,n,α,L. Finally we recall that w0,loc,h

n,α,L =−∇h�−1h ∂3w

0,loc,3n,α,L .

In the next step, we establish localization properties on �0,∞n,α,L and �

0,locn,α,L.

Those localization properties will enable us to prove that the function uα +�

0,∞n,α,L +�

0,locn,α,L approximates the solution to the (NS) system associated with

the Cauchy data �00,n,α,L.

Localization Properties of the Approximate SolutionIn this step, we prove localization properties on �

0,∞n,α,L and �

0,locn,α,L, namely

the fact that �0,∞,appn,α,L escapes to infinity in the space variable, while �

0,loc,appn,α,L

remains localized (approximately), and we also prove that �0,loc,appn,α,L remains

small near x3 = 0. Let us recall that, as claimed by (2.18), (2.19) and (2.20),those properties are true for their respective initial data. A first part of theselocalization properties derives from the following result.

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Anisotropy in the Weak Stability of the Navier–Stokes System 79

Proposition 2.25 Under the assumptions of Proposition 2.19, the control ofthe value of v at the point x3 = 0 is given by

∀r ∈ [1,∞] , ‖v(·,0)‖Lr(R+;B

2r2,1(R

2))� ‖v0(·,0)‖B0

2,1(R2)+‖v0(·,0)‖2

L2(R2).

(2.110)Moreover, we have for all η in ]0,1[ and γ in {0,1},‖(γ − θh,η)v‖A0 ≤ ∥∥(γ − θh,η)v0

∥∥B0 expT1(‖v0‖B0)+ηT2(‖v0‖Sμ) , (2.111)

where θh,η is the truncation function defined by (2.13).

Proof In order to establish Proposition 2.25, let us start by pointing out thatthe proof of Lemma 1.1 of [12] claims that for all x3 in R,(

�hk(v(t, ·,x3) · ∇hv(t, ·,x3))

∣∣�hkv(t, ·,x3)

)L2

� dk(t,x3)‖∇hv(t, ·,x3)‖2L2‖�h

kv(t, ·,x3)‖L2 ,(2.112)

where (dk(t,x3))k∈Z is a generic element of the sphere of �1(Z).Taking x3 = 0, we deduce by an L2 energy estimate in R2

1

2

d

dt‖�h

kv(t, ·,0)‖2L2 + c22k‖�h

kv(t, ·,0)‖2L2

� dk(t)‖∇hv(t, ·,0)‖2L2‖�h

kv(t, ·,0)‖L2 ,

where (dk(t))k∈Z belongs to the sphere of �1(Z), which after divisionby ‖�h

kv(t, ·,0)‖L2 and time integration leads to

‖�hkv(·,0)‖L∞(R+;L2)+ c22k‖�h

kv(·,0)‖L1(R+;L2)

≤ ‖�kv0(·,0)‖L2 +C∫ ∞

0dk(t)‖∇hv(t, ·,0)‖2

L2dt .(2.113)

By summation over k and in view of (2.79), we obtain Inequality (2.110) ofProposition 2.25.

Now to go to the proof of Inequality (2.111), let us define vγ ,ηdef= (γ −θh,η)v

and write that

∂tvγ ,η−�hvγ ,η+ divh(v⊗ vγ ,η

)= Eη(v)=3∑

i=1

Eiη(v) with

E1η(v)

def= −2η(∇hθ)h,η∇hv−η2(�hθ)h,ηv ,

E2η(v)

def= ηv · (∇hθ)h,ηv and

E3η(v)

def= −(γ − θh,η)∇h�−1h

∑1≤�,m≤2

∂�∂m(v�vm

).

(2.114)

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80 H. Bahouri, J.-Y. Chemin and I. Gallagher

Lemma 2.23 applied with s= 0, s′ = 1/2, a= vγ ,η, Q(v,a)= divh(v⊗ a), f =Eη(v) and β = 0 reduces the problem to the proof of the following estimate:

‖Eη(v)‖L1(R+;B0) � ηT2(‖v0‖Sμ) . (2.115)

Actually, in view of Inequality (2.73) applied with r = 1 and s = −1 (resp.with r= 2 and s=−1/2) this will follow from

‖Eη(v)‖L1(R+;B0) � η(‖v‖L1(R+;B1)+‖v‖2

L2(R+;B1/2)

). (2.116)

Product laws in anisotropic Besov spaces and the scaling properties ofhomogeneous Besov spaces give

‖(∇hθ)h,η∇hv(t)‖B0 � ‖(∇hθ)h,η‖B12,1(R

2)‖∇hv(t)‖B0

� ‖∇hθ‖B12,1(R

2)‖v(t)‖B1 .

Along the same lines, we get

‖(�hθ)h,ηv(t)‖B0 � ‖(�hθ)h,η‖B02,1(R

2)‖v(t)‖B1

� 1

η‖�hθ‖B0

2,1(R2)‖v(t)‖B1 .

Consequently

‖E1η(v)‖L1(R+;B0) � η‖v‖L1(R+;B1) . (2.117)

The same arguments enable us to deal with the term E2η(v) and to prove that

‖E2η(v)‖L1(R+;B0) � η‖v‖2

L2(R+;B1/2). (2.118)

Let us finally study the term E3η(v) which is most challenging. To this end, we

make use of the horizontal paraproduct decomposition:

av = Thva+Th

av+Rh(a,b) with Tha b

def=∑

k

Shk−1a�h

kb

and Rh(a,b)def=

∑k

�hka�h

kb ,

where �hk

def= ϕ(2−kξh) with ϕ a smooth compactly supported (in R2 \ {0})function which has value 1 near B(0,2−N0)+C, where C is an adequate annulus.

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Anisotropy in the Weak Stability of the Navier–Stokes System 81

This allows us to write

E3η(v)=

3∑�=1

E3,�η (v) with

E3,1η (v)

def= Th∇hpθh,η with ∇hp=∇h�−1

h

∑1≤�,m≤2

∂�∂m(v�vm) ,

E3,2η (v)

def= −∑

1≤�,m≤2

[Thγ−θh,η

,∇h�−1h ∂�∂m

]v�vm and

E3,3η (v)

def=∑

1≤�,m≤2

∇h�−1h ∂�∂mTh

v�vmθh,η,

(2.119)

where Tha b = Th

a b + Rh(a,b). Combining the laws of product with scalingproperties of Besov spaces, we obtain

‖Th∇hp(t)θh,η‖B0 � ‖∇hp(t)‖B−1‖θh,η‖B2

2,1(R2)

� η sup1≤�,m≤2

‖v�(t)vm(t)‖B0‖θ‖B22,1(R

2)

� η‖v(t)‖2B1/2‖θ‖B2

2,1(R2) .

Along the same lines we get

‖∇h�−1h ∂�∂mTh

v�(t)vm(t)θh,η‖B0 � ‖Thv�(t)vm(t)θh,η‖B1

� ‖v�(t)vm(t)‖B0‖θh,η‖B22,1(R

2)

� η‖v(t)‖2B1/2‖θ‖B2

2,1(R2) .

We deduce that

‖E3,1η (v)+E3,3

η (v)‖L1(R+;B0) � η‖v‖2L2(R+;B1/2)

. (2.120)

Now let us estimate E3,2η (v). By definition, we have[

Thγ−θh,η

,∇h�−1h ∂�∂m

]v�vm =

∑k

Ek,η(v) with

Ek,η(v)def= [

Shk−N0

(γ − θh,η),�hk∇h�−1

h ∂�∂m]�h

k(v�vm) .

Then by commutator estimates (see for instance Lemma 2.97 in [2])

‖�vj Ek,η(v(t))‖L2 � ‖∇θh,η‖L∞‖�h

k�vj (v

�(t)vm(t))‖L2 .

Noticing that ‖∇θh,η‖L∞ = η‖∇θ‖L∞ , we get by virtue of the laws of product

‖E3,2η (v)‖L1(R+;B0) � η‖v‖2

L2(R+;B1/2),

which ends the proof of Estimate (2.115) and thus of of Proposition 2.25.

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82 H. Bahouri, J.-Y. Chemin and I. Gallagher

A similar result holds for the solution w3 of

(Tβ) ∂tw3+ v · ∇hw3−�hw

3−β2∂23w

3 = 0 and w3|t=0 =w3

0 ,

where β is any non-negative real number. In the following statement, all theconstants are independent of β.

Proposition 2.26 Let v and w3 be as in Proposition 2.22. The control of thevalue of w3 at the point x3 = 0 states as follows. For any r in [2,∞],

‖w3(·,0)‖Lr(R+;B

2r2,1(R

2))≤ T2(‖(v0,w3

0)‖Sμ)(‖w3

0(·,0)‖1−2μ

4(1−μ)

B02,1(R

2)+β

). (2.121)

Moreover, with the notations of Theorem 2.14, we have for all η in ]0,1[ and γ

in {0,1},‖(γ − θh,η)w

3‖A0 ≤ ∥∥(γ − θh,η)w30

∥∥B0 expT1(‖v0‖B0)+ηT2(‖(v0,w3

0)‖Sμ) .(2.122)

The proof of Proposition 2.26 is very similar to that of Proposition 2.25 and isleft to the reader.

Propositions 2.25 and 2.26 easily imply the following result, using thespecial form of �

0,∞n,α,L and �

0,locn,α,L recalled in (2.107) and (2.109), and thanks

to (2.18), (2.19) and (2.20).

Corollary 2.27 The vector fields �0,locn,α,L and �

0,∞n,α,L satisfy the following: �0,loc

n,α,L

vanishes at x3 = 0, in the sense that for all r in [2,∞],lim

L→∞ limα→0

limsupn→∞

‖�0,locn,α,L(·,0)‖

Lr(R+;B2r2,1(R

2))= 0, (2.123)

and there is a constant C(α,L) such that for all η in ]0,1[,limsup

n→∞

(‖(1− θh,η)�

0,locn,α,L‖A0 +‖θh,η�

0,∞n,α,L‖A0

)≤ C(α,L)η .

Proof In view of (2.109) and under Notation (2.11)

�0,locn,α,L =�

0,loc,appn,α,L +ψ

0,locn,α,L with

�0,loc,appn,α,L = [

v0,locn,α,L+ h0

nw0,loc,hn,α,L ,w0,loc,3

n,α,L

]h0

n,

where v0,locn,α,L solves (NS2D)x3

with data v0,loc0,n,α,L, w0,loc,3

n,α,L solves the transport-

diffusion equation (Th0n) defined in Subsection 62 with data w0,loc,3

0,n,α,L

and w0,loc,hn,α,L = −∇h�−1

h ∂3w0,loc,3n,α,L . Combining Property (2.18) together with

Propositions 2.25 and 2.26, we infer that

limL→∞ lim

α→0limsup

n→∞‖�0,loc,app

n,α,L (·,0)‖Lr(R+;B

2r2,1(R

2))= 0,

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Anisotropy in the Weak Stability of the Navier–Stokes System 83

which ends the proof of (2.123) invoking (2.17) and (2.109). The argument is

similar for the other estimates.

Conclusion of the Proof of Theorem 2.15Now, with the above notations, we look for the solution to the (NS) systemassociated with the Cauchy data �0

0,n,α,L in the form:

�0n,α,L

def= uα +�0,∞n,α,L+�

0,locn,α,L+ψn,α,L .

In particular the two vector fields �0,locn,α,L and �

0,∞n,α,L satisfy Corollary 2.27, and

furthermore, thanks to the Lebesgue theorem,

limη→0

‖(1− θη)uα‖L2(R+;B1) = 0. (2.124)

Given a small number ε > 0, to be selected later on, we choose L, α and η =η(α,L,u0) so that thanks to Corollary 2.27 and (2.124), for all r in [2,∞], andfor n large enough,

‖�0,locn,α,L(·,0)‖Lr(R+;B2/r

2,1 (R2))+‖(1− θh,η)�

0,locn,α,L‖A0 +‖(1− θη)uα‖L2(R+;B1)

+‖θh,η�0,∞n,α,L‖A0 ≤ ε .

(2.125)For the sake of simplicity, denote in the sequel

(�0,∞ε ,�0,loc

ε ,ψε)def= (�

0,∞n,α,L,�0,loc

n,α,L,ψn,α,L) and �appε

def= uα+�0,∞ε +�0,loc

ε .

By straightforward computations, one can verify that the vector field ψε

satisfies the following equation, with null Cauchy data:

∂tψε−�ψε+ div(ψε⊗ψε+�app

ε ⊗ψε+ψε⊗�appε

)=−∇qε+Eε ,

with Eε = E1ε +E2

ε and

E1ε

def= div(�0,∞

ε ⊗ (�0,locε + uα)+ (�0,loc

ε + uα)⊗�0,∞ε

+�0,loc⊗ (1− θη)uα + (1− θη)uα⊗�0,loc)

,

E2ε

def= div(�0,loc

ε ⊗ θηuα + θηuα⊗�0,locε

).

(2.126)

The heart of the matter consists of proving that

limε→0

‖Eε‖F0 = 0. (2.127)

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84 H. Bahouri, J.-Y. Chemin and I. Gallagher

Indeed, exactly as in the proof of Theorem 2.14, this ensures that ψε belongsto L2(R+;B1), with

limε→0

‖ψε‖L2(R+;B1) = 0,

and allows us to conclude the proof of Theorem 2.15.So let us focus on (2.127). The term E1

ε is the easiest, thanks to the separationof the spatial supports. Let us first write E1

ε = E1ε,h+E1

ε,3 with

E1ε,h

def= divh

((�0,loc

ε + uα)⊗�0,∞,hε +�0,∞

ε ⊗ (�0,loc,hε + uh

α)

+ (1− θη)uα⊗�0,loc,h+�0,loc⊗ (1− θη)uhα

)and

E1ε,3

def= ∂3

((�0,loc

ε + uα)�0,∞,3ε +�0,∞

ε (�0,loc,3ε + u3

α)

+ (1− θη)uα�0,loc,3+�0,loc(1− θη)u

).

Now using as usual the action of derivatives and the fact that B1 is an algebra,we infer that

‖E1ε,h‖L1(R+;B0)+‖E1

ε,3‖L1(R+;B1,−1/22,1 )

≤ ‖θh,η�0,∞ε ‖L2(R+;B1)‖�0,loc

ε

+ uα‖L2(R+;B1)+‖(1− θh,η)(�0,locε + uα)‖L2(R+;B1)‖�0,∞

ε ‖L2(R+;B1)

+‖�0,locε ‖L2(R+;B1)‖u∞ε ‖L2(R+;B1) ,

where we denote by u∞ε the function (1− θη)uα . Thanks to (2.125) and to thea priori bounds on �0,∞

ε , �0,locε and uα , we easily get

limε→0

‖E1ε‖F0 = 0.

Next let us turn to E2ε . To this end, we use the following estimate (see, for

instance, Lemma 3.3 of [15]):

‖ab‖B1 � ‖a‖B1‖b(·,0)‖B12,1(R

2)+‖x3a‖B1‖∂3b‖B1 . (2.128)

Defining ulocε

def= θηuα , we get, applying Estimate (2.128),

‖E2ε‖F0 � ‖uloc

ε ‖L2(R+;B1)‖�0,locε (·,0)‖L2(R+;B1

2,1(R2))

+‖x3ulocε ‖L2(R+;B1)‖∂3�

0,locε ‖L2(R+;B1) .

Thanks to (2.125) as well as Inequality (2.26) of Theorem 2.14, we obtain

limε→0

‖E2ε‖F0 = 0.

This proves (2.127), hence Theorem 2.15.

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Anisotropy in the Weak Stability of the Navier–Stokes System 85

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Anisotropy in the Weak Stability of the Navier–Stokes System 87

[37] P.-L. Lions, The concentration-compactness principle in the calculus of variations.The limit case I, Revista. Matematica Iberoamericana 1 (1), 1985, 145–201.

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[41] M. Paicu, Equation anisotrope de Navier–Stokes dans des espaces critiques,Revista Matematica Iberoamericana 21 (1), 2005, 179–235.

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∗ Laboratoire d’Analyse et de Mathematiques Appliquees, Paris† Laboratoire Jacques Louis Lions, Paris‡ Universite Paris Diderot, Paris

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3

The Motion Law of Fronts for ScalarReaction-diffusion Equations with Multiple

Wells: the Degenerate CaseFabrice Bethuel∗ and Didier Smets†

Dedicated to the memory of our friend Abbas Bahri, with our deepestadmiration.

We derive a precise motion law for fronts of solutions to scalar one-dimensionalreaction-diffusion equations with equal-depth multiple wells, in the case when thesecond derivative of the potential vanishes at its minimizers. We show that,renormalizing time in an algebraic way, the motion of fronts is governed by asimple system of ordinary differential equations of nearest neighbor interactiontype. These interactions may be either attractive or repulsive. Our results are notconstrained by the possible occurrence of collisions nor splittings. They presentsubstantial differences with the results obtained in the case when the secondderivative does not vanish at the wells, a case which has been extensively studied inthe literature, and where fronts have been shown to move at exponentially smallspeed, with motion laws which are not renormalizable.

3.1 Introduction

This paper is a continuation of our previous works [4, 5] where we analyzedthe behavior of solutions v of the reaction-diffusion equation of gradient type

(PGL)ε∂vε

∂t− ∂2vε

∂x2=− 1

ε2∇V(vε),

where 0 < ε < 1 is a small parameter. In [5], we considered the case where thepotential V is a smooth map from Rk to R with multiple wells of equal depthwhose second derivative vanishes at the wells. The main result there, statedin Theorem 3.2 here, provides an upper bound for the speed of fronts. In thepresent paper we restrict ourselves to the scalar case, k = 1, and provide aprecise motion law for the fronts, showing in particular that the upper boundprovided in [5] is sharp. We assume throughout this paper that the potential V

88

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 89

is a smooth function from R to R which satisfies the following assumptions:

(A1) infV = 0 and the set of minimizers � ≡ {y ∈R,V(y)= 0} is finite,

with at least two distinct elements, that is

� = {σ1, . . . ,σq}, q≥ 2, σ1 < · · ·< σq.

(A2) There exists an integer θ > 1 such that for all i in {1, . . . ,q}, we have

V(u)= λi(u−σi)2θ + o

u→σi((u−σi)

2θ ),where λi > 0.

(A3) There exist constants α∞ > 0 and R∞ > 0 such that

u · ∇V(u)≥ α∞|u|2, if |u|> R∞.

Whereas assumption (A1) expresses the fact that the potential possesses atleast two minimizers, also termed wells, and (A3) describes the behavior atinfinity, and is of a more technical nature, assumption (A2), which is central inthe present paper, describes the local behavior near the minimizing wells. Thenumber θ is of course related to the order of vanishing of the derivatives nearzero. Since θ > 1, then V ′′(σi)= 0, and (A2) holds if and only if

dj

dujV(σi)= 0 for j= 1, . . . ,2θ − 1 and

d2θ

du2θV(σi) > 0,

with

λi = 1

(2θ)!d2θ

du2θV(σi).

A typical example of such potentials is given by V(u) = (1− u2)2θ = (1−u)2θ (1+ u)2θ which has two minimizers, +1 and −1, so that � = {+1,−1},minimizers vanishing at order 2θ . In this paper, the order of degeneracy is aninteger assumed to be the same at all wells: fractional or site-dependent orders(including non-degenerate) may presumably be handled with the same tools,however at the cost of more complicated statements.

We recall that equation (PGL)ε corresponds to the L2 gradient-flow of theenergy functional Eε which is defined for a function u : R �→R by the formula

Eε(u)=∫R

eε(u)=∫R

ε|u|22

+ V(u)

ε.

As in [4, 5], we consider only finite energy solutions. More precisely, we fix anarbitrary constant M0 > 0 and we consider the condition

(H0) Eε(u)≤M0 <+∞.

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90 Fabrice Bethuel and Didier Smets

Besides the assumptions on the potential, the main assumption is on the initialdata v0

ε (·)= vε(·,0), assumed to satisfy (H0) independently of ε. In particular,in view of the classical energy identity

Eε(vε(·,T2))+ ε

∫ T2

T1

∫R

∣∣∣∣∂vε∂t

∣∣∣∣2 (x, t)dxdt= Eε(vε(·,T1)) ∀0≤ T1 ≤ T2 ,

(3.1)we have

Eε (vε(·, t))≤M0, ∀t≥ 0.

This implies in particular that for every given t ≥ 0, we have V(vε(x, t))→ 0as |x| →∞. It is then quite straightforward to deduce from assumption (H0),(A1), (A2) as well as the energy identity (3.1), that vε(x, t)→ σ± as x→±∞,where σ± ∈� do not depend on t. In other words, for any time, our solutionsconnect two given minimizers of the potential.

3.1.1 Main Results: Fronts and Their Speed

The notion of fronts is central in the dynamics. For a map u : R �→R, the set

D(u)≡ {x ∈R, dist(u(x),�)≥μ0},is termed throughout the front set of u. The constant μ0 which appears in itsdefinition is fixed once for all, sufficiently small so that

λi

2(u−σi)

2θ ≤ V(u)≤ 1

θV ′(u)(u−σi)≤ 4V(u)≤ 8λi(u−σi)

2θ , (3.2)

for each i ∈ {1, . . . ,q} and whenever |u−σi| ≤μ0. The front set corresponds tothe set of points where u is “far” from the minimizers σi, and hence wheretransitions from one minimizer to the other may occur. A straightforwardanalysis yields the following.

Lemma 3.1 (see e.g. [4]) Assume that u verifies (H0). Then there exist � pointsx1, . . . ,x� in D(u) such that

D(u)⊂ �∪k=1[xk− ε,xk+ ε],

with a bound � ≤ M0η0

on the number of points, η0 being some constantdepending only on V.

In view of Lemma 3.1, the measure of the front sets is of order ε, andcorresponds to a small neighborhood of order ε of the points xi. Notice thatif (uε)ε>0 is a family of functions satisfying (H0) then it is well known that the

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 91

family is locally bounded in BV(R,R) and hence locally compact in L1(R,R).Passing to a subsequence if necessary, we may assert that

uε → u in L1loc(R),

where u takes values in � and is a step function. More precisely, there existan integer � ≤ M0

η0, � points a1 < · · · < a� and a function ı : { 1

2 , . . . , 12 + �} →

{1, . . . ,q} such that

u = σı(k+ 12 )

on (ak,ak+1),

for k= 0, . . . ,�, and where we use the convention a0 :=−∞ and a�+1 :=+∞.The points ak, for k = 1 . . . ,�, are the limits as ε shrinks to 0 of the points xi

provided by Lemma 3.1 (the number and the positions of which are of courseε dependent), so that the front set D(uε) shrinks as ε tends to 0 to a finite set.In the sequel, we shall refer to step functions with values into � as steep frontchains and we will write

u = u (�, ı , {ak})to determine them unambiguously.

We go back to equation (PGL)ε and consider a family of functions (vε)ε>0

defined on R×R+ which are solutions to the equation (PGL)ε and satisfy theenergy bound (H0). We set

Dε(t)=D(vε(·, t)).The evolution of the front set Dε(t) when ε tends to 0 is the main focus of ourpaper. The following result1 has been proved in [5].

Theorem 3.2 ([5]) There exist constants ρ0 > 0 and α0 > 0, depending onlyon the potential V and on M0, such that if r ≥ α0ε, then

Dε(t+�t)⊂Dε(t)+[−r,r], for every t≥ 0, (3.3)

provided 0≤�t≤ ρ0r2(

) θ+1θ−1 .

As a matter of fact, it follows from this result that the average speed of thefront set at that length-scale should not exceed

cave � r

(�t)max≤ ρ−1

0 r−(ω+1)εω, (3.4)

where

ω= θ + 1

θ − 1. (3.5)

1 which holds also more generally for systems.

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92 Fabrice Bethuel and Didier Smets

Notice that 1 <ω<+∞ and that the upper bound provided by (3.4) decreaseswith θ , that is, the more degenerate the minimizers of V are, the higher thepossible speed allowed by the bound (3.4). In contrast, the speed is at mostexponentially small in the case of non-degenerate potentials (see e.g. [9], [4]and the references therein). One aim of the present paper is to show that theupper bound provided by the estimate (3.4) is in fact optimal2 and actually toderive a precise motion law for the fronts. An important fact, on which ourresults are built, is the following observation3:

Equation (PGL)ε is renormalizable.

This assertion means that, rescaling time in an appropriate way, the evolutionof fronts in the asymptotic limit ε→ 0 is governed by an ordinary differentialequation which does not involve the parameter ε. More precisely, we acceleratetime by the factor ε−ω and consider the new time s = εωt. In the acceleratedtime, we consider the map

vε(x,s)= vε(x,sε−ω), and set Dε(s)=D(vε(·,s)). (3.6)

It follows from Theorem 3.2 that for given r ≥ α0ε,

Dε(s+�s)⊂Dε(s)+[−r,r], for every s≥ 0, (3.7)

provided that 0≤�s≤ ρ0rω+2.Concerning the initial data, we will assume that there exists a steep front

chain v (�0, ı0, {a0k}) such that

(H1)

{v0ε −→ v (�0, ı0, {a0

k}) in L1loc(R),

Dε(0)−→ {a0k}1≤k≤�0 , locally in the sense of the Hausdorff distance,

as ε → 0. Let us emphasize that assumption (H1) is not restrictive, sinceit follows assuming only the energy bound (H0) and passing possibly to asubsequence (see above). In our first result, we will impose the additionalcondition

(Hmin) |ı0(k+ 12 )− ı0(k− 1

2 )| = 1 for 1≤ k≤ �0.

This assumption could be rephrased as a “multiplicity one” condition: itmeans that the jumps consist of exactly one transition between consecutiveminimizers σi and σi±1. To each transition point a0

k we may assign a sign,

2 at least in the scalar case considered here.3 which to our knowledge has not been observed before, even using formal arguments.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 93

denoted by †k ∈ {+,−}, in the following way:

†k =+ if σı0(k+ 12 )= σı0(k− 1

2 )+ 1 and †k =− if σı0(k+ 1

2 )= σı0(k− 1

2 )− 1.

We consider next the system of ordinary differential equations

Skd

dsak = �+k(

ak+1− ak)ω+1 −

�−k(ak− ak−1

)ω+1 ,(S)

for 1≤ k ≤ �0, where we implicitly set a0 =−∞ and a�0+1 =+∞, Sk standsfor the energy of the corresponding stationary front, namely

Sk =∫ σ

ı0(k+ 12 )

σı0(k− 1

2 )

√2V(u)du, (3.8)

and where we have set,4 for k= 1, . . . ,�0,

�±k =

⎧⎪⎪⎨⎪⎪⎩− 2ω

(λı0(k± 1

2 )

)− 1θ−1 Aθ if †k =−†k±1,

− 2ω(λı0(k± 1

2 )

)− 1θ−1 Bθ if †k = †k±1 .

(3.9)

In (3.9), λı0(k+ 12 )

is defined in (A2) and the constants Aθ < 0 and Bθ > 0,depending only on θ , are defined in (A.9) of Appendix A, they are related tothe unique solutions of the two singular boundary value problems⎧⎨⎩ − d2U

dx2+U2θ−1 = 0 on (−1,1),

U(−1)=±∞, U(1)=+∞.

Note in particular that (S) is fully determined by the pair (�0, ı0), and we shalltherefore sometimes refer to it as S�0,ı0 . Our first result is Theorem 3.3.

Theorem 3.3 Assume that the initial data (vε(0))0<ε<1 satisfy conditions (H0),(H1), and (Hmin), and let 0< Smax ≤+∞ denote the maximal time of existencefor the system S�0,ı0 with initial data ak(0)= a0

k . Then, for 0 < s < Smax,

vε(s)−→ v (�0, ı0, {ak(s)}) (3.10)

in L∞loc(R \∪�0k=1{ak(s)}), as ε→ 0. In particular,

Dε(s)−→∪�0k=1{ak(s)} (3.11)

locally in the sense of the Hausdorff distance, as ε→ 0.

4 In view of our definition of a0 and a�0+1 the quantities �−0 and �+�0

need not be well defined.

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94 Fabrice Bethuel and Didier Smets

We consider now the more general situation where (Hmin) is not verified,and for 1 ≤ k ≤ �0 we denote by m0

k the algebraic multiplicity of a0k , that is,

we set

m0k = ı(k+ 1

2)− ı(k− 1

2). (3.12)

The case m0k = 0 corresponds to ghost fronts, whereas |m0

k | ≥ 2 correspondsto multiple fronts. The total number of fronts that will eventually emerge fromsuch initial data is given by

�1 =�0∑

k=1

|m0k |,

and their ordering is obtained by splitting multiple fronts according to the orderin �. More precisely, we define the function ı1 by⎧⎪⎨⎪⎩

ı1(12 )= ı0(

12 ),

ı1(M0k + p+ 1

2 )= ı0(k+ 12 )+ p, for p= 0, . . . , |m0

k |− 1 if m0k > 0,

ı1(M0k + p+ 1

2 )= ı0(k+ 12 )− p, for p= 0, . . . , |m0

k |− 1 if m0k < 0,

(3.13)

where k= 1, . . . ,�0 and M0k :=∑k−1

p=1 |m0p|. We say that (�1, ı1) is the splitting of

(�0, ı0).

Definition 3.4 A splitting solution of (S) with initial data (�0, ı0, {a0k}) on the

interval [0,S) is a solution a≡ (a1, . . . ,a�1) : (0,S)→R�1 of S�1,ı1 such that

lims→0+

ak(s)= a0j for k=M0

j , . . . ,M0j +|m0

j |− 1,

for any j= 1, . . . ,�0, where (�1, ı1) is the splitting of (�0, ı0).

We are now in a position to complete Theorem 3.3 by relaxingassumption (Hmin).

Theorem 3.5 Assume that the initial data (v0ε )0<ε<1 satisfy conditions (H0)

and (H1). Then there exist a subsequence εn → 0, and a splitting solutionof (S) with initial data (�0, ı0, {a0

k}), defined on its maximal time of existence[0,Smax), and such that for any 0 < s < Smax

vεn(s)−→ v (�1, ı1, {ak(s)}) (3.14)

in L∞loc(R \∪�1k=1{ak(s)}), as n→+∞. In particular,

Dεn(s)−→∪�1j=1{ak(s)} (3.15)

locally in the sense of the Hausdorff distance, as n→+∞.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 95

Remark 3.6 Local existence of splitting solutions can be established indifferent ways (including in particular using Theorem 3.5 !). To our knowledge,uniqueness is not known, unless of course |m0

k | ≤ 1 for all k, this is the mainreason why convergence is only obtained for a subsequence in Theorem 3.5whereas it was for the full sequence in Theorem 3.3.

So far, our results are constrained by the maximal time of existence Smax ofthe differential equation (S), which is related to the occurrence of collisions.To pursue the analysis past collisions, we first briefly discuss some propertiesof the system of equations (S), we refer to Appendix B for more details. Thesystem (S) describes nearest neighbor interactions with an interaction law ofthe form ±d−(ω+1), d standing for the distance between fronts. The sign ofthe interactions is crucial, since the system may contain both repulsive forcesleading to spreading and attractive forces leading to collisions, yielding themaximal time of existence Smax. In order to take signs into account, we set

εk+ 12= sign(�k+ 1

2)=− †k †k+1, for k= 0, . . . ,�0− 1. (3.16)

The case εk+ 12= −1 corresponds to repulsive forces between ak and ak+1,

whereas the case εk+ 12= +1 corresponds to attractive forces between ak

and ak+1, leading to collisions. As a matter of fact, in this last case ak+1

corresponds to the anti-front of ak. In order to describe the magnitude ofthe forces, we introduce the subsets J± of {1, . . . ,�0} defined by J± = {k ∈{1, . . . ,�0− 1}, such that εk+ 1

2=∓1} and the quantities{

da(s)= inf{|ak(s)− ak+1(s)|, for k ∈ 1, . . . ,�0− 1},d±a (s)= inf{|ak(s)− ak+1(s)|, for k ∈ J±}. (3.17)

Proposition 3.7 There are positive constants S1, S2, S3 and S4 depending onlyon the coefficients of the equation (S), such that for any time s ∈ [0,Smax) wehave ⎧⎨⎩d

+a (s)≥

(S1s+S2d

+a (0)

ω+2) 1ω+2 ,

d−a (s)≤(S3d

−a (0)

ω+2−S4t) 1ω+2 .

(3.18)

If for every k= 1, . . . ,�0 we have εk+ 12=−1, then Smax =+∞. Otherwise, we

have the estimate

Smax ≤ S3

S4

(d−a (0)

)ω+2 ≡K0(d−a (0)

)ω+2. (3.19)

This result shows that the maximal time of existence for solutions to (S)is related to the value of d−a (0), the minimal distance between fronts and

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96 Fabrice Bethuel and Didier Smets

anti-fronts at time 0. By the semi-group property, the same can be said aboutd−a (s), that is

Smax− s � d−a (s)ω+2.

On the other hand, in view of (S), d−a (s) provides an upper bound for the speedsak(s) in case of collision, that is

| d

dsak(s)|� d−a (s)−(ω+1).

It follows that∫ Smax

0| d

dsak(s)|ds �

∫ Smax

0(Smax− s)−

ω+1ω+2 ds <+∞

and therefore that the trajectories are absolutely continuous up to the collisiontime. Also, since d+a remains bounded from below by a positive constant, eachfront can only enter into collision with its anti-front (but there could be multiplecopies of both). From a heuristic point of view, it is therefore rather simpleto extend solutions past the collision time: it suffices to remove the collidingpairs from the collection of points, so that the total number of points has beendecreased by an even number. More precisely, we have the following.

Corollary 3.8 Let �1, ı1, a≡ (a1, . . . ,a�1) and Smax be as in Theorem 3.5. Then,there exists �2 ∈N such that �1−�2 ∈ 2N∗, and there exist �2 points b1 < · · ·<b�2 such that for all k= 1, . . . ,�1

lims→S−max

ak(s)= bj(k) for some j(k) ∈ {1, . . . ,�2}.

Moreover, if we set ı2(12 )= ı1(

12 ) and

ı2(q+ 12 )= ı1(k(q)+ 1

2 ), where k(q)=max{k ∈ {1, . . . ,�1} s.t. j(k)= q},for q= 1, . . . ,�2, then

ı2(q+ 12 )− ı2(q− 1

2 ) ∈ {+1,−1,0}for all q= 1, . . . ,�2.

We stress that Corollary 3.8 is obtained from Theorem 3.5 using onlyproperties of the system of ODEs (S), in particular Proposition 3.7.

We are now in position to state our last result.

Theorem 3.9 Under the assumptions of Theorem 3.5, we have as n→+∞,

vεn(Smax)−→ v (�2, ı2, {bk}) in L∞loc(R \∪�2k=1{bk}), (3.20)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 97

where �2, ı2 and b1 < · · · < b�2 are given by Corollary 3.8. In particular thesequence (vεn(Smax))n∈N, considered as initial data, satisfies the assumptions(H0) and (H1) with �0 := �2 and {a0

k} := {b0k}.

We may therefore apply Theorem 3.5 to the sequence of initial data(vεn(Smax))n∈N, and therefore, using the semi-group property of (3.1), extendthe analysis past Smax. Notice that since the multiplicities given by ı2 are equalto either ±1 or 0, no further subsequences are needed to pass through thecollision times. Finally, since the total number of fronts is decreased at least by2 at each collision time, the latter are finitely many.

Some comments on the results Motion of fronts for one-dimensional scalarreaction-diffusion equations has already quite a long history. Until recently,most efforts have been devoted to the case where the potential possessesonly two wells with non-vanishing second derivative: such potentials are oftenreferred to as Allen–Cahn potentials. Under suitable preparedness assumptionson the initial datum, the precise motion law for the fronts has been derivedby Carr and Pego in their seminal work [9] (see also Fusco and Hale [10]).They showed that the front points are moved, up to the first collision time,according to a first-order differential equation of nearest neighbor interactiontype, with interaction terms proportional to exp(−ε−1(aε

k+1(t)−aεk(t))). These

results present substantial differences from the results in the present paper, inparticular we wish to emphasize the following points:

• only attractive forces leading eventually to the annihilation of fronts withanti-fronts forces are present;

• the equation is not renormalizable. Indeed, the various forcesexp(−ε−1(aε

k+1(t)−aεk(t))) for different values of k may be of very different

orders of magnitude, and hence not commensurable.

Besides this, the essence of their method is quite different: it relies on acareful study of the linearized problem around the stationary front, in particularfrom the spectral point of view. This type of approach is also sometimestermed the geometric approach (see e.g. [8]). At least two other methodshave been applied successfully to the Allen–Cahn equation. First, the methodof subsolutions and supersolutions turns out to be extremely powerful andallowed us to handle larger classes of initial data and also to extend the analysispast collisions: this is for instance achieved by Chen in [8]. Another directionis given by the global energy approach initiated by Bronsard and Kohn [7]. Werefer to [4] for more references on these methods.

Several ideas and concepts presented here are influenced by our earlier workon the motion of vortices in the two-dimensional parabolic Ginzburg–Landau

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98 Fabrice Bethuel and Didier Smets

equation [2, 3]. As a matter of fact, this equation yields another remarkableexample of renormalizable slow motion, as proved by Lin or Jerrard andSoner ([13, 11]). Our interest in the questions studied in this paper wascertainly driven by the possibility of finding an analogous situation in onespace dimension.

This paper belongs to a series of papers we have written on the slow motionphenomenon for reaction-diffusion equation of gradient type with multiplewells (see [4, 5, 6]). Common to all of these papers is a general approachbased on the following ingredients.

• A localized version of the energy identity (see Subsection 3.1.3). Fronts arethen handled as concentration points of the energy, so that the evolutionof local energies yields also the motion of fronts. Besides dissipation, thislocalized energy identity contains a flux term, involving the discrepancyfunction, which has a simple interpretation for stationary solutions. Usingtest functions which are affine near the fronts, the flux term does not see thecore of the front, only its tail.

• Parabolic estimates away from the fronts.• Handling the time derivative as a perturbation of the one-dimensional

elliptic equations, hence allowing elementary tools as Gronwall’s identities.

Parallel to this paper, we have also revisited the scalar non-degenerate casein [6], considering in particular the case were there are more than twowells, leading as mentioned to repulsive forces which are not present in theAllen–Cahn case. Several tools are shared by the two papers, for instancewe rely on related definitions and properties of regularized fronts, and theproperties of the ordinary differential equations are quite similar. From atechnical point of view, differences appear at the level of the magnitudes ofenergies as well as of the parameter δ involved in the definition of regularfronts, and more crucially on the nature of the parabolic estimates of thefront sets. Whereas in [6] we rely essentially on linear estimates, in thedegenerate case considered here our estimates are truly non-linear, obtainedmainly through extensive use of the comparison principle.

Finally, it is presumably worth mentioning that the situation in higherdimension is very different: the dynamics is dominated by mean-curvatureeffects. The phenomena considered in the present paper are therefore of lowerorder, and do not appear in the limiting equations.

Among the problems left open in our paper, we would like to emphasizeagain the question of uniqueness of splitting solutions for (S), as well as thepossibility of interpreting our convergence results in terms of a Gamma-limit

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 99

involving a renormalized energy (see e.g [15] for related results on theGinzburg–Landau equation).

3.1.2 Regularized Fronts

The notion of regularized fronts is central in our description of the dynamicsof equation (PGL)ε. It is intended to describe in a quantitative way chainsof stationary solutions which are well-separated and suitably glued together.It also allows us to pass from front sets to front points, a notion which ismore accurate and therefore requires improved estimates. Recall first that fori ∈ {1, . . . ,q− 1}, there exists a unique (up to translations) solution ζ+i to thestationary equation with ε = 1,

− vxx+V ′(v)= 0 on R, (3.21)

with, as conditions at infinity, v(−∞) = σi and v(+∞) = σi+1. Set, fori ∈ {1, . . . ,q− 1}, ζ−i (·) ≡ ζi(−·), so that ζ−i is the unique (up to translations)solution to (3.21) such that v(+∞)= σi and v(−∞)= σi+1. A remarkable yetelementary fact, related to the scalar nature of the equation, is that there are noother non-trivial finite energy solutions to equation (3.21) than the solutions ζ±iand their translates: in particular there are no solutions connecting minimizerswhich are not nearest neighbors. For i = 1, . . . ,q − 1, we fix a point zi inthe interval (σi,σi+1) where the potential V restricted to [σi,σi+1] achievesits maximum and we set Z = {z1, . . . ,zq−1}. Again, since we consider onlythe one-dimensional scalar case, any solution ζi takes once and only once thevalue zi.

We next describe a local notion of well-preparedness.5 For an arbitrary r>0,we denote by Ir the interval [−r,r].Definition 3.10 Let L > 0 and δ > 0. We say that a map u verifying (H0)

satisfies the preparedness assumption WPLε (δ) if the following two conditions

are fulfilled.

• (WPIL

ε (δ))

We have

D(u)∩ I2L ⊂ IL (3.22)

and there exists a collection of points {ak}k∈J in IL, with J = {1, . . . ,�}, suchthat

D(u)∩ I2L ⊂ ∪k∈J

Ik, where Ik = [ak− δ,ak+ δ]. (3.23)

5 By local, we mean with respect to the interval [−L,L]. In contrast the related notion introducedin [6] is global on the whole of R.

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100 Fabrice Bethuel and Didier Smets

For k ∈ J, there exists a number i(k) ∈ {1, . . . ,q− 1} such that u(ak) = zi(k)

and a symbol †k ∈ {+,−} such that∥∥∥∥u(·)− ζ†ki(k)

( ·− ak

ε

)∥∥∥∥C1ε (Ik)

≤ exp

(−δ

ε

), (3.24)

where ‖u‖C1ε (Ik)

= ‖u‖L∞(Ik)+ ε‖u′‖L∞(Ik).

• (WPOL

ε (δ))

Set L = I2L \�∪

k=1Ik. We have the energy estimate∫

L

eε (u(x))dx≤ CwM0

(εδ

)ω. (3.25)

In the above definition Cw > 0 denotes a constant whose exact value is fixedonce for all by Proposition 3.22, and which depends only on V . ConditionWPIL

ε (δ) corresponds to an inner matching of the map with stationary fronts,it is only really meaningful if δ� ε. In the sequel we always assume that

L

2≥ δ≥ α1ε, (3.26)

where α1 is larger than the α0 of Theorem 3.2 and also sufficiently large so thatif WPIL

ε (δ) holds then the points ak and the indices i(k) and †k are uniquelyand therefore unambigously determined and the intervals Ik are disjoints. Inparticular, the quantity dε,L

min(s), defined by

dε,Lmin(s) :=min

{aε

k+1(s)− aεk(s), k= 1, . . . ,�(s)− 1

}if �(s) ≥ 2, and dε,L

min(s) = 2L otherwise, satisfies dε,Lmin(s) ≥ 2δ. Condition

WPOLε (δ) is in some weak sense an outer matching: it is crucial for some

of our energy estimates and its form is motivated by energy decay estimatesfor stationary solutions. Note that condition WPIL

ε(δ) makes sense on itsown, whereas condition WPOL

ε(δ) only makes sense if condition WPILε(δ) is

fulfilled. Note also that the larger δ is, the stronger condition WPILε(δ) is. The

same is not obviously true for condition WPOLε(δ), since the set of integration

L increases with δ. As a matter of fact, the constant Cw in (3.25) is chosensufficiently big6 so that WPOL

ε(δ) also becomes stronger when δ is larger. Wenext specify Definition 3.10 for the maps x �→ vε(x,s).

Definition 3.11 For s ≥ 0, we say that the assumption WPLε(δ,s) (resp.

WPILε(δ,s)) holds if the map x �→ vε(x,s) satisfies WPL

ε(δ) (resp. WPILε(δ)).

6 In view of WPILε(δ), how big it needs to be is indeed related to energy decay estimates for the

fronts ζi.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 101

When assumption WPILε(δ,s) holds, then all symbols will be indexed

according to s. In particular, we write7 �(s)= �, J(s)= J, and aεk(s)= ak. The

points aεk(s) for k∈ J(s), are now termed the front points. Whereas in [6] we are

able, due to parabolic regularization, to establish under suitable conditions thatWPL

ε(δ,s) is fulfilled for a length of the same order as the minimal distancebetween the front points, this is not the case in the present situation. Moreprecisely, two orders of magnitude for δ will be considered, namely

δε

log=1

ρwε

∣∣∣log(

4M20ε

L

)∣∣∣ and δε

loglog=ω

ρwε log

(1

ρw

∣∣∣log(

4M20ε

L

)∣∣∣) .

(3.27)In (3.27), the constant ρw (given by Lemma 3.25) depends only on V . The mainproperty for our purposes is that δ

ε

loglog/ε and δε

log/δε

loglog both tend to +∞ asε/L tends to 0.

In many places, it is useful to rely on a slightly stronger version of theconfinement condition (3.22), which we assume to hold on some interval oftime. More precisely, for positive L,S we consider the condition

(CL,S) Dε(s)∩ I4L ⊂ IL, ∀ 0≤ s≤ S.

For given L0 > 0 and S > 0, it follows easily from assumption (H1) andTheorem 3.2 that there exists L ≥ L0 for which the first condition in (CL,S)

is satisfied. Under condition (CL,S), the estimate

Eε(vε(s), I3L \ I 32 L)≤ Ce

( εL

)ω, ∀ s ∈ [εωL2,S], (3.28)

where Ce > 0 depends only on V , follows from the following regularizingeffect, which was obtained in [5].

Proposition 3.12 ([5]) Let vε be a solution to (PGL)ε, let x0 ∈ R, r > 0 and0≤ s0 < S be such that

vε(y,s) ∈ B(σi,μ0) for all (y,s) ∈ [x0− r,x0+ r]× [s0,S], (3.29)

for some i ∈ {1, . . . ,q}. Then we have for s0 < s≤ S

ε−ω

∫ x0+3r/4

x0−3r/4eε(vε(x,s))dx≤ 1

10C

⎛⎝1+(

εωr2

s− s0

) θθ−1

⎞⎠(1

r

(3.30)

7 In principle and at this stage, all those symbols depend also upon ε. Since eventually � and Jwill be ε-independent, at least for ε sufficiently small, we do not explicitly index them with ε.

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102 Fabrice Bethuel and Didier Smets

as well as

|vε(y,s)−σi| ≤ 1

10Cε

1θ−1

((1

r

) 1θ−1 +

(εω

s− s0

) 12(θ−1)

), (3.31)

for y ∈ [x0− 3r/4,x0+ 3r/4], where the constant C > 0 depends only on V.

Our first ingredient is the following.

Proposition 3.13 There exists α1 > 0, depending only on M0 and V , such that ifL≥ α1ε and if (CL,S) holds, then each subinterval of [0,S] of length εω+2

(L/ε

)contains at least one time s for which WPL

ε(δε

log,s) holds.

The idea behind Proposition 3.13 is that, (PGL)ε being a gradient flow, on asufficiently large interval of time one may find some time where the dissipationof energy is small. Using elliptic tools, and viewing the time derivative as aforcing term, one may then establish property WPL

ε(δε

log,s) (see Sections 3.2and 3.3).

The next result expresses the fact that the equation preserves to some extentthe well-preparedness assumption.

Proposition 3.14 Assume that (CL,S) holds, that εωL2 ≤ s0 ≤ S is such thatWPL

ε(δε

log,s0) holds, and assume moreover that

dε,Lmin(s0)≥ 16

( L

ρ0ε

) 1ω+2

ε. (3.32)

Then WPLε(δ

ε

loglog,s) holds for all times s0+ ε2+ω ≤ s≤ T ε0 (s0), where

T ε0 (s0)=max

{s ∈ [s0+ ε2+ω,S] s.t.

dε,Lmin(s

′)≥ 8( L

ρ0ε

) 1ω+2

ε ∀s′ ∈ [s0+ εω+2,s]}

.

For such an s we have J(s)= J(s0) and for any k ∈ J(s0) we have σi(k± 12 )(s)=

σi(k± 12 )(s0) and †k(s)= †k(s0).

Given a family of solution (vε)0<ε<1, we introduce the additional condition

d∗min(s0)≡ liminfε→0

dε,Lmin(s0) > 0, (3.33)

which makes sense if WPLε(α1ε,s0) holds and expresses the fact that the

fronts stay uniformly well-separated. The first step in our proof, which isstated in Proposition 3.19, is to establish the conclusion of Theorem 3.3 under

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 103

these stronger assumptions on the initial datum. From the inclusion (3.7) andProposition 3.14 we will obtain:

Corollary 3.15 Assume also that (CL,S) holds, let s0 ∈ [0,S] and assume thatWPL

ε (α1ε,s0) holds for all ε sufficiently small and that (3.33) is satisfied.Then, for ε sufficiently small,

WPLε (δ

ε

loglog,s) and dε,Lmin(s)≥

1

2d∗min(s0) (3.34)

are satisfied for any

s ∈ Iε(s0)≡[s0+ 2L2εω,s0+ρ0

(d∗min(s0)

8

)ω+2 ]∩ [0,S],

as well as the identities J(s) = J(s0), σi(k± 12 )(s) = σi(k± 1

2 )(s0) and †k(s) =

†k(s0), for any k ∈ J(s0).

Hence, the collection of front points {aεk(s)}k∈J is well defined, and the

approximating regularized fronts ζ†ki(k) do not depend on s (otherwise than

through their position), on the full time interval Iε(s0).

3.1.3 Paving the Way to the Motion Law

As in [4], we use extensively the localized version of (3.1), a tool which turnsout to be perfectly adapted to tracking the evolution of fronts. Let χ be anarbitrary smooth test function with compact support. Set, for s≥ 0,

Iε(s,χ)=∫R

eε (vε(x,s))χ(x)dx. (3.35)

In integrated form the localized version of the energy identity is written as

Iε(s2,χ)−Iε(s1,χ)+∫ s2

s1

∫R

ε1+ωχ(x)|∂svε(x,s)|2 dxds

= ε−ω

∫ s2

s1

FS(s,χ ,vε)ds, (3.36)

where the term FS is given by

FS(s,χ ,vε)=∫R×{s}

([εvε

2

2− V(vε)

ε

]χ(x)

)dx≡

∫R×{s}

ξε(vε(·,s))χdx.

(3.37)The last integral on the left-hand side of Identity (3.36) stands for localdissipation, whereas the right-hand side is a flux. The quantity ξε is defined

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104 Fabrice Bethuel and Didier Smets

for a scalar function u by

ξε(u)≡ εu2

2− V(u)

ε, (3.38)

and is referred to as the discrepancy term. It is constant for solutions to thestationary equation−uxx+ε−2V ′(u)= 0 on some given interval I and vanishesfor finite energy solutions on I =R. Notice that |ξε(u)| ≤ eε(u). We set for twogiven times s2 ≥ s1 ≥ 0 and L≥ 0

dissipLε [s1,s2] = ε

∫I 5

3 L×[s1ε

−ω ,s2ε−ω]|∂vε∂t|2dxdt= ε1+ω

∫I 5

3 L×[s1,s2]

|∂vε∂s|2 dxds.

(3.39)Identity (3.36) then yields the estimate, if we assume that suppχ ⊂ I 5

3 L,∣∣∣∣Iε(s2,χ)−Iε(s1,χ)− ε−ω

∫ s2

s1

FS(s,χ ,vε)ds

∣∣∣∣≤ dissipLε [s1,s2]‖χ‖L∞(R).

(3.40)We will show that under suitable assumptions, the right-hand side of (3.40)is small (see Step 3 in the proof of Proposition 3.19), so that the term

ε−ω

∫ s2

s1

FS(s,χ ,vε)ds provides a good approximation of Iε(s2,χ)−Iε(s1,χ).

On the other hand, it follows from the properties of regularized maps provedin Section 3.2.2 (see Proposition 3.22) that if WPL

ε(δε

loglog,s) holds then∣∣∣∣∣Iε(s,χ)−∑k∈J

χ(aεk(s))Si(k)

∣∣∣∣∣≤ CM0

(( ε

δε

loglog

)ω‖χ‖∞+ ε‖χ ′‖∞)

, (3.41)

where Si(k) stands for the energy of the corresponding stationary front. Set

Fε(s1,s2,χ)≡ ε−ω

∫ s2

s1

FS(s,χ ,vε)ds≡∫ s2

s1

ε−ωξε(vε(·,s))χ(·)ds.

Combining (3.40) and (3.41) shows that, if WPLε(δ

ε

loglog,s) holds for any s ∈(s1,s2), then we have

|∑k∈J

[χ(aε

k(s2))−χ(aεk(s1))

]Si(k)−Fε(s1,s2,χ)|

≤ CM0

((log |log

ε

L|)−ω‖χ‖∞+ ε‖χ ′‖∞

)+ dissipL

ε [s1,s2]‖χ‖∞.

(3.42)If the test function χ is chosen to be affine near a given front point ak0 and

zero near the other front points in the collection, then the first term on theleft-hand side yields a measure of the motion of ak0 between times s1 and s2,whereas the second, namely Fε(s1,s2,χ), is hence a good approximation of

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 105

the measure of this motion, provided we are able to estimate the dissipationdissipL

ε [s1,s2]. Our previous discussion suggests that

aεk0(s2)− aε

k0(s1)� 1

χ ′(aεk0)Si(k0)

Fε(s1,s2,χ).

It turns out that the computation of Fε(s1,s2,χ) can be performed withsatisfactory accuracy if the test function χ is affine (and hence has vanishingsecond derivatives) close to the front set, this is the object of the nextsubsections.

3.1.4 A First Compactness Result

A first step in deriving the motion law for the fronts is to obtain roughbounds from above for both dissipL

ε [s1,s2] and Fε(s1,s2,χ). To obtain these,and under the assumptions of Corollary 3.15, notice that if χ vanishes on theset {aε

k(s0)}k∈J + [−d∗min(s0)/4,d∗min(s0)/4], then from the inequality |ξε(u)| ≤eε(u), from Corollary 3.15 and from (3.30) of Proposition 3.12, we derive thatfor s1 ≤ s2 in Iε(s0),

|Fε(s1,s2,χ)| ≤ Cd∗min(s0)−ω‖χ‖L∞(R)(s2− s1). (3.43)

Going back to (3.36), and choosing the test function χ so that χ ≡ 1 on I 53 L

with compact support on I2L, Estimate (3.43) combined with (3.41) yields inturn a first rough upper bound on the dissipation dissipL

ε [s1,s2]. Combiningthese estimates we obtain the following.

Proposition 3.16 Under the assumptions of Corollary 3.15, for s1≤ s2 ∈ Iε(s0)

we have

|aεk(s1)− aε

k(s2)| ≤ C(

d∗min(s0)−(ω+1)

(s2− s1)

+M0((log | log

ε

L|)−ωd∗min(s0)+ ε

)). (3.44)

As an easy consequence, we deduce the following compactness property,setting

I∗(s0)=(

s0,s0+ρ0(d∗min(s0)

8

)ω+2)∩ (0,S).

Corollary 3.17 Under the assumptions of Corollary 3.15, there exists asubsequence (εn)n∈N converging to 0 such that for any k ∈ J the functionaεn

k (·) converges uniformly on any compact interval of I∗(s0) to a Lipschitzcontinuous function ak(·).

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106 Fabrice Bethuel and Didier Smets

3.1.5 Refined Estimates Off the Front Set and the Motion Law

In order to derive the precise motion law, we have to provide an accurateasymptotic value for the discrepancy term off the front set. In other words, fora given index k ∈ J we need to provide a uniform limit of the function ε−ωξε

near the points

k+ 12(s)≡ aε

k(s)+ aεk+1(s)

2and aε

k− 12(s)≡ aε

k−1(s)+ aεk(s)

2.

We notice first that vε takes values close to σi(k+ 12 )

near aε

k+ 12(s). In view of

Estimate (3.30), we introduce the functions

wε(·,s)=wkε(·,s)= vε−σi(k+ 1

2 )and Wε =Wk

ε ≡ ε− 1

θ−1 wkε

= ε− 1

θ−1

(vε−σi(k+ 1

2 )

). (3.45)

As a consequence of inequality (3.31) and Corollary 3.15 we have a uniformbound.

Lemma 3.18 Under the assumptions of Corollary 3.15, we have

|Wε(x,s)| ≤ C(d(x,s)

)− 1θ−1 (3.46)

for any x ∈ (ak(s)+δε

loglog,ak+1(s)−δε

loglog) and any s ∈ Iε(s0), where we haveset d(x,s) := dist(x, {aε

k(s),aεk+1(s)}) and where C > 0 depends only on V and

M0. Moreover, we also have⎧⎪⎪⎨⎪⎪⎩−sign(†k)Wε

(aε

k(s)+ δε

loglog

)≥ 1

C

(δε

loglog

)− 1θ−1

,

sign(†k+1)Wε

(aε

k+1(s)− δε

loglog

)≥ 1

C

(δε

loglog

)− 1θ−1

.

(3.47)

We describe next on a formal level how to obtain the desired asymptoticsfor ε−ωξε, as ε→ 0, near the point ak+ 1

2(s). Going back to the limiting points

{ak(s)}k∈J defined in Proposition 3.16, we consider the subset of R×R+

Vk(s0)=⋃

s∈I∗(s0)

(ak(s),ak+1(s))×{s}. (3.48)

It follows from the uniform bounds established in Lemma 3.18, that, passingpossibly to a further subsequence, we may assume that

Wεn ⇀W∗ in Lploc(Vk(s0)), for any 1≤ p <∞.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 107

On the other hand, thanks to estimate (3.46), for a given point (x,s) ∈ Vk(s0)

we expand (PGL)ε near (x,s) as

εω∂Wε

∂s− ∂2Wε

∂x2+ 2θλi(k+ 1

2 )W2θ−1

ε =O(ε1

θ−1 ). (3.49)

Passing to the limit εn → 0, we expect that for every s ∈ I∗(s0), W∗ solves⎧⎨⎩−∂2W∗∂x2

(s, ·)+ 2θλi(k+ 12 )W2θ−1

∗ (s, ·)= 0 on (ak(s),ak+1(s)),

W∗(ak(s))=−sign(†k)∞ and W∗(ak+1(s))= sign(†k)∞,(3.50)

the boundary conditions being a consequence of the asymptotics (3.47). It turnsout, in view of Lemma 3.59 of Appendix A, that the boundary value problem(3.50) has a unique solution. By scaling, and setting rk(s)= 1

2 (ak+1(s)−ak(s)),we obtain⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

W∗(x,s)=±rk(s)− 1

θ−1

(λi(k+ 1

2 )

)− 12(θ−1) ∨

u+(

x− ak+ 12

rk(s)

), if †k =−†k+1,

W∗(x,s)=±rk(s)− 1

θ−1

(λi(k+ 1

2 )

)− 12(θ−1) �

u

(x− ak+ 1

2

rk(s)

), if †k = †k+1,

where∨u+

(resp.�u) are the unique solutions to the problems{−Uxx+ 2θ U2θ−1 = 0 on (−1,+1),

U(−1)=+∞ (resp. U(−1)=−∞) and U(+1)=+∞.(3.51)

Still on a formal level, we deduce therefore the corresponding values of thediscrepancy⎧⎪⎪⎨⎪⎪⎩

ε−ωξε(vε)� ξ(W∗)=−λ− 1

θ−1

i(k+ 12 )

rk(s)−(ω+1)Aθ if †k =−†k+1,

ε−ωξε(vε)� ξ(W∗)= λ− 1

θ−1

i(k+ 12 )

rk(s)−(ω+1)Bθ if †k = †k+1,

(3.52)

where the numbers Aθ and Bθ are positive, depend only on θ , and correspond

to the absolute value of the discrepancy of∨u+

and�u respectively. Notice that

the signs in (3.52) are different, the first case yields attractive forces whereasthe second yields repulsive ones. Inserting this relation in (3.42) and arguingas for (3.44), we will derive the motion law.

The previous formal discussion can be put on a sound mathematical ground,relying on comparison principles and the construction of appropriate upper andlower solutions (see Section 3.5). This leads to the central result of this paper.

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108 Fabrice Bethuel and Didier Smets

Proposition 3.19 Assume that conditions (H0) and (H1) are fulfilled. Let 0 <

S < Smax be given and set

L0 := 3max

{|a0

k |, 1≤ k≤ �0 ,

(S

ρ0

) 1ω+2

}.

Assume that WPIL0ε (α1ε,0) holds as well as (3.33) at time s= 0. Then J(s)=

{1, . . . ,�0} and the functions aεk(·) are well defined and converge uniformly on

any compact interval of (0,S) to the solution ak(·) of (S) supplemented withthe initial condition ak(0)= a0

k .

Notice that the combination of assumptions WPILε(α1ε,0), (H1) and (3.33)

at s= 0 implies the multiplicity-one condition (Hmin). Whereas the conclusionof Proposition 3.19 is similar to the one of Theorem 3.3, the assumptionsof Proposition 3.19 are more restrictive. Indeed, on one hand we assumethe well-preparedness condition WPIL

ε, and on the other hand we impose(3.33) which is far more constraining than (Hmin): it excludes in particularthe possibility of having small pairs of fronts and anti-fronts. Our next effortsare hence devoted to handling this type of situation: Proposition 3.19, throughrescaling arguments, will nevertheless be the main building block for that task.

In order to prove Theorems 3.3, 3.5 and 3.9 we need to relax the assumptionson the initial data, in particular we need to analyze the behavior of data withsmall pairs of fronts and anti-fronts, and show that they are going to annihilateon a short interval of time. For that purpose we will consider the followingsituation, corresponding to confinement of the front set at initial time. Assumethat for a collection of points {bε

q}q∈J0 in R we have

Dε(0)∩ I5L ⊂ ∪q∈J0

[bεq−r,bε

q+r] ⊂ Iκ0L and bεp−bε

q ≥ 3R for p = q∈ J0,

(3.53)for some κ0 ≤ 1

2 and α1ε≤ r≤ R/2≤ L/4. It follows from (3.7) that if 0≤ s≤ρ0(R− r)ω+2 then

Dε(s)∩ I4L ⊂ ∪k∈J0

(bεk −R,bε

k +R)⊂ I2κ0L, where the union is disjoint.

Consider next 0 ≤ s ≤ ρ0(R − r)ω+2 such that WPLε(α1ε,s) holds, so

that the front points {aεk(s)}k∈J(s) are well defined. For q ∈ J0, consider

Jq(s) = {k ∈ J(s),aεk(s) ∈ (bε

q − R,bεq + R)}, set �q = "Jq, and write Jq(s) =

{kq,kq+1, . . . ,kq+�q−1}, where k1 = 1, and kq = �1 + ·· · + �q−1 + 1, for q ≥ 2.Our next result shows that, after a small time, only the repulsive forces surviveat the scale given by r, provided the different lengths are sufficiently distinct.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 109

Proposition 3.20 There exist positive constants α∗ and ρ∗, depending only onV and M0, such that if (3.53) holds and

κ−10 ≥ α∗, r ≥ α∗ε

(L

ε

) 2ω+2

, R≥ α∗r, (3.54)

then at time

sr = ρ∗rω+2

condition WPLε(α1ε,sr) holds and, for any q ∈ J0 and any k,k′ ∈ Jq(sr) we

have †k(sr)= †′k(sr), or equivalently for any k ∈ Jq(sr) \ {kq(sr)+ �q(sr)− 1},we have

εk+ 12(sr)= †k(sr) †k+1 (sr)=+1. (3.55)

Moreover, we have

dε,Lmin(sr)≥ r, (3.56)

and if "Jq(sr)≤ 1 for every q ∈ J0, then we actually have dε,Lmin(sr)≥ R.

The proofs of Theorems 3.3, 3.5 and 3.9 are then deduced from Proposi-tions 3.19 and 3.20.

The paper is organized as follows. We describe in Section 3.2 someproperties of stationary fronts, as well as for solutions to some perturbations ofthe stationary equations. In Section 3.3 we describe several properties relatedto the well-preparedness assumption WPL

ε , in particular the quantizationof the energy, how it relates to dissipation, and its numerous implicationsfor the dynamics. We provide in particular the proofs to Proposition 3.13,Proposition 3.14 and Corollary 3.15. In Section 3.4, we prove the compactnessresults stated in Proposition 3.16 and Corollary 3.17. Section 3.5, provides anexpansion of the discrepancy term off the front set, from a technical point ofview it is the place where the analysis differs most from the non-degeneratecase. Based on this analysis, we show in Section 3.6 how the motion lawfollows from prepared data establishing the proof of Proposition 3.19. InSection 3.7 we analyze the clearing-out of small pairs of front–anti-front and,more generally, we present the proof of Proposition 3.20. Finally, in Section 3.8we present the proofs of the main theorems, namely Theorems 3.3, 3.5 and 3.9.Several results concerning the first- or second-order differential equationsinvolved in the analysis of this paper are given in separate appendices, inparticular the proof of Proposition 3.7.

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110 Fabrice Bethuel and Didier Smets

3.2 Remarks on Stationary Solutions

3.2.1 Stationary Solutions on R with Vanishing Discrepancy

Stationary solutions are described using the method of separation of variables.For u, the solution to (3.21), we multiply (3.21) by u and verify that ξ isconstant. We restrict ourselves to solutions with vanishing discrepancy

ξ = 1

2u2−V(u)= 0, (3.57)

and solve Equation (3.57) by separation of variables. Let γi be defined on(σi,σi+1) by

γi(u)=∫ u

zi

ds√2V(s)

, for u ∈ (σi,σi+1), (3.58)

where we recall that zi is a fixed maximum point of V in the interval (σi,σi+1).The map γi is one-to-one from (σi,σi+1) to R, so we may define its inversemap ζ+i : R→ (σi,σi+1) by

ζ+i (x)= γ−1i (x) as well as ζ−i (x)= γ−1

i (−x) for x ∈R. (3.59)

In view of the definition (3.59), we have ζ±i (0) = zi, ζ+i′(0) = √

2V(zi) >

0, whereas a change of variable shows that ζi has finite energy given by theformula (3.8). We verify that ζ+i

( ·ε

)and ζ−i

( ·ε

)solve (3.57) and hence (3.21).

The next elementary result then directly follows from uniqueness in ODEs.

Lemma 3.21 Let u be a solution to (3.21) such that (3.57) holds, and such thatu(x0) ∈ (σi,σi+1), for some x0 ∈R, and some i ∈ 1, . . .q− 1. Then, there existsa ∈R such that u(x)= ζ+i (x− a) or u(x)= ζ−i (x− a) ,∀x ∈R.

We provide a few simple properties of the functions ζ±i which enter directlyinto our arguments. We expand V near σi for u≥ σi as√

V(u)=√λi(u−σi)

θ (1+O(u−σi)), as u→ σi.

Integrating, we are led to the expansion

γi(u)=−θ − 1√2λi

(u−σi)−θ+1(1+O(u−σi)), as u→ σi,

and therefore also to the expansions

ζ±i (x)= σi+(√

2λi|x|θ − 1

)− 1θ−1

(1+ o(1)), as x→∓∞.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 111

Similarly,

ζ±i (x)= σi+1−(√

2λi+1|x|θ − 1

)− 1θ−1

(1+ o(1)), as x→±∞,

and corresponding asymptotics for the derivatives can be derived as well (e.g.using the fact that the discrepancy is zero).

For 0 < ε < 1 given, and i = 1, . . . ,q − 1, consider the scaled function

ζ±i,ε = ζ±i( ·ε

)which is a solution to

−uxx+ ε−2V ′(u)= 0,

hence a stationary solution to (PGL)ε. Straightforward computations based onthe previous expansions show that⎧⎪⎪⎪⎨⎪⎪⎪⎩

eε(ζ±i,ε

)(x)= (2λi)

− 1θ−1 (θ − 1)

2θθ−1 1

ε

∣∣ xε

∣∣−(ω+1)+ oxε→∓∞

(1ε

∣∣ xε

∣∣−(ω+1))

,

eε(ζ±i,ε

)(x)= (2λi+1)

− 1θ−1 (θ − 1)

2θθ−1 1

ε

∣∣ xε

∣∣−(ω+1)+ oxε→±∞

(1ε

∣∣ xε

∣∣−(ω+1))

,

(3.60)with ω defined in (3.5). Hence there is some constant C > 0 independent of rand ε such that

Si ≥∫ r

−reε(ζ±i,ε

)dx≥Si−C

(εr

)ω. (3.61)

3.2.2 On the Energy of Chains of Stationary Solutions

If u satisfies condition WPILε (δ) and (H0), we set

ELε (u)=

∑k∈J

Si(k) and ELε (u)=

∫I2L

eε(u(x))dx. (3.62)

Proposition 3.22 We have⎧⎪⎨⎪⎩ELε (u)≥ EL

ε (u)−CfM0

(εδ

)ωif WPIL

ε(δ) holds,

ELε (u)≤ EL

ε (u)+ (Cw+Cf)M0

(εδ

)ωif WPL

ε(δ) holds.(3.63)

Moreover, for any smooth function χ with compact support in I2L we have∣∣∣∣∣Iε(χ)−∑k∈J

χ(ak)Si(k)

∣∣∣∣∣≤ (Cw+Cf)M0

((εδ

)ω ‖χ‖∞+ ε‖χ ′‖∞)

,

if WPLε(δ) holds, (3.64)

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112 Fabrice Bethuel and Didier Smets

where Iε(χ)=∫

I2Leε(u)χ(x)dx. The constant Cf which appears in (3.63) and

(3.64) depends only on V , and the constant Cw appears in the definition ofcondition WPL

ε.

Proof We estimate the integral of |eε(u)− eε(ζ†ki(k)(·− ak))| on Ik as

ε

2

∫Ik

|u2− (ζ†ki(k),ε(·− ak))

2|dx

≤ ε‖u− ζ†ki(k),ε(·− ak)‖L∞(Ik)

[Eε(u)

12 +Eε(ζ †k

i(k),ε)12

]√δ

ε

and likewise we obtain

ε−1∫

Ik

|V(u)−V(ζ†ki(k),ε(·− ak))|dx≤ C

δ

ε‖u− ζ

†ki(k),ε(·− ak)‖L∞(Ik).

It suffices then to invoke WPILε(δ) and WPOL

ε(δ) as well as the decay estimates(3.61) to derive (3.63), using the fact that since δ≥ α1ε, negative exponentialsare readily controlled by negative powers. Estimate (3.64) is derived in a verysimilar way, the error in ε‖χ ′‖∞ being a consequence of the approximation of∫χeε(ζ

†ki(k),ε(·− ak)) by χ(ak)Si(k).

This result shows that, if δ is sufficiently large, the energy is close to a setof discrete values, namely the finite sums of Sk. We will therefore refer to thisproperty as the quantization of the energy, it will play an important role laterwhen we will obtain estimates on the dissipation rate of energy.

3.2.3 Study of the Perturbed Stationary Equation

Consider a function u defined on R satisfying the perturbed differentialequation

uxx = ε−2V ′(u)+ f , (3.65)

where f ∈ L2(R), and the energy bound (H0). We already know, thanks toLemma 3.21 that if f = 0 then u is of the form ζ±i,ε(· − a). Our results below,summarized here in loose terms, show that if f is sufficiently small on somesufficiently large interval, then u is close to a chain of translations of thefunctions ζ±i,ε suitably glued together on that interval.

Following the approach of [4], we first recast Equation (3.65) as a systemof two differential equations of first order. For that purpose, we set w= εux sothat (3.65) is equivalent to the system

ux = 1

εw and wx = 1

εV ′(u)+ εf ,

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 113

which we may write in a more condensed form as

Ux = 1

εG(U)+ εF on R, (3.66)

where we have set U(x) = (u(x),w(x)) and F(x) = (0, f (x)), and where Gdenotes the vector field G(u,w) = (w,V ′(u)). Notice that the energy bound(H0) and assumption (A3) together imply a global L∞ bound on u. In turn,this L∞ bound implies a Lipschitz bound, denoted C0, for the non-linearityG(u,w).

Lemma 3.23 Let u1 and u2 satisfy (3.65) with forcing terms f1 and f2, andassume that both satisfy the energy bound (H0). Denote by U1,U2,F1,F2 thecorresponding solutions and forcing terms of (3.66). Then, for any x,x0 in somearbitrary interval I,

|(U1−U2)(x)|≤(|(U1−U2)(x0)|+ ε

32√

2C0‖F1−F2‖L2(I)

)exp

(C0|x− x0|

ε

).

(3.67)

Proof Since (U1 − U2)x = G(U1) − G(U2) + ε(F1 − F2) we obtain theinequality

|(U1−U2)x| ≤ C0

ε|U1−U2|+ ε|F1−F2|.

It follows from Gronwall’s inequality that

|(U1−U2)(x)| ≤ exp(

C0|x−x0|ε

)|(U1−U2)(x0)|

+ |∫ x

x0

ε|(F1−F2)(y)|exp(

C0|y−x0|ε

)dy|.

Claim (3.67) then follows from the Cauchy–Schwarz inequality.

We will combine the previous lemma with the following one.

Lemma 3.24 Let u be a solution of (3.65) satisfying (H0). Then

supx,y∈I

|ξε(u)(x)− ξε(u)(y)| ≤√

2M0ε12 ‖f‖L2(I),

where I ⊂R is an arbitrary interval.

Proof This is a direct consequence of the equalityd

dxξε(u)= εf

d

dxu, the

Cauchy–Schwarz inequality, and the definition of the energy.

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114 Fabrice Bethuel and Didier Smets

Lemma 3.25 Let u be a solution of (3.65) satisfying (H0). Let L > 0 andassume that

D(u)∩ I2L ⊆ IL.

There exists a constant 0 < κw < 1, depending only on V, such that if

M0ε

L+M

120 ε

32 ‖f‖L2(I 3

2 L) ≤ κw, (3.68)

then the condition WPILε (δ) holds where

δ

ε:=− 2

ρwlog

(M0

ε

L+M

120 ε

32 ‖f‖L2(I 3

2 L)

), (3.69)

and where the constant ρw depends on only M0 and V . Moreover, κw issufficiently small so that 2|logκw|/ρw ≥ α1, where α1 was defined in (3.26).

Proof If D(u)∩ I2L =∅ then there is nothing to prove. If not, we first claim thatthere exists a point a1 ∈ IL such that u(a1)= zi(1) for some i(1) ∈ {1, . . . ,q−1}.Indeed, if not, and since the endpoints of I2L are not in the front set, thefunction u would have a critical point with a critical value in the complementof ∪jB(σj,μ0). At that point, the discrepancy would therefore be larger thanC/ε for some constant C > 0 depending on only V (through the choice ofμ0). On the other hand, since |ξε| ≤ eε, by averaging there exists at leastone point in I 3

2 L where the discrepancy of u is smaller in absolute value thanM0/(3L). Combined with the estimate of Lemma 3.24 on the oscillation of thediscrepancy, we hence derive our first claim, provided κw in (3.68) is chosensufficiently small. Wet set †1 = sign(u′(a1)), u1 = u and u2 = ζ

†1i(1),ε(· − x1)).

SinceV(u1(a1))= V(u2(a1))= V(zi(1)),

and since

|ξε(u1)(a1)− ξε(u2)(a1)| = |ξε(u1)(a1)| ≤M0/(3L)+√2M0ε

12 ‖f‖L2(I 3

2 L),

we obtain∣∣ε(u′1)2(a1)− ε(u′2)2(a1)

∣∣≤M0/(L)+ 2√

2M0ε12 ‖f‖L2(I 3

2 L).

Since also

|u′1(a1)+ u′2(a1)| ≥ |u′2(a1)| = |√

2V(zi(1))

ε2| ≥ C/ε,

it follows that∣∣ε(u′1− u′2)(a1)∣∣≤ C

(M0

ε

L+√

M0ε32 ‖f‖L2(I 3

2 L)

),

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 115

for a constant C> 0 which depends on only V . We may then apply Lemma 3.23to u1 and u2 with the choice x0 = a1, and for which we thus have, with thenotations of Lemma 3.23,

|(U1−U2)(x0)| ≤ C

(M0

ε

L+√

M0ε32 ‖f‖L2(I 3

2 L)

).

Estimate (3.67) then yields (3.24) on I1 = [a1 − δ,a1 + δ], for the choice of δgiven by (3.69) with ρw = 4(C0+1), where C0 depends on only M0 and V andwas defined above Lemma 3.23.

If D(u)∩ (I 32 L \ [a1− δ,a1+ δ])= ∅, we are done, and if not we may repeat

the previous construction (the boundary points of [a1−δ,a1+δ] are not part ofthe front set), until after finitely many steps we cover the whole front set.

We turn to the outer condition8 WPOLε .

Lemma 3.26 Let u be a solution of (3.65) verifying (H0), and assume that forsome index i ∈ {1, . . . ,q}

u(x) ∈ B(σi,μ0) ∀x ∈ A,

where A is some arbitrary bounded interval. Set R= length(A), let 0 < ρ < R,and set B= {x ∈ A | dist(x,Ac) > ρ}. Then we have the estimate

Eε(u,B)≤ Co

(Eε(u,A \B)

1θ(ερ

)1+ 1θ +R

32 M

12θ0

(εR

)1+ 12θ ‖f‖L2(A)

),

where the constant Co depends on only V .

Proof Let 0 ≤ χ ≤ 1 be a smooth cut-off function with compact support in Aand such that χ ≡ 1 on B and |χ ′| ≤ 2/ρ on A. We multiply (3.65) by ε(u−σi)χ

2 and integrate on A. This leads to∫Aεu2

xχ2+ 1

εV ′(u)(u−σi)χ

2 =∫

A\B2εux(u−σi)χχ

′ −∫

Aεf (u−σi)χ

2.

We estimate the first term on the right-hand side above by∣∣∫A\B

2εux(u−σi)χχ′∣∣≤ (∫

Aεu2

xχ2) 1

2(∫

A\Bεθ (u−σi)

2θ) 1

2θ(∫

A\B|2χ ′| 2θ

θ−1) θ−1

≤ 1

2

∫Aεu2

xχ2+ 1

2ε1+ 1

θ(∫

A\B2

λieε(u)

) 1θ

(4

ρ

)2

(2ρ)θ−1θ

≤ 1

2

∫A

u2xχ

2+ 16λ− 1

θi

ρ

)1+ 1θ

Eε(u,A \B)1θ ,

8 for which several adaptations have to be carried out compared to the non-degenerate case.

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116 Fabrice Bethuel and Didier Smets

where we have used (3.2) and the fact that length(A \ B) = 2ρ. Similarly, weestimate ∣∣∫

Aεf (u−σi)χ

2∣∣≤ ε‖f‖L2(A)

(∫A(u−σi)

2θ) 1

2θ Rθ−12θ

≤ ε1+ 12θ ‖f‖L2(A)

(2

λi

)−1

M1

2θ0 R

θ−12θ .

Also, by (3.2) we have∫A

1

εV ′(u)(u−σi)χ

2 ≥ θ

∫B

1

εV(u).

Combining the previous inequalities, the conclusion follows.

Combining Lemma 3.25 with Lemma 3.26 we obtain the following.

Proposition 3.27 Let u be a solution to (3.65) satisfying assumption (H0), andsuch that D(u)∩ I3L ⊂ IL. There exist positive constants9 Cw and α1, dependingon only M0 and V, such that if α ≥ α1 and if

1. M0ε

L≤ 1

2exp(−ρw

2 α),

2. ‖f‖L2(I3L)≤ 1

2M− 1

20 ε

− 32 exp(−ρw

2 α),

3. ‖f‖L2(I3L)≤ Cw

2CoM

1− 12θ

0

( εL

)−1− 12θ L−

32 α−ω,

then WPLε(αε) holds.

Proof Direct substitution shows that assumptions 1 and 2 imply condition(3.68), provided α1 is choosen sufficiently large, and also imply conditionWPIL

ε(δ) for some δ ≥ αε given by (3.69). It remains to consider WPOLε(αε).

We invoke Lemma 3.26 on each of the intervals A = (ak + 12αε,ak+1 − 1

2αε),taking B= (ak+αε,ak+1−αε). In view of WPIL

ε(αε) and (3.61), we obtain

Eε(u,A \B)≤ Cα−ω,

and therefore

Eε(u,A \B)1θ α−1− 1

θ ≤ Cα−ω,

9 Recall that Cw enters in the definition of condition WPLε . A parameter named Cw already

appears in the statement of Proposition 3.22 We impose that its updated value here is be largerthan its original value in Proposition 3.22 (and Proposition 3.22 remains of course true with thisupdated value!).

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 117

where C depends on only V . Also, in view of assumption 3 we have

Co

∑k

R32 M

12θ0

(εR

)1+ 12θ ‖f‖L2(A) ≤ CoL

32 M

12θ0

(εL

)1+ 12θ ‖f‖L2(I3L)

≤ 1

2CwM0α

−ω,

provided α1 is sufficiently large (third requirement). It remains to estimateeε(u) on the intervals (−2L,a1) and (a�,2L). We first use Lemma 3.26 withA = (−3L,−L) (resp. A = (L,3L) and B = (− 5

2 L,− 32 L) (resp. B = ( 3

2 L, 52 L)).

This yields, using the trivial bound Eε(u,A \B)≤M0, the estimate

Eε(u, I 52 L \ I 3

2 L)≤ C

(M

1θ0

( εL

)1+ 1θ +M

12θ0

( εL

) 12θ

)≤ Cα−ω, (3.70)

in view of 1 and provided α1 is sufficiently large. We apply one last timeLemma 3.26, with A= (−2L− 1

2αε,a1− 12αε) (resp. A= (a�+ 1

2αε,2L+ 12αε))

and B = (−2L,a1 − αε) (resp. B = (a� + αε,2L)). Since A \ B ⊂ I 52 L \ I 3

2 L, itfollows from (3.70) and Lemma 3.26, combined with our previous estimates,that condition WPOL

ε(αε) is satisfied provided we choose Cw sufficientlylarge.

Remark 3.28 Notice that condition 1 in Proposition 3.27 is always sat-isfied when αε ≤ δ

ε

log, since L/ε ≥ 1. Also, for α = δε

log/ε, assumption 3in Proposition 3.27 is weaker than assumption 2 We therefore deduce thefollowing.

Corollary 3.29 Let u be a solution to (3.65) satisfying assumption (H0), andsuch that D(u)∩ I3L ⊂ IL. If

ε‖f‖L2(I3L)≤(

M0

L

) 12

, (3.71)

then WPLε(δ

ε

log) holds.

3.3 Regularized Fronts

In this section, we assume that vε is a solution of (PGL)ε which satisfies (H0)

and the confinement condition CL,S.

3.3.1 Finding Regularized Fronts

We provide here the proof of Proposition 3.13, which is deduced from thefollowing.

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118 Fabrice Bethuel and Didier Smets

Lemma 3.30 Given any s1 < s2 in [0,S], there exists at least one time s in[s1,s2] for which vε(·,s) solves (3.65) with

‖f‖2L2(I3L)

≡ εω−1‖∂svε(·,s)‖2L2(I3L)

≤ εω−1 dissip3Lε (s1,s2)

s2− s1≤ εω−1 M0

s2− s1.

(3.72)

Proof It is a direct mean value argument, taking into account the rescaling of(PGL)ε according to our rescaling of time.

Proof of Proposition 3.13 We invoke Lemma 3.30, and from (3.72) and theassumption s2 − s1 = εω+1L of Proposition 3.13, we derive exactly theassumption (3.71) in Corollary 3.29, from which the conclusion follows.

Following the same argument, but relying on Lemma 3.25 and Proposi-tion 3.27 rather than on Corollary 3.29, we readily obtain the following.

Proposition 3.31 For α1 ≤ α ≤ δε

log :

1. Each subinterval of [0,S] of size q0(α)εω+2 contains at least one time s at

which WPILε(αε,s) holds, where

q0(α)= 4M20 exp (ρwα) . (3.73)

2. Each subinterval of [0,S] of size q0(α,β)εω+2 contains at least one time sat which WPL

ε(αε,s) holds, where

β := L

εand q0(α,β)=max

(q0(α),

(2Co

Cw

)2( β

M0

)1− 1θα2ω

). (3.74)

3.3.2 Local Dissipation

For s ∈ [0,S], set ELε (s) = EL

ε (vε(s)) and, when WPILε(α1ε,s) holds, EL

ε (s) =EL

ε (vε(s)), ELε being defined in (3.62). We assume throughout that s1 ≤ s2 are

contained in [0,S], and in some places (in view of (3.28) that s2 ≥ L2εω.

Proposition 3.32 If s2 ≥ L2εω, we have

ELε (s2)+dissipL

ε (s1,s2)≤ELε (s1)+100CeL−(ω+2)(s2−s1)+Ce(1+M0)

(L

ε

)−ω

.

(3.75)

Proof Let 0 ≤ ϕ ≤ 1 be a smooth function with compact support in I2L, suchthat ϕ(x) = 1 on I 5

3 L, |ϕ′′| ≤ 100L−2. It follows from the properties of ϕ and

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 119

(3.28) that

Iε(s,ϕ)≤ ELε (s) for s ∈ (s1,s2) and Iε(s2,ϕ)≥ EL

ε (s2)−Ce

(L

ε

)−ω

,

which combined with (3.36) yields

ELε (s2)+ dissipL

ε (s1,s2)≤ ELε (s1)+Ce

(L

ε

)−ω

+ ε−ω

∫ s2

s1

FS(s,ϕ,vε)ds,

where FS is defined in (3.37). The estimate (3.75) is then obtained invokingthe inequality |ξε| ≤ eε to bound the term involving FS: combined with (3.28)for times s≥ L2εω and with assumption (H0) for times s≤ L2εω.

If WPLε(δ,s1) and WPIL

ε(δ′,s2) hold, for some δ,δ′ ≥ α1ε and s2 ≥ L2εω,

then combining inequality (3.75) with the first inequality (3.63) applied tovε(s2) as well as the second applied to vε(s1), we obtain

ELε (s2)+ dissipL

ε (s1,s2)

≤ ELε (s2)+CfM0

( ε

δ′)ω+ dissipL

ε (s1,s2)

≤ ELε (s1)+ 100CeL−(ω+2)(s2− s1)+CfM0

( ε

δ′)ω+Ce(1+M0)

( εL

)ω≤ EL

ε (s1)+ (Cw+Cf)M0

(εδ

)ω+CfM0

( ε

δ′)ω+ 100CeL−(ω+2)(s2− s1)

+Ce(1+M0)( ε

L

)ω.

(3.76)We deduce from this inequality an estimate for the dissipation between s1 ands2 and an upper bound on EL

ε (s2).

Corollary 3.33 Assume that WPLε(δ,s1) and WPIL

ε(δ′,s2) hold, for some

δ,δ′ ≥ α1ε and s2 ≥ L2εω, and that ELε (s1)= EL

ε (s2). Then

dissipLε [s1,s2] ≤ (Cw+Cf)M0

(εδ

)ω+CfM0

( ε

δ′)ω+ 100CeL−(ω+2)(s2− s1)

+Ce(1+M0)( ε

L

)ω,

ELε (s2)−EL

ε (s2)≤ (Cw+Cf)M0

(εδ

)ω+ 100CeL−(ω+2)(s2− s1)

+Ce(1+M0)( ε

L

)ω.

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120 Fabrice Bethuel and Didier Smets

3.3.3 Quantization of the Energy

Let s ∈ [0,S] and δ ≥ α1ε, and assume that vε satisfies WPLε(δ,s). The front

energy ELε (s), by definition, may take only a finite number of values, and

is hence quantized. We emphasize that, at this stage, ELε (s) is only defined

assuming condition WPILε(δ,s) holds. However, the value of EL

ε (s) does notdepend on δ, provided that δ≥ α1ε, so that it suffices ultimately to check thatcondition WPIL

ε(α1ε,s) is fulfilled.Since Eε(s) may take only a finite number of values, let μ1 > 0 be the

smallest possible difference between two distinct such values. Let L0 ≡L0(s1,s2) > 0 be such that

100CeL−(ω+2)0 (s2− s1)= μ1

4(3.77)

and finally choose α1 sufficiently large so that((2Cf+Cw)M0+Ce(1+M0)

)α−ω

1 ≤ μ1

4. (3.78)

As a direct consequence of (3.76), (3.77), (3.78) and the definition of μ1 weobtain the following result.

Corollary 3.34 For s1 ≤ s2 ∈ [0,S] with s2 ≥ εωL2, assume that WPLε(α1ε,s1)

and WPILε(α1ε,s2) hold and that L≥ L0(s1,s2). Then we have EL

ε (s2)≤ ELε (s1).

Moreover, if ELε (s2) < EL

ε (s1), then ELε (s2)+μ1 ≤ EL

ε (s1).

In the opposite direction we have the following.

Lemma 3.35 For s1 ≤ s2 ∈ [0,S], assume that WPILε(α1ε,s1) and

WPILε(α1ε,s2) hold and that L≥ L0(s1,s2). Assume also that

s2− s1 ≤ ρ0

(1

8dε,L

min(s1)

)ω+2

. (3.79)

Then we have ELε (s2) ≥ EL

ε (s1). In the case of equality, we have J(s1)= J(s2)

and

σi(k± 12 )(s1)= σi(k± 1

2 )(s1), for any k ∈ J(s1) and dε,L

min(s2)≥ 1

2dε,L

min(s1). (3.80)

Proof It a consequence of the bound (3.7) in Theorem 3.2 on the speed ofthe front set combined with assumption (3.79). Indeed, this implies that forarbitrary s ∈ [s1,s2], the front set at time s is contained in a neighborhood ofsize dε,L

min(s1)/8 of the front set at time s1. In view of the definition of dε,Lmin(s1),

and of the continuity in time of the solution, this implies that for all k0 ∈ J(s1)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 121

the set

Ak0 ={

k ∈ J(s2) such that aεk(s2) ∈

[aε

k0(s1)− 1

4dε,L

min(s1),aεk0(s1)+ 1

4dε,L

min(s1)]}

is non-empty, since it must contain a front connecting σi(k0− 12 )(s1) to

σi(k0+ 12 )(s1). In particular, summing over all fronts in Ak0 , we obtain∑

k∈Ak0

SLi(k) ≥SL

i(k0),

with equality if and only if "Ak0 = 1. Summing over all indices k0, we are ledto the conclusion.

3.3.4 Propagating Regularized Fronts

We discuss in this subsection the case of equality ELε (s1)=EL

ε (s2). We assumethroughout that we are given δ

ε

log ≥ δ> α1ε and two times s1 ≤ s2 ∈ [εωL2,S]such that

C(δ,L,s1,s2)

{WPL

ε(δ,s1) and WPILε (δ,s2) hold ,

ELε (s1)= EL

ε (s2), with L≥ L0(s1,s2).

Under that assumption, our first result shows that vε remains well-prepared onalmost the whole time interval [s1,s2], though with a smaller δ.

Proposition 3.36 There exists α2 ≥α1, depending only on V, M0 and Cw, withthe following property. Assume that C(δ,L,s1,s2) holds with α2ε ≤ δ ≤ δ

ε

log,then property WPL

ε(�log(δ),s) holds for any time s ∈ [s1+ ε2+ω,s2], where

�log(δ)= ω

ρwε

(log

δ

ε

). (3.81)

The proof of Proposition 3.36 relies on the following.

Lemma 3.37 Assume that C(δ,L,s1,s2) holds with δ ≥ α1ε. We have theestimate, for s ∈ [s1+ εω+2,s2],∫

I 32 L

|∂tvε(x,sε−ω)|2dx≤ Cε−3dissipLε [s,s− εω+2].

Proof of Lemma 3.37 Differentiating equation (PGLε) with respect to time,we are led to

|∂t(∂tvε)− ∂xx(∂tvε) | ≤ C

ε2|∂tvε|.

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122 Fabrice Bethuel and Didier Smets

It follows from standard parabolic estimates, working for x∈ I2L on the cylinder�ε(x) = [x− ε,x+ ε] × [t − ε2, t], where t := sε−ω, that for any point y ∈[x− ε

2 ,x+ ε2 ] we have

|∂tvε(y, t)| ≤ Cε−32 ‖∂tvε‖L2(�ε(x)).

Taking the square of the previous inequality, and integrating over [x− ε2 ,x+ ε

2 ],we are led to∫ x+ ε

2

x− ε2

|∂tvε(y, t)|2dy≤ Cε−2∫[x−2ε,x+2ε]×[t−ε2,t]

|∂tvε(y, t)|2dy.

An elementary covering argument then yields∫I 3

2 L

|∂tvε(y, t)|2dy≤ Cε−2‖∂tvε‖2L2(I 5

3 L×[t−ε2,t]) ≤ Cε−3dissipL

ε [s,s− εω+2].

Proof of Proposition 3.36 In view of Proposition 3.31, Corollary 3.34, andassumption C(δ,L,s1,s2), we may assume, without loss of generality, that

s2− s1 ≤ 2q0(δ/ε,L/ε). (3.82)

Let s ∈ (s1 + εω+2,s2), and consider once more the map u = vε(·,s), so thatu is a solution to (3.65), with source term f = ∂tvε(·,sε−ω). It follows fromLemma 3.37, combined with the first of Corollary 3.33 on the dissipation, that

‖f‖2L2(I 3

2 L)≤ Cε−3

[(Cw+ 2Cf)M0

(εδ

)ω+ 100CeL−(ω+2)(s2− s1)

+Ce(1+M0)( ε

L

)ω].

Notice that (3.82) combined with the assumption δ≤ δε

log yields

100CeL−(ω+2)(s2− s1)≤ C(εδ

)ω.

We deduce from Lemma 3.25, imposing on α2 the additional conditionω

ρw(logα2)≥ α1, that WPIL

ε((�log(δ),s) holds. It remains to show that

WPOLε(�log(δ),s) holds likewise. To that aim, we invoke (3.33) which we

use with the choice s1 = s1 and s2 = s. This yields, taking once more (3.82)into account,

ELε (s)−EL

ε (s)≤ (C+Cw)(εδ

)ω.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 123

Combining this relation with (3.61) and the first inequality of (3.63), we deducethat∫

eε(vε(s))ds≤ (C+Cw)(εδ

)ω+C

�log(δ)

≤ CwM0

�log(δ)

,

(3.83)provided α2 is chosen sufficiently large.

In view of (3.79) and (3.7), we introduce the function

q1(α) :=(q0(α)

ρ0

) 1ω+2

,

which represents therefore the maximum displacement of the front set inthe interval of time needed (at most) to find two consecutive times at whichWPIL

ε(αε) holds.From Proposition 3.36 and Lemma 3.35 we deduce the following.

Corollary 3.38 Let s ∈ [εωL2,S] and α2 ≤ α ≤ δε

log, and assume thatWPL

ε(αε,s) holds as well as dε,Lmin(s) ≥ 16q1(α)ε. Then WPL

ε(�log(αε),s′)holds for any s+ ε2+ω ≤ s′ ≤ T ε

0 (α,s), where

T ε0 (α,s)=max

{s+ ε2+ω ≤ s′ ≤ S s.t. dε,L

min(s′′)≥ 8q1(α)ε

∀s′′ ∈ [s+ εω+2,s′]} .

We complete this section presenting the following proof.

Proof of Proposition 3.14 This follows directly from Corollary 3.38 with thechoice α = δ

ε

log, noticing that �log(δε

log)= δε

loglog.

Proof of Corollary 3.15 If we assume moreover that s0 ≥ εωL2 and thatWPL

ε(δε

log,s0) holds, then it is a direct consequence of the inclusion (3.7) andProposition 3.14, taking into account the assumption (3.33). If we assume onlythat s0 ≥ 0 and that WPL

ε(α1ε,s0) holds, then it suffices to consider the firsttime s′0 ≥ s0 + εωL2 at which WPL

ε(δε

log,s′0) holds and to rely on Proposi-tion 3.14 likewise. Indeed, since s′0 − s0 ≤ εωL2 + εω+1L by Proposition 3.13,we may apply Corollary 3.34 and Lemma 3.35 for s1 = s0 and s2 = s′0, whichyields EL

ε (s0)=ELε (s

′0) and therefore also the same asymptotics for dε,L

min at timess0 and s′0.

3.4 First Compactness Results for the Front Points

The purpose of this section is to provide the proofs of Proposition 3.16 andCorollary 3.17.

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124 Fabrice Bethuel and Didier Smets

Proof of Proposition 3.16 As mentioned, we choose the test functions (inde-pendently of time) so that they are affine near the front points for any s∈ Iε(s0).More precisely, for a given k0 ∈ J we impose the following conditions on thetest functions χ ≡ χk0 in (3.42):⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

χ has compact support in [aεk(s0)− 1

3d∗min(s0),a

εk(s0)+ 1

3d∗min(s0)],

χ is affine on the interval [aεk(s0)− 1

4d∗min(s0),a

εk(s0)+ 1

4d∗min(s0)],

with χ ′ = 1 there

‖χ‖L∞(R) ≤ Cd∗min(s0),‖χ ′‖L∞(R) ≤ C and ‖χ ′′‖L∞(R) ≤ Cd∗min(s0)−1.(3.84)

It follows from Corollary 3.15 that, for ε sufficiently small, we are in positionto claim (3.42) and (3.43) for arbitrary s1 and s2 in the full interval I∗(s0).Combined with the first estimate of Corollary 3.33, with δ = δ′ = δ

ε

loglog, thisyields the conclusion (3.44).

Proof of Corollary 3.17 The family of functions (vε)0<ε<1 is equi-continuouson every compact subset of the interval I∗(s), so that the conclusion followsfrom the Arzela–Ascoli theorem.

3.5 Refined Asymptotics Off the Front Set

3.5.1 Relaxations Towards Stationary Solutions

Throughout this section, we assume that we are in the situation described byCorollary 3.15, in particular L is fixed and ε will tend to zero. Our main purposeis then to provide rigorous mathematical statements and proofs concerning theproperties of the function Wεn =Wk

εndefined in (3.45), for given k ∈ J, which

have been presented, most of them in a formal way, in Subsection 3.1.5. Wenotice first that we may expand V ′ near σ≡ σi(k+ 1

2 )as

V ′(σ+ u)= 2θλu2θ−1 (1+ ug(u)) , (3.85)

where g is some smooth function on R and where we have set for the sake ofsimplicity λ = λi(k+ 1

2 ). We work on the sets Vk(s0) defined in (3.48) and on

their analogs at the ε level:

Vεk (s0)= ∪

s∈Iε(s0)Jε(s)×{s} ≡ ∪

s∈Iε(s0)

(aε

k(s)+ δε

loglog,aεk+1(s)− δ

ε

loglog

)×{s}.(3.86)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 125

We will therefore work only with arbitrary small values of u. Let u0 > 0 besufficiently small so that |ug(u)| ≤ 1/4 on (−u0,u0) and V ′(σ+ u) is strictlyincreasing on (−u0,u0), convex on (0,u0) and concave on (−u0,0). For smallvalues of ε, the value of u in (3.85), in view of (3.46) in Lemma 3.18, will notexceed u0, and we may therefore assume for the considerations in this sectionthat ug(u) = u0g(u0), if u ≥ u0 and −ug(u) = u0g(u0), if u ≤ −u0. Equation(PGL)ε translates into the following equation for Wε:

Lε(Wε)≡ εω∂Wε

∂s− ∂2Wε

∂x2+λfε(Wε)= 0, (3.87)

where we have set

fε(w)= 2θw2θ−1(

1+ ε1

θ−1 wg(ε1

θ−1 w))

. (3.88)

Notice that our assumption yields in particular

|fε(w)| ≥ 3

2θ |w|2θ−1. (3.89)

The analysis of the parabolic equation (3.87) is the core of this section. Asmentioned, our results express convergence to stationary solutions. We firstprovide a few properties concerning these stationary solutions: the first lemmadescribes stationary solutions involved in the attractive case, whereas thesecond lemma is used in the repulsive case.

Lemma 3.39 Let r > 0 and 0 < ε < 1. There exist unique solutions∨u+ε,r (resp.

∨u−ε,r) to⎧⎨⎩−

dUdx2

+λfε(U)= 0 on (−r,r),

U(−r)=+∞ (resp. U(−r)=−∞) and U(r)=+∞(resp. U(r)=−∞).

Moreover, we have

C−1r−1

θ−1 ≤ ∨u+ε,r ≤ C (r−|x|)− 1

θ−1 and C−1r−1

θ−1 ≤−∨u−ε,r ≤ C (r−|x|)− 1

θ−1 ,(3.90)

for some constant C > 0 depending only on V .

Lemma 3.40 Let r > 0 and 0 < ε < 1 be given. There exists a unique solution�uε,r to

− dUdx2

+λfε(U)= 0 on (−r,r), U(−r)=−∞ and U(r)=+∞.

These and related results are standard and have been considered since theworks of Keller [12] and Osserman [14] in the 1950s, at least regarding

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126 Fabrice Bethuel and Didier Smets

existence. The convexity/concavity assumptions are sufficient for uniqueness.We refer to Lemma 3.59 in Appendix A for a short discussion of the case of apure power non-linearity.

We set rε(s)= rεk+ 1

2(s)= 1

2(aε

k+1(s)− aεk(s)). Our aim is to provide suffi-

ciently accurate expansions of Wε and the renormalized discrepancy ε−ωξε onneighborhoods of the points aε

k+ 12(s), for instance the intervals

k+ 12(s)= aε

k+ 12(s)+[−7

8rε(s),

7

8rε(s)] = [aε

k(s)+1

8rε(s),aε

k+1(s)−1

8rε(s)].(3.91)

We first turn to the the attractive case †k =−†k+1. We may assume additionallythat

k ∈ {1, . . . ,�− 1} and †k =−†k+1 = 1, (3.92)

the case †k =−†k+1 =−1 being handled similarly.

Proposition 3.41 If (3.92) holds and ε is sufficiently small, then for any s ∈Iε(s0) and every x ∈$ε

k+ 12(s) we have the estimate

|Wε(x,s)−λ− 1

2(θ−1)∨u+rε(s)(x)| ≤ Cε

min( 1ω+2 , ω−1

2(θ−1) ). (3.93)

The repulsive case corresponds to †k = †k+1 and we may assume as abovethat

k ∈ {1, . . . ,�− 1} and †k = †k+1 = 1. (3.94)

Proposition 3.42 If (3.94) holds and ε is sufficiently small, then for any s ∈Iε(s0) and every x ∈$ε

k+ 12(s) we have the estimate

|Wε(x,s)−λ− 1

2(θ−1)�urε(s)(x)| ≤ Cε

min( 1ω+2 , ω−1

2(θ−1) ). (3.95)

Combining these results with parabolic estimates, we obtain estimates forthe discrepancy.

Proposition 3.43 If ε is sufficiently small, then for any s ∈ Iε(s0) and everyx ∈$ε

k+ 12(s) we have the estimate

|ε−ωξε(vε)−λ− 1

2(θ−1)

i(k+ 12 )

rε(s)−(ω+1)γk+ 12| ≤ Cε

1θ2 , (3.96)

where ⎧⎨⎩γk+ 12= Aθ if †k =−†k+1,

γk+ 12= Bθ if †k = †k+1.

(3.97)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 127

For the outer regions, corresponding to k = 0 and k = �, estimates forthe discrepancy are directly deduced from the crude estimates provided byProposition 3.12. Proposition 3.43 provides a rigorous ground to the formalcomputation (3.52) of the introduction, and hence allows us to derive theprecise motion law. The proofs of Propositions 3.41 and 3.42, however, arethe central part of this section. Note that by no means are the estimatesprovided in Propositions 3.41, 3.42 and 3.43 optimal, our goal was only toobtain convergence estimates, valid for all ε sufficiently small, uniformly on∪s∈Iε(s0)$

ε

k+ 12(s)×{s}.

3.5.2 Preliminary Results

We first turn to the proof of Lemma 3.18, which provides first properties of Wε.

Proof of Lemma 3.18 Let x ∈ (ak(s) + δε

loglog,ak+1(s) − δε

loglog) and any s ∈Iε(s0), and recall that d(x,s) := dist(x, {aε

k(s),aεk+1(s)}). In view of Proposi-

tion 3.13, and in particular of estimate (3.31), it suffices to show that

vε(y,s)∈B(σi,μ0) for all (y,s)∈[

x− d(x,s)

2,x+ d(x,s)

2

]×[s−εωd(x,s)2,s].

By Theorem 3.2, on such a time scale the front set moves at most by a distance

d :=(εωd(x,s)2

ρ0

) 1ω+2

≤ ρ− 1

ω+20 (

ε

δε

loglog

ω+2 d(x,s)≤ d(x,s)

4,

provided ε/L is sufficiently small. More precisely, Theorem 3.2 only providesone inclusion, forward in time, but its combination with Corollary 3.15provides both forward and backward inclusions (for times in the intervalIε(s0)), from which the conclusion then follows.

For the analysis of the scalar parabolic equation (3.87), we will extensivelyuse the fact that the map fε is non-decreasing on R, allowing comparisonprinciples. The desired estimates for Wε will be obtained using appropriatechoices of sub- and super-solutions. The construction of these functionsinvolves a number of elementary solutions. First, we use the functions W±

ε ,independent of the space variable x and solve the ordinary differential equation⎧⎨⎩εω

∂W±ε

∂s=−λfε(W±

ε ),

Wε(0)=±∞.(3.98)

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128 Fabrice Bethuel and Didier Smets

Using separation of variables, we may construct such a solution which verifiesthe bounds

0 <W+ε (s)≤ Cε

ω2(θ−1) [λs]− 1

2(θ−1) and 0≥W−ε (s)≥−Cε

ω2(θ−1) [λs]− 1

2(θ−1) ,(3.99)

so that it relaxes quickly to zero. We will also use solutions of the standardheat equation and rely in several places on the next remark.

Lemma 3.44 Let � be a non-negative solution to the heat equationεω∂s� − �xx = 0, and U be such that Lε(U) = 0. Then Lε(U + �) ≥ 0,and Lε(U−�)≤ 0.

Proof Notice that Lε(U±�) = λ(fε(U±�)− fε(U)), so that the conclusionfollows from the fact that fε is non-decreasing.

Next, let s be given in Iε(s0). By translation invariance, we may assumewithout loss of generality that

k+ 12(s)= 0. (3.100)

We set hε = (ε/2ρ0)1

ω+2 , and consider the cylinders

�extε (s)= J ext

ε (s)×[s− ε,s] and �intε (s)= J int

ε (s)×[s− ε,s], (3.101)

where J intε (s)= [−rεint(s),r

εint(s)], J ext

ε (s)= [−rεext(s),rεext(s)] with

rεext(s)= rε(s)+ 2hε and rεint(s)= rε(s)− 2hε.

If ε is sufficiently small, in view of (3.7) we have the inclusions, with Vεk (s0)

defined in (3.86) ,

�intε (s)⊂ ε(s)≡ Vε

k (s0)∩ ([s− ε,s]×R)⊂�extε (s).

As a matter of fact, still for ε sufficiently small, we have for any τ ∈ [s− ε,s],{−rεext(s)+ hε ≤ aεk(τ )+ δ

ε

loglog ≤−rεint(s)− hε,

rεint+ hε ≤ aεk+1(τ )− δ

ε

loglog ≤ rεext(s)− hε(s0).(3.102)

We also consider the parabolic boundary of �extε (s):

∂p�extε (s)= [−rεext(s),r

εext(s)]× {s− ε}∪ {−rεext}× [s− ε,s] ∪ {rεext}× [s− ε,s]

= ∂�extε (s) \ [−rεext(s),r

εext(s)]× {s},

and define ∂p�intε (s) accordingly. Finally, we set

∂p ε(s)= ∂( ε(s)) \ [aεk(s)+ δ

ε

loglog,aεk+1(s)− δ

ε

loglog]× {s}.A first application of the comparison principle leads to the following bounds.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 129

Proposition 3.45 For x ∈ J intε (s),⎧⎪⎨⎪⎩

Wε(x,s)≤ ∨u+ε,rεint

(x)+Cεω−1

2(θ−1) ,

Wε(x,s)≥ ∨u−ε,rεint

(x)−Cεω−1

2(θ−1) .(3.103)

Proof We work on the cylinder �intε (s) and consider there the comparison map

Wsupε (y,τ)= ∨

u+ε,rεint

(y)+Wε(τ − (s− ε)) for (y,τ) ∈�intε (s).

Since the two functions on the r.h.s. of the definition of Wsupε are positive

solutions to (3.87) and since fε is super-additive on R+, that is, since

fε(a+ b)≥ fε(a)+ fε(b) provided a≥ 0,b≥ 0, (3.104)

we deduce that

(Wsup

ε (y,τ))≥ 0 on �int

ε (s) with Wsupε (y,τ)=+∞ for (y,τ) ∈ ∂p�

intε ,

so that Wsupε (x,s)≥Wε on ∂p�intε . It follows that Wsup

ε (y,τ) ≥Wε on �intε ,

which, combined with (3.99), immediately leads to the first inequality. Thesecond is derived similarly.

At this stage, the constructions are somewhat different in the case ofattractive and repulsive forces, so we need to distinguish the two cases.

3.5.3 The Attractive Case

We assume here that †k =−†k+1. Without loss of generality, we may assumethat

†k =−†k+1 = 1, (3.105)

the case †k=−†k=−1 being handled similarly. The purpose of this subsectionis to provide the proof of Proposition 3.41. We split the proof into separatelemmas, the main efforts being devoted to the construction of subsolutions.We start with the following lower bound.

Lemma 3.46 Assume that (3.105) holds. Then, for x ∈ Jε(s− ε2 ), we have the

lower bound

Wε(x,s− ε

2)≥−Cε

ω−12(θ−1) .

Proof In view of (3.47), we notice that

Wε(y,τ)≥ 0 on ∂p ε(s) \ [ak(s− ε)+ δε

loglog,ak+1(s− ε)− δε

loglog]× {s− ε}.

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130 Fabrice Bethuel and Didier Smets

We consider next the function Wε defined for τ ≥ s − ε by Wε(y,τ) =W−

ε (τ − (s − ε)). Since Wε < 0, and since Wε(s − ε) = −∞, we obtainWε ≤Wε on ∂p ε(s), so that, by the comparison principle, we are led toWε ≤Wε on ε(s), leading to the conclusion.

Proposition 3.47 Assume that (3.105) holds. We have the lower bound forx ∈ Jε(s):

Wε(x,s)≥ ∨u+ε,rεext

(x)−Cε− 1

3θ−1 exp

(−π2 ε−ω+1

32(rε(s))2

). (3.106)

Proof On Jε(s− ε2 ) we consider the map ϕε defined by

ϕε(x)= inf{Wε(x,s− ε

2)−∨

u+ε,rεext

(x),0} ≤ 0. (3.107)

Invoking (3.102) and estimates (3.90) for∨u+ε,rεext

, we obtain, for x ∈ Jε(s− ε2 ),

0≤ ∨u+ε,rεext

(x)≤ Ch− 1

θ−1ε , (3.108)

which, combined with Lemma 3.46, yields

|ϕε(x)| ≤ Ch− 1

θ−1ε for x ∈ Jε(s− ε

2). (3.109)

Combining (3.108), estimate (3.90) of Lemma 3.39 and estimate (3.47) ofLemma 3.18, we deduce that, if ε is sufficiently small then

ϕε(aεk(s−

ε

2)+ δ

ε

loglog)= ϕε(aεk+1(s−

ε

2)− δ

ε

loglog)= 0. (3.110)

We extend ϕε by 0 outside the set Jε(s− ε2 ), and consider the solution �ε to⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

εω∂�ε

∂τ− ∂�ε

∂x2= 0 on �ext

ε (s)∩{τ ≥ s− ε

2},

�ε(x,s− ε

2)= ϕε(x) for x ∈ J ext

ε (s− ε

2),

�ε(±rεext(s),τ)= 0 for τ ∈ (s− ε

2,s).

(3.111)

Notice that �ε ≤ 0. We consider next on �extε (s)∩{τ ≥ s− ε

2 } the function W infε

defined by

W infε (y,τ)= ∨

u+ε,rεext

(y)+�ε(y,τ).

It follows from Lemma 3.44 that Lε(W infε ) ≤ 0, so that W inf

ε is a subsolution.Since W inf

ε ≤Wε on ∂p( ε(s)∩{τ ≥ s− ε

2 })

it follows in particular that

W infε ≤Wε on Jε(s). (3.112)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 131

To complete the proof, we rely on the next linear estimates for �ε.

Lemma 3.48 We have the bound, for y ∈ J extε and τ ∈ (s− ε

2 ,s),

|�ε(y,τ)| ≤ C exp

(−π2ε−ω

(τ − (s− ε2 )

16(rε(s))2

)‖ϕε‖L∞(Jε(s− ε

2 )).

We postpone the proof of Lemma 3.48 and complete the proof ofProposition 3.47.

Proof of Proposition 3.47 completed Combining Lemma 3.48 with (3.109), weare led, for x ∈ Jε(s), to

|�ε(x,s)| ≤ Ch− 1

θ−1ε exp

(−π2 ε−ω+1

32(rε(s))2

). (3.113)

The conclusion then follows, invoking (3.112).

Proof of Lemma 3.48 Consider on the interval [−2rε(s),2rε(s)] the function

ψ(x) defined by ψ(x)= cos

4rε(s)x

), so that −ψ = π2

16(rε(s))−2ψ , ψ ≥ 0,

ψ(−2rε(s))=ψ(2rε(s))= 0 and ψ(x)≥ 1/2 for x ∈ [−rεext(s),rεext(s)]. Hence,

we obtain

εω%τ −%xx = 0 on �extε (s)∩{τ ≥ s− ε

2},

where %(x,τ)= exp

(−π2ε−ω

τ − (s− ε2 )

16rε(s)2

)ψ(x).

On the other hand, for (y,τ) ∈ ∂p(�ext

ε (s)∩{τ ≥ s− ε2 })

we have

|�ε(y,τ)| ≤ ‖ϕε‖L∞(Jε(s− ε2 ))

2%(y,τ)

and the conclusion follows therefore from the comparison principle for the heatequation.

Proof of Proposition 3.41 completed Combining the upper bound (3.103) ofProposition 3.45 with the lower bound (3.106) of Proposition 3.47, we are led,for ε sufficiently small, to

∨u+ε,rεext

(x)−Aε ≤Wε(x,s)≤ ∨u+ε,rεint

(x)+Aε, (3.114)

where we have set

Aε = Cεω−1

2(θ−1) . (3.115)

The conclusion (3.93) then follows from Proposition 3.63 of Appendix Acombined with the definition of hε and (A.7).

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132 Fabrice Bethuel and Didier Smets

3.5.4 The Repulsive Case

In this subsection, we assume throughout that †k = †k+1 and may assumemoreover that

†k = †k+1 = 1; (3.116)

the case †k = †k = −1 is handled similarly. The main purpose of thissubsection is to provide the proof of Proposition 3.42, the central part beingthe construction of accurate supersolutions, subsolutions being provided bythe same construction. We assume as before that (3.100) holds, and use ascomparison map Uε defined on I trs

ε (s)≡ (−rεext(s),rεint(s)) by

Uε(·)≡ �uε,rε(s) (·+ 2hε) ,

so that Uε(x) → +∞ as x → rεint(s), Uε(x) → −∞ as x → −rεext(s) and

|Uε(−rε(s))| ≤ Ch− 1

θ−1ε .

Proposition 3.49 For x∈ (ak(s)+δε

loglog,rεint(s)) we have the inequality, whereC > 0 denotes some constant,

Wε(x,s)≤ Uε(x)+Cε− 1

3θ−1 exp

(−π2 ε−ω+1

16(rε(s))2

). (3.117)

Proof As for (3.107), write for x ∈ I trsε (s)∩Jε(s− ε)

ψε(x)= sup{Wε(x,s− ε)−Uε,0} ≥ 0.

We notice that

ψε(ak(s− ε)+ δε

loglog)=ψε(rεint(s))= 0.

Indeed, for the first relation, we argue as in (3.110), whereas for the second, we

have Uε(rεint(s))=�uε,rε(s)(rε(s))=+∞. We extend ψε by 0 outside the interval

I trsε (s)∩Jε(s− ε) and derive, arguing as for (3.109),

|ψε(x)| ≤ Ch− 1

θ−1ε ≤ Cε

− 13θ−1 for x ∈R. (3.118)

We introduce the cylinder �transε (s) ≡ (−rεext(s),r

εint(s)) × (s − ε,s) and the

solution %ε to⎧⎪⎪⎪⎨⎪⎪⎪⎩εω

∂%ε

∂τ− ∂%ε

∂x2= 0 on �trans

ε (s),

�ε(x,s− ε)=ψε(x) for x ∈ (−rεext(s),rεint(s)) and

%ε(−rεext(s),τ)=%ε(rεint(s),τ)= 0 for τ ∈ (s− ε,s),

(3.119)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 133

so that %ε ≥ 0. Arguing as for (3.113), we obtain for τ ∈ (s− ε,s)

|%ε(y,τ)| ≤ Cε− 1

3θ−1 exp

(−π2ε−ω (τ − (s− ε))

16(rε(s))2

). (3.120)

We consider on �transε (s) the function W trans

ε defined by

W transε (y,τ)= Uε(y)+%ε(y,τ).

It follows from Lemma 3.44 that Lε(W transε )≥ 0, that is W trans

ε is a supersolutionfor Lε on �trans

ε (s). Consider next the subset transε (s) of �trans

ε defined by

transε (s)≡ ∪

τ∈(s−ε,s)(ak(τ )+ δ

ε

loglog,rεint(s))×{τ }.

We claim that

W transε ≥Wε on ∂p

transε (s). (3.121)

Indeed, by construction, we have W transε = +∞ on rεint(s) × (s − ε,s) and

W transε (x,s− ε) ≥Wε(x,s− ε) for x ∈ (ak(s− ε)+ δ

ε

loglog,rεint(s)). Finally on

∪τ∈(s−ε,s){ak(τ )+ δε

loglog} × {τ }, the conclusion (3.121) follows from estimate(3.47) of Lemma 3.18. Combining inequality (3.121) with the comparisonprinciple, we are led to

W transε ≥Wε on trans

ε (s). (3.122)

Combining (3.122) with (3.120), we are led to (3.117).

Our next task is to construct a subsolution. To that aim, we rely on thesymmetries of the equation, in particular the invariance x→−x and the almostoddness of the non-linearity. To be more specific, we introduce the operator

Lε(u)≡ εω∂u

∂τ−∂2u

∂x2+λfε(u)=0, with fε(u)=2θu2θ−1

(1−ε

1θ−1 ug(−ε

1θ−1 u)

),

which has the same properties as Lε, and consider the stationary solution�uε,rε(s)

for Lε defined on (−rε(s),rε(s)) by

− ∂2�uε,rε(s)

∂x2+λfε(

�uε,rε(s))= 0,

�uε,rε(s)(−rε(s))=+∞

and�uε,rε(s)(r

ε(s))=−∞,

so that −�uε,rε(s) is a stationary solution to Lε. Consider the function Wε

defined by

Wε(x,τ)=−Wε(−x,τ) (3.123)

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134 Fabrice Bethuel and Didier Smets

and observe that Lε(Wε)= 0. Finally, we define on the interval (−rεint(s),rεext(s))

the function

Vε(x)≡ �uε,rε(s) (2hε− x) ,

so that Vε(x)→−∞ as x→−rεint(s) and Vε(x)→+∞ as x→ rεext(s).

Proposition 3.50 For x ∈ (−rεint(s),ak+1(s)− δε

loglog) we have the inequality

Wε(x,s)≥Vε(x)−Cε− 1

3θ−1 exp

(−π2 ε−ω+1

16(rε(s))2

). (3.124)

Proof We argue as in the proof of Proposition 3.49, replacing Lε by ε, Wε by

Wε, and Uε by Uε =−�uε,rε(s) (·− 2hε(s0)). Inequality (3.124) for Wε is then

obtained by inverting relation (3.123) and from the corresponding estimateon Wε.

Proof of Proposition 3.42 completed Combining (3.117) with (3.124), we areled to

Uε(x)− Aε ≤Wε(x,s)≤Vε(x)+ Aε, (3.125)

where we have set Aε = Cε− 1

3θ−1 exp

(−π2 ε−ω+1

16(rε(s))2

). The proof is then

completed with the same arguments as in the proof of Proposition 3.41

3.5.5 Estimating the Discrepancy

Linear EstimatesThe purpose of this section is to provide the proof of Proposition 3.43. So farProposition 3.41 and Proposition 3.42 provide a good approximation of Wε

on the level of the uniform norm. However, the discrepancy involves also afirst-order derivative, for which we rely on the regularization property of thelinear heat equation. To that aim, set⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

�≡ (−1,1)×[0,1], �1/2 ≡(−1

2,1

2

)×[

3

4,1

],

and more generally for & > 0

�& ≡ (−&,&)×[0,&2], �1/2&≡(−1

2&,

1

2&

)×[

3

4&2,&2

].

The following standard result (see e.g. [2] Lemma A. 7 for a proof) is usefulin our context.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 135

Lemma 3.51 Let u be a smooth real-valued function on �. There exists aconstant C > 0 such that

‖ux‖L∞(�1/2) ≤ C(‖ut − uxx‖L∞(�)+‖u‖L∞(�)).

We deduce from this result the following scaled version.

Lemma 3.52 Let & > 0 and let u be defined on �&. Then we have for someconstant C > 0 independent of &

‖ux‖2L∞(�

1/2& )

≤ C[‖ut − uxx‖L∞(�&)‖u‖L∞(�&)+&−2‖u‖2

L∞(�&)

]. (3.126)

Proof The argument is parallel to the proof of Lemma A.1 in [1], whichcorresponds to its elliptic version. Set h= ut−uxx, let (x0, t0) be given in �

1/2& ,

and let 0 < μ ≤ &

2 be a constant to be determined in the course of the proof.We consider the function

v(y,τ)= u(2μy+ x0, 4μ2(τ − 1)+ t0)

),

so that v is defined on � and satisfies there

vt− vyy =μ2h((2μy+ x0, 4μ2(τ − 1)+ t0)

)on �.

Applying Lemma 3.51 to v, we are led to

|vy(0,1)| ≤ C(μ2‖h

(2μy+ x0, 4μ2(τ − 1)+ t0)

)‖L∞(�)+‖v‖L∞(�)

)≤ C

(μ2‖h‖L∞(�&)+‖u‖L∞(�&)

),

so that, going back to u, we obtain

μ|ux(x0, t0)| ≤ C(μ2‖h|L∞(�&)+‖u‖L∞(�&)

). (3.127)

We distinguish two cases.

Case 1: ‖u‖L∞ ≤ &2‖h‖L∞ . In this case we apply (3.127) with μ=(‖u‖L∞

‖h‖L∞

) 12

.

This yields

|uy(x0, t0)| ≤ 2C‖u‖1/2L∞‖h‖1/2

L∞ .

Case 2: ‖u‖L∞ ≥ &2‖h‖L∞ . In this case we apply (3.127) with μ=&. We obtain

|ux(x0, t0)| ≤ C(&‖h‖L∞(�&)+&−1‖u‖L∞(�&)

)≤ C

(‖h‖1/2

L∞(�&)‖u‖1/2

L∞(�&)+ r−1‖u‖L∞(�&)

).

(3.128)

In both cases, we obtain the desired inequality.

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136 Fabrice Bethuel and Didier Smets

Estimating the Derivative of Wε

Consider the general situation where we are given two functions U and Uε

defined for (x, t) ∈ �& and such that L0(U) = 0 and Lε(Uε) = 0, where s :=ε−ωt, so that, in view of (3.88),

|∂t(U−Uε)− ∂xx(U−Uε)|≤ C

[|U−Uε|(|U|2θ−2+|Uε|2θ−2)+ ε

1θ−1 |Uε|2θ )

]on �&.

We deduce from (3.126) applied to the difference U−Uε that we have (we usethe notation ‖ · ‖ = ‖ · ‖L∞(�&) for simplicity)

‖(U−Uε)x ‖2L∞(�

1/2& )

≤ C ‖U−Uε‖2(‖U‖2θ−2+‖Uε‖2θ−2+&−2

)+Cε

1θ−1 ‖U−Uε‖‖Uε‖2θ .

Similarly, applying (3.126) to U and Uε we obtain

‖(U+Uε)x ‖2L∞(�

1/2& )

≤ C(‖U‖2θ +‖Uε‖2θ +&−2(‖U‖2+‖Uε‖2

)+ ε

1θ−1 (‖Uε‖2θ+1+‖U‖‖Uε‖2θ )),

so that

‖(U2−U2ε

)x ‖2

L∞(�1/2& )

≤ C[‖U−Uε‖2Rε

1(U,Uε)+‖U−Uε‖Rε2(U,Uε)

],

(3.129)where we have set⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Rε1(U,Uε)= (‖U‖2θ−2+‖Uε‖2θ−2+&−2)(‖U‖2θ +‖Uε‖2θ

+&−2(‖U‖2+‖Uε‖2)+ ε1

θ−1 (‖Uε‖2θ+1+‖U‖‖Uε‖2θ )),

Rε2(U,Uε)= ε

1θ−1 ‖Uε‖2θ (‖U‖2θ +‖Uε‖2θ +&−2(‖U‖2+‖Uε‖2)

+ ε1

θ−1 (‖Uε‖2θ+1+‖U‖‖Uε‖2θ )).

We now apply the discussion to our original situation. Thanks to the generalinequality (3.129), we are in position to establish the following.

Proposition 3.53 If (3.105) holds and ε is sufficiently small, then for any s ∈Iε(s0) and every x ∈$ε

k+ 12(s) we have the estimate

|(Wε)2x(x)−λ

− 1(θ−1) (

∨u+rε(s))

2x(x)| ≤ Cε

1θ2 .

Proof We apply inequality (3.129) on the cylinder �& with &= 116 d∗min(s0) and

to the functions U(y,τ)=Wε(y+ x,εωτ + s) and Uε(y,τ)= ∨U+rε(s)(y+ x). We

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 137

first estimate R1 and R2. Since we have

|U(y,τ)|+ |Uε| ≤ Cd∗min(s0)− 1

θ−1 , for (y,τ) ∈�&,

it follows that

Rε1 (U,Uε)≤ d∗min(s0)

−4− 2θ−1 and Rε

2 (U,Uε)≤ ε1

θ−1 d∗min(s0)−4− 4

θ−1 .

Invoking inequality (3.126) of Lemma 3.52, and combining it with (A.7) andthe conclusion of Proposition 3.41, we derive the conclusion using a crudelower bound for the power of ε.

Similarly we obtain the following.

Proposition 3.54 If (3.94) holds and ε is sufficiently small, then for any s ∈Iε(s0) and every x ∈$ε

k+ 12(s) we have the estimate

|(Wε)2x(x)−λ

− 1(θ−1) (

�urε(s))

2x(x)| ≤ Cε

1θ2 . (3.130)

Proof of Proposition 3.43 completed The proof of Proposition 3.43 follows bycombining Proposition 3.53 in the attractive case and Proposition 3.54 in therepulsive case with the estimates (A.10).

3.6 The Motion Law for Prepared Data

In this section, we present the proof of Proposition 3.19.

Proof of Proposition 3.19,Step 1 First, by definition of L0, assumption (H1) and estimate (3.7), it

follows that for fixed L ≥ L0, and for all ε sufficiently small (depending onlyon L),

Dε(s)∩ I4L ⊂ IL ∀0≤ s≤ S,

so that (CL,S) holds.Step 2 Since the assumptions of Corollary 3.33 are met with the choice s0=0

and L= L0, we obtain that for ε sufficiently small, WPL0ε (δ

ε

loglog,s) holds and

dε,Lmin(s) ≥ 1

2 d∗min(0) = 12 min{a0

k+1 − a0k , k = 1, . . . ,�0 − 1}, for all s ∈ Iε(0), as

well as the identities J(s)= J(0), σi(k± 12 )(s)=σi(k± 1

2 )(0) and †k(s)= †k(0), for

any k ∈ J(0).Step 3 We claim that for any s1 ≤ s2 ∈ I∗(0), we have

limsupε→0

(dissipLε (s1,s2))= 0. (3.131)

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138 Fabrice Bethuel and Didier Smets

Indeed, let L≥ L0 be arbitrary. We know from Step 1 that (CL,S) holds providedε is sufficiently small. By Proposition 3.13, for ε sufficiently small there existtwo times sε1 and sε2 such that 0 < sε1 ≤ s1 ≤ s2 ≤ sε2, |si − sεi | ≤ εω+1L andWPL

ε (δε

log,sεi ) holds for i = 1,2. From the second step and assumption (H1)

we infer that ELε (s

ε1) = E

L0ε (sε1) = E

L0ε (sε2) = EL

ε (sε2). Invoking Corollary 3.33

we are therefore led to the inequality

dissipL0ε (s1,s2)≤ dissipL

ε (sε1,sε2)≤CM0

δε

log

+CL−(ω+2)(s2−s1+2εω+1L).

Since L≥ L0 was arbitrary the conclusion (3.131) follows by letting first ε→ 0and then L→∞.

Step 4 In view of Corollary 3.17 we may find a subsequence (εn)∈N tendingto 0 such that the functions aεn

k (·)n∈N converge uniformly as n→ 0 on compactsubsets on I (0). Consider the cylinder

C∗k+ 1

2≡ [a0

k +1

4d∗min(0), a0

k+1−1

4d∗min(0)]× I∗(0).

It follows from Step 2 and Proposition 3.43 that

εn−ωξεn(vεn)→ λ

− 12(θ−1)

i(k+ 12 )

rk+ 12(s)−(ω+1)γk as εn → 0, for k= 1, . . .�0− 1

(3.132)uniformly on every compact subset of C∗

k+ 12, where γk is defined in (3.97) and

where rk+ 12(s)= ak+1(s)− ak(s).

Step 5 As in (3.84), we consider a test function χ ≡ χk with the followingproperties:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩χ has compact support in [a0

k −1

3d∗min(0),a

0k +

1

3d∗min(0)],

χ is affine on the interval [a0k −

1

4d∗min(0),a

0k +

1

4d∗min(0)], with χ ′ = 1 there

‖χ‖L∞(R) ≤ Cd∗min(0),‖χ ′‖L∞(R) ≤ C and ‖χ ′′‖L∞(R) ≤ Cd∗min(0)−1.

It follows from the definition of χk that χ ′′k = 0 outside a, and so isε−ωξε(vε)χ

′′k . It follows from (3.132) that for s1 ≤ s2 ∈ I∗(0),

Fεn(s1,s2,χk)→(∫ a0

k− 14 d∗min(0)

a0k− 1

3 d∗min(0)χ ′′(x)dx

)(∫ s2

s1

λ− 1

2(θ−1)

k− 12

rk− 12(s)−

1θ−1 γk− 1

2ds

)

+(∫ a0

k+ 13 d∗min

a0k+ 1

4 d∗min(0)χ ′′(x)dx

)(∫ s2

s1

λ− 1

2(θ−1)

k+ 12

rk+ 12(s)−

1θ−1 γk+ 1

2ds

)(3.133)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 139

as εn → 0. Since the above two integrals containing χ ′′ are identically equal to1 and −1 respectively, we finally deduce from (3.42) combined with (3.131)and (3.133), letting εn tend to 0, that for s1 ≤ s2 ∈ I∗(0) we have

[ak(s1)− ak(s2)]Si(k)

=∫ s2

s1

(λ− 1

2(θ−1)

i(k− 12 )

rk− 12(s)−(ω+1)γk− 1

2−λ

− 12(θ−1)

i(k+ 12 )

rk+ 12(s)−(ω+1)γk+ 1

2

)ds,

which is nothing other than the integral formulation of the system (S). Sincethe latter possesses a unique solution, the limiting points are unique andtherefore convergence of the aε

k for s ∈ I∗(s) holds for the full family (vε)ε>0.Step 6 We use an elementary continuation method to extend the convergence

from I∗(0) to the full interval (0,S). Indeed, as long as d∗min(s) remains boundedfrom below by a strictly positive constant (which holds, by definition of Smax,as long as s < S) we may take s as a new origin of times (Step 2 yieldsWPL0

ε (α1ε,s)) and use Steps 1 to 5 to extend the stated convergence past s.The proof is here completed.

3.7 Clearing-out

The purpose of this section is to provide a proof of Proposition 3.20. We areled to consider the situation where for some length L≥ 0 we have

Dε(0)∩ [−5L,5L] ⊂ [−κ0L,κ0L] (3.134)

for some (small) constant κ0 ≤ 12 . It follows from Theorem 3.3 that

CL,S holds, where S= ρ0

(L

2

)ω+2

,

and that for s ∈ [0,S] we have

Dε(s)∩ [−4L,4L] ⊂ [−κ0(s)L,κ0(s)L], (3.135)

where

κ0(s) := κ0+(

s

ρ0

) 1ω+2 1

L. (3.136)

For those times s ∈ [0,S] for which the preparedness assumption WPILε(α1ε,s)

holds, we set {dε,+

min(s)=min{|aεk+1(s)− aε

k(s)|, k ∈ J+(s)}, and

dε,−min(s)=min{|aε

k+1(s)− aεk(s)|, k ∈ J−(s)},

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140 Fabrice Bethuel and Didier Smets

with J±(s) = {k ∈ {1, . . . ,�(s) − 1}, s.t.εk+ 12= ∓1}, so that dε

min(s) =min{dε,+

min(s),dε,−min(s)}, with the convention that the quantities are equal to L

in the case when the defining set is empty.At first, we will focus on the case J−(s) = ∅. The following result provides

an upper bound in terms of dε,−min(s) for a dissipation time for the quantized

function ELε . This phenomenon is related to the cancellation of a front with its

anti-front, and is the main building block for the proof of Proposition 3.20.

Proposition 3.55 There exist κ1 > 0, α3 > 0, and Kcol > 0, all depending ononly V and M0, with the following properties. If (3.134) holds, if s0 ∈ (εωL2,S)is such that κ0(s0) ≤ κ1, WPL

ε(α3ε,s0) holds, J−(s0) is non-empty, and s0 +Kcold

ε,−min(s0)

ω+2 < S, then there exists some time T ε,+col (s0) ∈ (s0,S) such that

WPILε(α3ε,T ε,+

col (s0)) holds,

ELε (T

ε,+col (s0))≤ EL

ε (s0)−μ1, (3.137)

where μ1 is a constant introduced in Lemma 3.34, and

T ε,+col (s0)− s0 ≤Kcol

(dε,−

min(s0))ω+2

. (3.138)

We postpone the proof of Proposition 3.55 until after Section 3.7.1, wherewe will analyze in more detail the attractive and repulsive forces at work atthe ε level. We will then prove Proposition 3.55 in Section 3.7.2, and finallyProposition 3.20 in Section 3.7.3.

3.7.1 Attractive and Repulsive Forces at the ε Level

In this subsection we consider the general situation where CL,S holds, for somelength L≥ 0 and some S > 0.

In order to deal with the attractive and repulsive forces underlying annihila-tions or splittings, we set

Fk+ 12(s)=−ω−1Bk+ 1

2

(aε

k+1(s)− aεk(s)

)−ω

and consider the positive functionals

F εrep(s)=

∑k∈J+(s)

Fk+ 12(s), F ε

att(s)=−∑

k∈J−(s)

Fk+ 12(s), (3.139)

with the convention that the quantity is equal to +∞ in the case when thedefining set is empty. For some constants 0 < κ2 ≤ κ3 depending on only M0

and V we have ⎧⎨⎩ κ2F εatt(s)

− 1ω ≤ dε,−

min(s)≤ κ3F εatt(s)

− 1ω ,

κ2F εrep(s)

− 1ω ≤ dε,+

min(s)≤ κ3F εrep(s)

− 1ω .

(3.140)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 141

Let s0 ∈ [εωL2,S] be such that

WPLε(α2ε,s0) holds and dε,L

min(s0)≥ 16q1(α2)ε. (3.141)

We consider as in Corollary 3.38 the stopping time

T ε0 (α2,s0)=max

{s0+ ε2+ω ≤ s≤ S s.t. dε,L

min(s′)≥ 8q1(α2)ε

∀s′ ∈ [s0+ εω+2,s]} ,

and for simplicity we will write T ε0 (s0) ≡ T ε

0 (α2,s0). In view of (3.141) andthe statement of Corollary 3.38,

WPLε(α1ε,s) holds ∀ s ∈ Iε

0(s0)≡ [s0+ ε2+ω,T ε0 (s0)].

The functionals F εatt and F ε

rep are in particular well defined and continuouson the interval of time Iε

0(s0) with J+(s) = J+(s0) and J−(s) = J−(s0)

for all s in that interval. Note that the attractive forces are dominant whendε,−

min(s) ≤ dε,+min(s) and by contrast the repulsive forces are dominant when

dε,+min(s)≤ dε,−

min(s).We first focus on the attractive case, and for s ∈ Iε

0(s0), we introduce thenew stopping times

T ε1 (s)= inf{s≤ s′ ≤ T ε

0 (s0), Fatt(s′)≥ υω

1 Fatt(s) or s′ = T ε0 (s0)},

where υ1 = 10κ23κ

−22 , so that υ1 > 10 and T ε

1 (s)≤ T ε0 (s0). In view of (3.140),

we have1

10

(κ2

κ3

)3

dε,−min(s)≤ dε,−

min(T ε1 (s)), (3.142)

and if T ε1 (s) < T ε

0 (s0) then

dε,−min(T ε

1 (s))≤ 1

10

κ2

κ3dε,−

min(s)≤1

10dε,−

min(s). (3.143)

The next result provides an upper bound on T ε1 (s) − s. Central to our

argument is Proposition 3.19, which we use combined with various argumentsby contradiction. We have the following proposition.

Proposition 3.56 There exists β0 > 0, depending on only V and M0, with thefollowing properties. If J−(s0) = ∅, s ∈ Iε

0(s0) and

β0 ε ≤ dε,−min(s)≤ dε,+

min(s), (3.144)

then we haveT ε

1 (s)− s≤K0(dε,−

min(s))ω+2

, (3.145)

where K0 is defined in (3.19), and moreover if T ε1 (s) < S then

dε,−min(T ε

1 (s))≤ dε,+min(T ε

1 (s)). (3.146)

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142 Fabrice Bethuel and Didier Smets

Proof Up to a translation of times we may first assume that s= 0, which easesthe notations somewhat. We then argue by contradiction and assume that theconclusion is false, that is, there does not exist any such constant β0, no matterhow large it is chosen, such that the conclusion holds. Taking β0 = n, thismeans that given any n ∈ N∗ there exists some 0 < εn ≤ 1, a solution vn to(PGL)εn such that Eεn(vn)≤M0, such that WP

L0εn (α1εn,0) holds, such that

nεn ≤ dεn,−min (0)= dεn

min(0)≤ dεn,+min (0), (3.147)

and such that one of the conclusions fails, that is such that either

T n1 ≡ T εn

1 (0) >K0 (dε,−min(0))

ω+2, (3.148)

ordε,−

min(T n1 ) > dε,+

min(T n1 ). (3.149)

Setting Sn0 =K0(d

εn,−min (0))ω+2, relation (3.148) may be rephrased as

Fnatt(s)≤ υω

1 Fnatt(0) and dεn

min(s)≥ 8q1(α2)εn for any s ∈ [0,Sn0], (3.150)

where the superscripts n refer to the corresponding functionals computed forthe map vn. Passing possibly to a subsequence, we may therefore assumethat at least one of the properties (3.150) or (3.149) holds for any n ∈ N∗.Also, passing possibly to a further subsequence, we may assume that the totalnumber of fronts of vn(0) inside [−L,L] is constant, equal to a number �,denote an

1(s), . . . ,an�(s) the corresponding front points, for s ∈ [0,T n

1 ], and setd−n (s)= dεn,−

min (s),d+n (s)= dεn,+min (s),dn(s)= dεn

min(s).In order to obtain a contradiction we shall make use of the scale invariance

of the equation: if vε is a solution to (PGL)ε then the map vε(x, t)= vε(rx,r2t)is a solution to (PGL)ε with ε = r−1ε. As scaling factor rn, we choose rn =dεn,−

min (0)≥ nεn and set

vn(x, t)= vn(rnx,r2nt), vn(x,τ)= vn(x, ε−ω

n τ), (3.151)

so that vn is a solution to (PGL)εn satisfying WPLnεn(α2εn,0) with Ln = r−1

n Land

εn = (rn)−1εn = (dεn,−

min (0))−1εn ≤ 1

n, hence we have εn → 0 as n→+∞.

The points an1(s) = r−1

n an1(r

−(2+ω)n s), . . . , an

�(s) = r−1n an

�(r−(2+ω)n s) are the front

points of vn. Let d−n , d+n , dn be the quantities corresponding to dε,−min,dε,+

min,dεmin

for vn, so that

d−n (s)= r−1n d−n (r

−(2+ω)n s), d+n (s)= r−1

n d+n (r−(2+ω)n s),

and dn(s)= r−1n dn(r

−(2+ω)n s),

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 143

and notice that d−n (0) = dn(0) = 1. We next distinguish the following twocomplementary cases.

Case 1: (3.150) holds for all n ∈ N∗. It follows from assumption (3.150) thatWPLn

εn(α1εn,τ) holds for every τ ∈ (0, Sn

1), where Sn1 = r−(2+ω)

n Sn0 = K0. Let

k0 ∈ {1, . . . ,�} be such that

ank0+1(0)− an

k0(0)= dεn,−

min (0).

Using a translation if necessary, we may also assume that ank0(0)= 0 so that

ank0+1(0)= dεn,−

min (0). We denote by Fnatt the functional F ε

att computed for thefront points of vn, so that

Fnatt(r

−(2+ω)n s)= r(2+ω)

n Fnatt(s).

By construction we have

ank0(0)= 0 and an

k0+1(0)= 1= d−n (0). (3.152)

Since εn → 0 as n→∞, we may implement part of the already-establishedasymptotic analysis for (PGL)ε on the sequence (vn)n∈N. First, passing pos-sibly to a subsequence, we may assume that for some subset J ⊂ J(0) thepoints {ak(0)}k∈J converge to some finite limits {a0

k}k∈J , whereas the pointswith indices in J(0) \ J diverge either to +∞ or to −∞. We choose L ≥ 1 sothat

∪k∈J{a0

k} ⊂ [−L

2,L

2]. (3.153)

In view of (3.152), we have ak0(0) = 0, ak0+1 = 1 and inf{|ak+1(0)− ak(0)|,k ∈ J} = 1. We are then in position to apply the convergence result statedin Proposition 3.19 to the sequence (vn(·))n∈N. It states that the front points(an

k(τ ))k∈J0 which do not escape to infinity converge to the solution (ak(·))k∈J

of the ordinary differential equation (S) supplemented with the correspondinginitial values (ak(0))k∈J , uniformly in time on every compact subset of(0, Smax), where Smax denotes the maximal time of existence for the solution.In particular, we have⎧⎨⎩ d−n (τ )→ d−a (τ ), uniformly on every compact subset of (0, Smax),

limsupn→+∞

Fnatt(τ )≥ Fatt(a(τ )) for every τ ∈ (0, Smax),

the presence of the limsup being related to the fact that some points mightescape to infinity so that the limiting values of the functionals are possiblysmaller. We use next the properties of the differential equation (S) established

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144 Fabrice Bethuel and Didier Smets

in Appendix B. We first invoke Proposition 3.7 which asserts that Smax ≤ K0

and that

Fatt(a(τ ))→+∞ as τ → Smax.

Hence, there exists some τ1 ∈ (0, Smax) ⊂ (0,K0) such that, if n is sufficientlylarge, then

Fnatt(τ1) > υω

1 Fnatt(0).

Going back to the original time scale, this yields Fnatt(r

ω+2n τ1) > υω

1 Fnatt(0).

Since rω+2n τ1 ∈ (0,r2+ω

n K0)= (0,Sn0) this contradicts (3.150) and completes the

proof in Case 1.

Case 2: (3.149) holds for all n ∈ N∗. We consider an arbitrary index j ∈ J+.As above, translating the origin, we may assume without loss of generalitythat an

j (0)= 0. We also define the map vn as in Case 1, according to the samescaling as described in (3.151), the only difference being that the origin hasbeen shifted differently. With similar notations, we have

anj (0)= 0 and an

j+1(0)≥ 1= d−n (0).

Passing possibly to a further subsequence, we may assume that the front pointsat time 0 converge to some limits in R denoted ak(0). We are hence again inthe position to apply the convergence result of Proposition 3.19, so that thefront points (an

k(s))k∈Jj which do not escape to infinity converge to the solution(ak(·))k∈Jj of the ordinary differential equation (S) supplemented with thecorresponding initial values (ak(0))k∈Jj , uniformly in time on every compact

subset of (0, S′max), where S′max denotes the (new) maximal time of existencefor the solution. It follows from assumption (3.172), Theorem 3.2 and scalingthat 0 < T1 ≡ liminf T n

1 ≤ S′max. We claim that, for any τ ∈ (0, T1), and forsufficiently large n, we have

|anj (τ )− an

j+1(τ )| ≥κ2

2κ3. (3.154)

This is actually a property of the differential equation (S). We have indeed, inview of Proposition 3.65, 0 < Frep(a(τ ))≤ Frep(a(0)), so that it follows from(3.140) that

|aj(τ )− aj+1(τ )| ≥ κ2

κ3,

which yields (3.154) taking the convergence into account. Since (3.154) holdsfor any j, we deduce that

dεn,+min (T n

1 )≥ κ2

2κ3dεn,−

min (0)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 145

and therefore by (3.149) we have

dεnmin(T n

1 )= dεn,−min (T n

1 )≥ dεn,+min (T n

1 )≥ κ2

2κ3dεn,−

min (0)≥ κ2

2κ3nε. (3.155)

For n sufficiently large, this implies that T n1 < T n

0 , and therefore from (3.143)we have

dεn,−min (T n

1 )≤ 1

10

κ2

κ3dεn,−

min (0),

which is in contradiction with (3.155).

We turn now to the case where dε,+min(s) ≤ dε,−

min(s). In order to handle therepulsive forces at work, for s ∈ Iε

0(s0) we introduce the new stopping times

T ε2 (s)= inf{s≤ s′ ≤ T ε

0 (s0), F εrep(s

′)≤ υω2 F ε

rep(s) or s′ = T ε0 (s0)},

where υ2 = κ22

10κ23

, so that υ2 < 1. Notice that, in view of (3.140), we have, if

T ε2 (s) < T ε

0 (s0),

dε,+min(T ε

2 (s))≥ υ−12

κ2

κ3dε,+

min(s)≥ 10dε,+min(s). (3.156)

With S1 introduced in Proposition 3.7, we set

K1 = S−ω1

(2κ3

κ2υ2

)ω+2

. (3.157)

Proposition 3.57 There exists β1 > 0, depending on only V and M0, with thefollowing properties. If J+(s0) = ∅, s ∈ Iε

0(s0) and

β1 ε ≤ dε,+min(s)≤ dε,−

min(s), (3.158)

then we haveT ε

2 (s)− s≤K1(dε,+

min(s))ω+2

, (3.159)

and if T ε2 (s) < S then T ε

2 (s) < T ε0 (s0) and for any s ∈ [s,T ε

2 (s)], we have

dεmin(s)≥

1

2S2dε,+

min(s) (3.160)

and

F εatt (s)

− 1ω ≤F ε

att(s)− 1

ω + 1

κ3(dε,+

min(s)), (3.161)

where S2 is defined in Proposition 3.7 and κ3 is defined in (3.140).

Proof The argument shares strong similarities with the proof of Proposi-tion 3.56; we therefore just sketch its main points, in particular relyingimplicitly on the notations introduced there, as far as this is possible. By

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146 Fabrice Bethuel and Didier Smets

translation in time we also assume that s = 0 and argue by contradictionassuming that for any n ∈ N∗ there exists some 0 < εn ≤ 1, a solution vn to(PGL)εn such that Eεn(vn) ≤M0,WPL

εn(α1εn,0) holds, such that nεn ≤ d+n (0),

and such that either we have for any s ∈ (0,Sn1), where Sn

1 = K1d+n (0)ω+2,dn(s)≥ 8q(α2)εn and

κω3 (d

+n (s

′))−ω ≥ Fnrep(s

′)≥ υω2 Fn

rep(0)≥ υω2 κ

ω2 (d

+n (0))

−ω (3.162)

or there is some τn ∈ (0,T n2 ) such that

dε,+min(τn) <

1

2S2dε,+

min(s) (3.163)

or

Fnatt (τn)

− 1ω <Fn

att(0)− 1

ω + 1

32κ3(d+n (s)). (3.164)

As in (3.151), but with a different scaling rn, we set

rn = dεn,+min (0)≥ nεn, vn(x, t)= vn(rnx,r2

nt), and vn(x,s)= vn(x, ε−ωn s). (3.165)

We verify that vn is a solution to (PGL)εn with εn = (rn)−1εn → 0 as n→∞

and that the points ank(τ ) = r−1

n ank(r

−(2+ω)n τ) for k ∈ J are the front points of

vn. We distinguish three cases, which together were all possible cases, takingsubsequences if necessary.

Case 1: (3.162) holds, for any n ∈ N. It follows that WPLnεn(α1εn,τ) holds

for every τ ∈ (0, Sn1 ), where Sn

1 = r−(2+ω)n Sn

1 = K1. Let j be an arbitraryindex in J+. Translating the origin if necessary, we may assume thatan

j (0) = 0 so that anj+1(0) ≥ d+n (0) ≥ nεn and hence an

j+1(0) − anj (0) ≥ 1.

Since εn → 0 as n→∞, we may implement part of the already-establishedasymptotic analysis for (PGL)ε on the sequence (vn)n∈N. First, passingpossibly to a subsequence, we may assume that for some subset J ⊂ J(0)the points {ak(0)}k∈J converge to some finite limits {a0

k}k∈J , whereas the pointswith indices in J(0) \ J diverge either to +∞ or to −∞. We choose L ≥ 1 sothat (3.153) holds. It follows from Proposition 3.19 that for τ ∈ (0,K1), wehave

|anj+1(τ )− an

j (τ )|→ |aj+1(τ )− aj(τ )| ≥(S1τ +S2d

+a (0)

ω+2) 1ω+2

= (S1τ +S2)1

ω+2

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 147

as n → ∞, where the last inequality is a consequence of Proposition 3.7.Taking the infimum over J+, we obtain, for n sufficiently large,

d+n (τ )= infj∈J+

|anj+1(τ )− an

j (τ )| ≥1

2(S1τ +S2)

1ω+2 ≥ 1

2(S1τ)

1ω+2 ,∀τ ∈ (0,K1).

(3.166)On the other hand, going back to (3.162), with the same notation as inProposition 3.56, we are led to the inequality

d+n (τ )≤ κ3κ−12 υ−1

2 for τ ∈ (0,K1). (3.167)

In view of our choice (3.157) of K1, relations (3.166) and (3.167) arecontradictory for τ close to K1, thus yielding a contradiction in Case 1.

Case 2: (3.162) does not hold, but (3.163) holds for any n ∈ N. The argumentis almost identical, we conclude again thanks to (3.166) but keeping S2 insteadof S1τ in its last inequality.

Case 3: (3.162) does not hold but (3.164) holds for any n ∈N. As in the proofof Proposition 3.56, we conclude that 0 < T2 ≡ liminfn→+∞ T n

2 . This situationis slightly more delicate than the ones analyzed so far, and we have also totrack the fronts escaping possibly to infinity. Up to a subsequence, we may

assume that the set J is decomposed as a disjoint union of clusters J = q∪i=1

Jp,

where each of the sets Jp is an ordered set of mp+1 consecutive points, that isJp = {kp,kp+1, . . .kp+mp} and such that the two following properties hold.

• There exists a constant C > 0 independent of n such that

|ankp(0)− akp+r(0)| ≤ C for any p ∈ {1, . . . ,q} and any r ∈ {kp, . . . ,mp}.

(3.168)• For 1≤ p1 < p2 ≤ q, we have an

kp2− an

kp1→+∞.

For a given p ∈ {1, . . . ,q}, translating the origin if necessary, we may assumethat an

kp(0) = 0, and passing possibly to a further subsequence, that the front

points at time 0 converge as n → +∞ to some limits denoted ap,k(0), fork ∈ {kp, . . . ,kp + mp}. Notice that, as an effect of the scaling, all other frontpoints diverge to infinity, in the chosen frame. We now apply Proposition 3.19to this cluster of points: it yields uniform convergence, for k ∈ {kp, . . . ,kp+mp}of the front points an

k(·) to the solution ap,k(·) of the differential equation (S)supplemented with the initial time conditions ap,k(0) defined above. If Fp

att

denotes the functional Fatt defined in (3.139) restricted to the points of thecluster Jp, we have in view of (B.17)

d

dτFp

att(τ )≥ 0, for any p= 1, . . . ,q, for any τ ∈ (0, T2).

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148 Fabrice Bethuel and Didier Smets

On the other hand, since the mutual distances between the distinct clustersdiverge towards infinity, and hence their mutual interaction energies tend tozero, one obtains, in view of the uniform convergence for each separate cluster,that

limn→+∞F

natt(τ )=

q∑p=1

Fpatt(τ )≥

q∑p=1

Fpatt(0)= lim

n→+∞Fnatt(0), for τ ∈ (0, T2).

Therefore, for n sufficiently large we are led to

Fnatt(T2)

− 1ω ≤ Fn

att(0)− 1

ω + 1

2κ3.

Scaling back to the original variables, this contradicts (3.164) and hencecompletes the proof.

From Propositions 3.56 and 3.57 we obtain the following.

Proposition 3.58 There exists K2 > 0, depending on only V and M0, with thefollowing properties. Assume that J−(s0) = ∅ and that s ∈ Iε

0(s0) satisfies

dε,Lmin(s)≥max(β0,β1)ε, and s+K2dε,−

min(s)ω+2 < S. (3.169)

Then there exists some time T −col(s) ∈ Iε

0(s0) such that

T ε,−col (s)− s≤K2dε,−

min(s)ω+2 (3.170)

and

dε,Lmin(T

ε,−col (s))≤max(β0,8q1(α2))ε. (3.171)

Proof We distinguish two cases.

Case I:

dε,Lmin(s)= dε,−

min(s)≤ dε,+min(s). (3.172)

In that case we will make use of Proposition 3.56 in an iterative argument. Inview of (3.169), we are in a position to invoke Proposition 3.56 at time s = sand set s1 = T ε

1 (s), so that in particular

s1− s≤K0dε,−min(s)

ω+2 and dε,−min(s1)≤ dε,+

min(s1). (3.173)

Notice that by (3.169) and (3.173) we have s1 < S.We distinguish two sub-cases.

Case I.1: s1=T ε0 (s0) or dε,−

min(s1)<β0ε. In that case, we simply set T ε,−col (s)= s1

and we are done if we require K2≥K0, by (3.173) and the definition of T ε0 (s0).

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 149

Case I.2: s1 < T ε0 (s0) and dε,−

min(s1) ≥ β0ε. In that case, we may applyProposition 3.56 at time s= s1 and set s2 = T ε

1 (s1), so that in particular

s2− s1 ≤K0dε,−min(s1)

ω+2 and dε,−min(s2)≤ dε,+

min(s2). (3.174)

Moreover, since in that case s1 = T ε1 (s) < T ε

0 (s0), it follows from (3.143) that

dε,−min(s1)≤ 1

10dε,−

min(s), (3.175)

and therefore from (3.174) we actually have

s2− s1 ≤K010−(ω+2)dε,−min(s)

ω+2 and dε,−min(s2)≤ dε,+

min(s2). (3.176)

We then iterate the process until we fall into Case I.1. If we have not reachedthat stage up to step m, then thanks to Proposition 3.56 applied at time s= sm

we obtain, with sm+1 := T ε1 (sm),

sm+1− sm ≤K0dε,−min(sm)

ω+2 and dε,−min(sm+1)≤ dε,+

min(sm+1). (3.177)

Moreover, since Case I.1 was not reached before step m, we have sp =T ε

1 (sp−1) < T ε0 (s0) for all p≤m, so that repeated use of (3.143) yields

dε,−min(sp)≤

(1

10

)p

dε,−min(s), ∀p≤m. (3.178)

From (3.177) we thus also have

sp+1− sp ≤K010−p(ω+2)dε,−min(s)

ω+2, ∀p≤m, (3.179)

and therefore by summation

sm+1− s≤K0(

m∑p=0

10−p(ω+2))dε,−min(s)

ω+2, (3.180)

so that in particular from (3.169) it holds that sm+1 < S if we choose K2 ≥ 2K0.It follows from (3.178) that Case I.1 is necessarily reached in a finite numberof steps, thus defining T ε,−

col (s), and from (3.180) we obtain the upper bound

T ε,−col (s)− s≤K0(

∞∑p=0

10−p(ω+2))dε,−min(s)

ω+2 ≤ 2K0dε,−min(s)

ω+2, (3.181)

from which (3.170) follows.

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150 Fabrice Bethuel and Didier Smets

Case II:dε,L

min(s)= dε,+min(s) < dε,−

min(s). (3.182)

Note that this implies that J+(s0) = ∅. We will show that Case II can be reducedto Case I after some controlled interval of time necessary for the repulsiveforces to push dε,+

min above dε,−min. More precisely, we define the stopping time

T εcros(s)= inf{T ε

0 (s)≥ s′ ≥ s, dε,−min(s

′)≤ dε,+min(s

′)}.As in Case I, we implement an iterative argument, but based this timeon Proposition 3.57. In view of (3.182) and (3.169), we may applyProposition 3.57 at time s= s and set s1 = T ε

2 (s), so that in particular

s1− s≤K1dε,+min(s)

ω+2 ≤K1dε,−min(s)

ω+2. (3.183)

Notice that by (3.169) and (3.183) we have s1 < S and therefore dε,+min(s1) ≥

10dε,+min(s)≥ β1, and by (3.161)

F εatt (s1)

− 1ω ≤F ε

att(s)− 1

ω + 1

κ3dε,+

min(s)

≤F εatt(s)

− 1ω + 1

10κ3dε,+

min(s1).(3.184)

We distinguish two sub-cases.

Case II.1: s1 ≥ T εcros(s). In that case we proceed to Case I which we

will apply starting at s1 instead of s and we set T ε,−col (s) := T ε,−

col (s1). Since,combining the first inequality of (3.184) with (3.140), we deduce that

dε,−min(s1)≤ κ3κ

−12 dε,−

min(s)+ dε,+min(s)≤

(κ3κ

−12 + 1

)dε,−

min(s), (3.185)

the equivalent of (3.181) becomes

T ε,−col (s1)− s1 ≤K0(

∞∑p=0

10−p(ω+2))dε,−min(s1)

ω+2

≤ 2K0dε,−min(s1)

ω+2

≤ 2K0(κ3κ

−12 + 1

)ω+2dε,−

min(s)ω+2,

(3.186)

and therefore it follows from (3.183) that

T ε,−col (s)−s≤T ε,−

col (s1)−s1+(s1−s)≤(K1+ 2K0

(κ3κ

−12 + 1

)ω+2)

dε,−min(s)

ω+2,

(3.187)and (3.170) follows if K2 ≥K1+ 2K0

(κ3κ

−12 + 1

)ω+2.

Case II.2: s1 < T εcros(s). In that case we proceed to construct s2 = T ε

2 (s1).Notice that combining the second inequality of (3.184) with (3.140), we

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 151

deduce that

dε,−min(s1)≤ κ3κ

−12 dε,−

min(s)+1

5dε,+

min(s1)≤ κ3κ2−1dε,−

min(s)+1

5dε,−

min(s1), (3.188)

so that

dε,+min(s1)≤ dε,−

min(s1)≤ 5

4κ3κ

−12 dε,−

min(s). (3.189)

We now explain the iterative argument. Assume that for some m ≥ 1 we havealready constructed s1, . . . ,sm, such that for 2≤ p≤m

sp < S, β1ε ≤ dε,+min(sp)≤ dε,−

min(sp), sp = T ε2 (sp−1).

First, repeated use of (3.156) yields

dε,+min(sp)≥ 10pdε,+

min(s), ∀1≤ p≤m, (3.190)

and actually

dε,+min(sp)≥ 10p−qdε,+

min(s), ∀1≤ q≤ p≤m. (3.191)

Hence, by repeated use of (3.161), we obtain

F εatt (sm)

− 1ω ≤F ε

att(s)− 1

ω + 1

κ3

⎛⎝dε,+min(s)+

m−1∑p=1

dε,+min(sp)

⎞⎠≤F ε

att(s)− 1

ω + 1

κ3

m−1∑p=0

10−pdε,+min(sm−1)

≤F εatt(s)

− 1ω + 2

κ3dε,+

min(sm−1)

≤F εatt(s)

− 1ω + 1

5κ3dε,+

min(sm).

Combining the latter with (3.140), we deduce that

dε,−min(sm)≤ κ3κ

−12 dε,−

min(s)+1

5dε,+

min(sm)≤ κ3κ2−1dε,−

min(s)+1

5dε,−

min(sm),

so that

dε,+min(sm)≤ dε,−

min(sm)≤ 5

4κ3κ

−12 dε,−

min(s). (3.192)

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152 Fabrice Bethuel and Didier Smets

Let sm+1 := T ε2 (sm). Then by (3.159) and (3.190)

sm+1− s≤K1

⎛⎝dε,+min(s)

ω+2+m∑

p=1

dε,+min(sp)

ω+2

⎞⎠≤K1

m−1∑p=0

10−(ω+2)(m−p)dε,+min(sm)

ω+2

≤ 2K1dε,+min(sm)

ω+2.

(3.193)

Combining (3.193) with (3.192), we are led to

sm+1− s≤ 2

(5κ3

4κ2

)ω+2

K1dε,−min(s)

ω+2

and therefore by (3.169) we have sm+1 < S.Combining (3.192) with (3.190), we obtain

0≤ dε,−min(sm)− dε,+

min(sm)≤ κ3

κ2dε,−

min(s)− 10m(dε,+min(s)),

and therefore necessarily

m≤ log10

(κ3dε,−

min(s)

κ2dε,+min(s)

).

It follows that the number m0 = sup{m ∈N∗,dε,−min(sm)≥ dε,+

min(sm)} is finite, andat that stage we proceed to Case I as in Case II.1 above, and the conclusionfollows likewise, replacing (3.185) by

dε,+min(sm0+1)≤ 9

4κ3κ

−12 dε,−

min(s)

which is obtained by combining

dε,+min(sm0+1)≤ κ3κ

−12 dε,−

min(s)+ dε,+min(sm0),

with

dε,+min(sm0)≤ dε,−

min(sm0)≤5

4κ3κ

−12 dε,−

min(s).

3.7.2 Proof of Proposition 3.55

We will fix the value of the constants κ1, α3 and Kcol in the course of the proof.Let s0 be as in the statement. We first require that

α3 ≥ α2 and that α3 ≥ 16q1(α2),

so that assumption WPLε(α3,s0) implies assumption 3.141 of Subsection 3.7.1.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 153

Next, we set s = s0 + εω+2 and we wish to make sure that the assumptionsof Proposition 3.55 are satisfied at time s. In view of the upper bound (3.7) onthe velocity of the front set, we deduce that

dε,Lmin(s)≥ dε,L

min(s0)−Cρ1

ω+20 ε ≥ α3ε−Cρ

1ω+20 ε ≥max(β0,β1)ε

provided we choose α3 sufficiently large. Also,

1

2dε,−

min(s0)≤ dε,Lmin(s0)−Cρ

1ω+20 ε ≤ dε,L

min(s)≤ dε,Lmin(s0)+Cρ

1ω+20 ε ≤ 2dε,L

min(s0),

(3.194)and therefore provided we choose

Kcol ≥ 2ω+2K2

it follows from the assumption s0+Kcoldε,Lmin(s0)

ω+2 < S that s+K2dε,Lmin(s)

ω+2 <

S. Therefore we may apply Proposition 3.58. Let T ε,−col (s) ∈ Iε

0(s0) be given byits statement, so that by (3.194)

T ε,−col (s)− s≤ 2ω+2K2dε,−

min(s0)ω+2,

and

dε,Lmin(T

ε,−col (s))≤max(β0,8q1(α2))ε. (3.195)

By Proposition 3.31, there exists some time T ε,+col (s0) ∈ [T ε,−

col (s),T ε,−col (s) +

q0(α3)εω+2] such that WPIL

ε(α3ε,T ε,+col (s0)) holds. In view of (3.194) and since

dε,−min(s0)≥ α3ε, it follows that

0≤ T ε,+col (s0)− s0 ≤ εω+2+ 2ω+2K2dε,−

min(s0)ω+2+ q0(α3)ε

ω+2

≤(

2ω+2K2+ 1+ q0(α3)

αω+23

)dε,−

min(s0)ω+2

≤Kcoldε,−min(s0)

ω+2

(3.196)

provided we finally fix the value of Kcol as

Kcol =(

2ω+2K2+ 1+ q0(α3)

αω+23

).

[Note that at this stage Kcol is fixed but its definition depends on α3 which hasnot yet been fixed. Of course when we fix α3 below we shall do it without anyreference to Kcol, in order to avoid impossible loops.]Next, we first claim that

ELε (s0)≥ EL

ε (s)≥ ELε (T

ε,+col (s0)).

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154 Fabrice Bethuel and Didier Smets

In view of Corollary 3.34, it suffices to check that L≥ L0(s0,T ε,+col (s0)), where

we recall that the function L0(·) was defined in (3.77). In view of (3.196), thisreduces to

100CeL−(ω+2)Kcoldε,−min(s0)

ω+2 ≤ μ1

4.

Since by (3.135) we have dε,−min(s0)≤ 2κ0(s0)L, it therefore suffices that

κ0(s0)≤ 1

2

(μ1

400CeKcol

) 1ω+2 ≡ κ1,

and we have now fixed the value of κ1.Next, we claim that actually

ELε (T

ε,+col (s0))≤ EL

ε (s0)−μ1.

Indeed this is so, otherwise by Corollary 3.34 we would have ELε (T

ε,+col (s0))=

ELε (s0), and therefore condition C(α3ε,L,s0,T ε,+

col (s0)) of Subsection 3.3.4would hold. Invoking Proposition 3.36, this would imply that conditionWPL(�log(α3ε),τ) holds for τ ∈ (s0+ εω+2,T ε,+

col (s0)), so that in particular

dεmin(T

ε,−col (s0))≥�log(α3ε).

It suffices thus to choose α3 sufficiently big so that

�log(α3ε) > max(β0,8q1(α2)))ε,

and the contradiction then follows from (3.195).

3.7.3 Proof of Proposition 3.20

We will fix the values of κ∗ and ρ∗ in the course of the proofs, as the smallestnumbers which satisfy a finite number of lower bound inequalities.

First, recall that it follows from (3.53) and (3.7) that if 0≤ s≤ ρ0(R− r)ω+2

then

Dε(s)∩I4L⊂ ∪k∈J0

(bεk−R,bε

k+R)⊂ I2κ0L, where the union is disjoint, (3.197)

and in particular CL,S holds, where

S := ρ0(R− r)ω+2 ≥ ρ0

(R

2

)ω+2

.

Having (3.77) in mind, and in view of (3.197) and (3.54), we estimate

100CeL−(ω+2)S≤ 100Ce

(R

2L

)ω+2

S≤ 100Ceα−(ω+2)∗ ≤ μ1

4,

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 155

where the last inequality follows provided we choose α∗ sufficiently large. Asa consequence, the function EL

ε is non-increasing on the set of times s in theinterval [εωL2,S] where WPL

ε(α1ε,s) holds.For such times s, the front points {aε

k(s)}k∈J(s) are well defined, and for q∈ J0,we have defined in the introduction Jq(s)= {k ∈ J(s),aε

k(s) ∈ [bεq−R,bε

q+R]},and we have set �q = "Jq and Jq(s)= {kq,kq+1, . . . ,kq+�q−1}, where k1 = 1, andkq = �1+·· ·+ �q−1+ 1, for q≥ 2.

Step 1. Annihilations of all the pairs of fronts–anti-fronts We claim that thereexists some time s ∈ (εωL2, 1

2 S) such that WPLε(δ

ε

log, s) holds and such that forany q ∈ J0, εk+ 1

2(s) = +1, for k ∈ Jq(s) \ {kq(s)+ �q(s)− 1} or "Jq(s) ≤ 1,

or equivalently that †k(s)= †k′(s) for k and k′ in the same Jq(s). In particular,dε,−

min(s)≥ 2R.

Proof of the claim If we require α∗ to be sufficiently large, then by (3.54) wehave that εωL2 + εω+1L ≤ S/2, and therefore by Proposition 3.13 we maychoose a first time

s0 ∈ [εωL2,εωL2+ εω+1L] such that WPLε(δ

ε

log,s0) holds.

Actually, we have

s0 ≤ εωL2+ εω+1L≤ 2εωL2 ≤ 2α−(ω+2)∗ rω+2 ≤ 2α−2(ω+2)

∗ Rω+2

≤ 1

ρ02ω+3α−2(ω+2)

∗ S. (3.198)

Note that by (3.7) we have the inclusion

Dε(s0)∩ IL ⊂N (b,r0),

where

r0 = r+(

s0

ρ0

) 1ω+2 ≤ 2r,

provided once more that α∗ is sufficiently large, and where for ρ > 0 we haveset N (b,ρ)=∪q∈J0 [bε

j −ρ,bεj +ρ]. In view of the confinement condition (3.53)

only two cases can occur:

(i) dε,−min(s0)≥ 3R− 2r0 or (ii) dε,−

min(s0)≤ 2r0.

If case (i) occurs, then, for any q ∈ J0, we have εk+ 12= +1, for any k ∈

Jq(τ1) \ {kq(s0)+ �q(s0)− 1}. Choosing s= s0, Step 1 is completed in the caseconsidered.

If instead case (ii) occurs, then we will make use of Proposition 3.55 toremove the small pairs of fronts–anti-fronts present at small scales. More

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156 Fabrice Bethuel and Didier Smets

precisely, assume that for some j≥ 0 we have constructed 0≤ sj ≤ S and rj > 0such that WPL

ε(δε

log,sj) holds, such that we have

Dε(sj)∩ IL ⊂N (b,rj), ELε (sj)≤ EL

ε (s0)− jμ1, (3.199)

as well as the estimates

r0≤ rj≤ γ jr0≤ R

2, sj≤ s0+(2ω+2Kcol+1)

γ j(ω+2)− 1

γ ω+2− 1rω+2

0 ≤ S

2, (3.200)

where γ :=(

2+ 2(Kcolρ0

) 1ω+2

), and moreover that case (ii) holds at step j,

that isdε,−

min(sj)≤ 2rj ≤ R. (3.201)

Let sj := T ε,+col (sj) be given by Proposition 3.55 (the confinement condition

holds in view of (3.197) and we have δε

log ≥ α3ε provided α∗ is sufficiently

large), and then let sj+1 ∈ [sj, sj + εω+1L] satisfying WPLε(δ

ε

log,sj+1) be givenby Proposition 3.13. In particular, we have

ELε (sj+1)≤ EL

ε (sj)≤ ELε (sj)−μ1 ≤ EL

ε (s0)− (j+ 1)μ1. (3.202)

Sincesj+1− sj ≤Kcol(2rj)

ω+2+ εω+1L≤ (2ω+2Kcol+ 1

)rω+2

j ,

we have, in view of (3.200),

sj+1 ≤ s0+(2ω+2Kcol+ 1

)[γ j(ω+2)− 1

γ ω+2− 1+ γ j(ω+2)

]rω+2

0

≤ s0+(2ω+2Kcol+ 1

) γ (j+1)(ω+2)− 1

γ ω+2− 1rω+2

0 ,

(3.203)

and by (3.7) Dε(sj+1)∩ IL ⊂N (b,rj+1), where

rj+1 = rj+ 2

(Kcol

ρ0

) 1ω+2

rj+ 1

ρ0ε

(L

ε

) 1ω+2 ≤

(2+ 2

(Kcol

ρ0

) 1ω+2

)rj = γ rj.

(3.204)In view of (3.198) and (3.54), we also have

γ j+1r0 ≤ 2γ j+1α−1∗ R (3.205)

and

s0+(2ω+2Kcol+ 1

) γ (j+1)(ω+2)− 1

γ ω+2− 1rω+2

0

≤[

2ω+3

ρ0α−(ω+2)∗ + 22ω+4

ρ0

(2ω+2Kcol+ 1

) γ (j+1)(ω+2)− 1

γ ω+2− 1

]α−(ω+2)∗ S.

(3.206)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 157

It follows from (3.203), (3.204), (3.205) and (3.206) that if α∗ is sufficientlylarge (depending on only M0, V and j), then (3.200) holds also for sj+1. Asabove we distinguish two cases:

(i) dε,−min(sj+1)≥ 3R− 2rj+1 or (ii) dε,−

min(sj+1)≤ 2rj+1.

If case (i) holds then by (3.200) we have dε,−min(sj+1) ≥ 2R, we set s = sj+1

which therefore satisfies the requirements of the claim, and we proceed toStep 2.

If case (ii) occurs then we proceed to construct sj+2 as above. The key factin this recurrence construction is the second inequality in (3.199), which, sinceEL

ε (sj) ≥ 0 independently of j, implies that the process has to reach case (i)in a number of steps less than or equal to M0/μ1. In particular, choosing theconstant α∗ sufficiently big so that the right-hand side of (3.205) is smaller thanR/2 for all 0≤ j≤M0/μ1 and so that the right-hand side of (3.206) is smallerthan S/2 for all 0≤ j≤M0/μ1 ensures that the construction was licit and thatthe process necessarily reaches case (i) before it could reach j=M0/μ1+1, sodefining s as above.

Step 2: Divergence of the remaining repulsing fronts at small scale At thisstage we have constructed s ∈ [εωL2, 1

2 S] which satisfies the requirements ofthe claim in Step 1. Moreover, note that in view of (3.198) and (3.200) wehave the upper bound

s≤⎛⎝2α−(ω+2)

∗ + 2ω+2(2ω+2Kcol+ 1)γ

M0μ1

(ω+2)− 1

γ ω+2− 1

⎞⎠rω+2. (3.207)

In order to complete the proof, we next distinguish two cases:

(i) "Jq(s)≤ 1, for any q ∈ J0. (ii) "Jq0(s) > 1, for some q0 ∈ J0.

If case (i) holds, then we actually have

dε,Lmin(s)≥ 2R. (3.208)

Since 2R ≥ 16q1(δε

log)ε when α∗ is sufficiently large, it follows from Corol-

lary 3.38 that WPLε(δ

ε

loglog,s) holds for any s + ε2+ω ≤ s ≤ T ε0 (δ

ε

log, s),where

T ε0 (δ

ε

log, s)=max{

s+ ε2+ω ≤ s≤ S s.t. dε,Lmin(s

′)≥ 8q1(δε

log)ε

∀s′ ∈ [s+ εω+2,s]} .

In particular, WPLε(δ

ε

loglog,s) holds for any s in s+ε2+ω≤ s≤T ε3 (δ

ε

log, s), where

T ε3 (δ

ε

log, s)=max{s+ ε2+ω ≤ s≤ S s.t. dε,L

min(s′)≥ R ∀s′ ∈ [s+ εω+2,s]} .

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158 Fabrice Bethuel and Didier Smets

In view of (3.208) and estimate (3.7), we obtain the lower bound

T ε3 (δ

ε

log, s)≥ s+ρ0Rω+2. (3.209)

Note that (3.209) and (3.54) also yield

T ε3 (δ

ε

log, s)≥ s+ρ0α−1∗ rω+2 ≥ ρ0α

−1∗ rω+2. (3.210)

Combining (3.207) and (3.210), we deduce in particular that

WPLε(α1ε,sr) holds and dε,L

min(sr)≥ R≥ r,

which is the claim of Proposition 3.20, provided that

ρ∗ ≥(

3+ 2ω+2(2ω+2Kcol+ 1)γ

M0μ1

(ω+2)− 1

γ ω+2− 1

)and ρ∗ ≤ ρ0α

−1∗ .

(3.211)It remains to consider the situation where case (ii) holds. In that case, we

havedε,−

min(s)≥ 2R and dε,+min(s)≤ 2γM0/μ1 r ≤ R,

so that we are in a situation suited for Proposition 3.57. We may actuallyapply Proposition 3.57 recursively with s0 := s and s≡ sk = (T ε

2 )k(s0), wheres0 = s+ εω+2, as long as dε,+

min(sk) remains sufficiently small with respect to

R (say e.g. dε,+min(sk) ≤ α

− 12∗ R provided α∗ is chosen sufficiently large), the

details are completely similar to the ones in Case II of Proposition 3.58 andare therefore not repeated here. If we denote by k0 the first index for whichdε,+

min(sk0) becomes larger than 2S2

r (in view of (3.160)) and k1 the last index

before dε,+min reaches α

− 12∗ R, then we have

sk0 ≤ Crω+2 and sk1 ≥1

Cα−(ω+2)/2∗ Rω+2 ≥ 1

Cα(ω+2)/2∗ rω+2,

for some constant C > 0 depending on only M0 and V , and the conclusion thatWPL

ε(α1ε,sr) holds follows as in case (i) above, choosing first ρ∗ sufficientlylarge (independently of α∗) and then α∗ sufficiently large (given ρ∗).

3.8 Proofs of Theorems 3.3, 3.5 and 3.9

3.8.1 Proof of Theorem 3.3

Theorem 3.3 being essentially a special case of Theorem 3.5, we go directly tothe proof of Theorem 3.5. Notice, however, that in Theorem 3.3 the solution tothe limiting system is unique, so that the result is not constrained by the needto pass to a subsequence.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 159

3.8.2 Proof of Theorem 3.5

We fix S < Smax and let L ≥ κ−1∗ L0, where L0 is defined in the statementof Proposition 3.19 and κ∗ in the statement of Proposition 3.20. We setR= 1

2 min{a0k+1−a0

k ,k= 1, . . . ,�0−1} and consider an arbitrary 0 < r < R/α∗.Since (H1) holds, there exists some constant εr > 0 such that, if 0 < ε ≤ εr,then (3.53) holds with bk ≡ a0

k for any k ∈ {1, . . . ,�0}. We are therefore inposition to make use of Proposition 3.20 and assert that for all such ε conditionWPL

ε(α1ε,sr) holds as well as (3.55) and (3.56). It follows in particular from(3.55) and (3.56) that for every k ∈ 1, . . . ,�0 we have "Jk(sr)= |m0

k |, where m0k

is defined in (3.12), and therefore "J(sr)=∑�0k=1 |m0

k | ≡ �1, in other words thenumber of fronts as well as their properties do not depend on ε nor on r.

We construct next the limiting splitting solution to the ordinary differentialequation and the corresponding subsequence proceeding backwards in timeand using a diagonal argument. For that purpose, we introduce an arbitrarydecreasing sequence {rm}m∈N∗ such that 0< r1≤R/α∗, and such that rm→ 0 asm→+∞. For instance, we may take rm = 1

m R/α∗, and we set sm = srm . Takingfirst m = 1, we find a subsequence {εn,1}n∈N∗ such that εn,1 → 0 as n →∞,and such that all points {aεn,1

k (s1)}k∈J converge to some limits {a1k(s1)}k∈J as

n→+∞. It follows from (3.56), passing to the limit n→+∞, that

d∗min(s1)≥ r1. (3.212)

We are therefore in position to apply the convergence result of Proposi-tion 3.19, which yields in particular that

Dεn,1(s)∩ I4L −→∪�1k=1{a1

k(s)} ∀s1 < s < S1max, (3.213)

as n →+∞, where {a1k(·)}k∈J is the unique solution of (S) with initial data

{a1k(s1)}k∈J on its maximal time of existence (s1,S1

max).Taking next m = 2, we may extract a subsequence {εn,2}n∈N∗ from the

sequence {εn,1}n∈N∗ such that all the points {aεn,2k (s2)}k∈J converge to some

limits {a2k(s2)}k∈J as n→+∞. Arguing as above, we may assert that

Dεn,2(s)∩ I4L −→∪�1k=1{a2

k(s)} ∀s2 < s < S2max, (3.214)

as n →+∞, where {a2k(·)}k∈J is the unique solution of (S) with initial data

{a2k(s2)}k∈J on its maximal time of existence (s2,S2

max). It follows from (3.55),namely that only repulsive forces are present at scale smaller than R, thatS2

max ≥ s1. Therefore, since we have extracted a subsequence, it follows from(3.213) and (3.214) that a2

k(s1) = a1k(s1) for all k ∈ J, and therefore also that

S2max = S1

max ≡ Smax and a2k(·)= a1

k(·)= ak(·) on (s2,Smax).

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160 Fabrice Bethuel and Didier Smets

We proceed similarly at each step m ∈ N∗, extracting a subsequence{εn,m}n∈N∗ from the sequence {εn,m−1}n∈N∗ such that all the points {aεn,m

k (sm)}k∈J

converge to some limits {amk (sm)}k∈J . Finally setting, for n ∈ N∗, εn = εn,n, we

obtain that

Dεn(s)∩ I4L −→∪�1k=1{ak(s)} ∀0 < s < Sm

max,

where {ak(·)}k∈J is a splitting solution of (S) with initial data {a0k}k∈J0 , on its

maximal time of existence (0,Smax). Since L≥ L0 was arbitrary, it follows that(3.15) holds.

It remains to prove that (3.14) holds. This is actually a direct consequence of(3.15), the continuity of the trajectories ak(·) and the regularizing effect of thefront set stated in Proposition 3.12 (e.g. (3.31) for the L∞ norm). As a matterof fact, it is standard to deduce from this the fact that the convergence towardsthe equilibria σq, locally outside the trajectory set, holds in any Cm norm, sincethe potential V was assumed to be smooth.

3.8.3 Proof of Theorem 3.9

As underlined in the introduction, Theorem 3.9 follows rather directly fromTheorem 3.5 and more importantly its consequence, Corollary 3.8 (whoseproof is elementary and explained after Proposition 3.7), combined withTheorem 3.2 and Proposition 3.12.

Let therefore L > L0 and δ > 0 be arbitrarily given; we shall prove that, atleast for ε ≡ εn sufficiently small,

Dε(Smax)∩ IL ⊂∪j∈{1,...,�2}[bj− δ,bj+ δ] (3.215)

and

|vε(x,Smax)−σı(j+ 12 )| ≤ C(δ,L)ε

1θ−1 (3.216)

for all j ∈ {0, . . . ,�2} and for all x ∈ (bj + δ,bj+1 − δ), where we have used theconvention that b0 = −L and b�2+1 = L. Since L can be arbitrarily big and δ

arbitrarily small, this will imply that assumption (H1) is verified at times Smax,which is the claim of Theorem 3.9.

Concerning (3.215), by Corollary 3.8 there exists

s− ∈ [Smax−ρ0

4

)ω+2

,Smax

)(3.217)

such that

∪k∈{1,...,�1}{ak(s−)} ⊂ ∪j∈{1,...,�2}[bj− 1

4δ,bj+ 1

4δ].

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 161

This, together with Theorem 3.5 implies that, for ε sufficiently small,

Dε(s−)∩ I2L ∩

(∪j∈{1,...,�2} [bj− 1

2δ,bj+ 1

2δ])c = ∅. (3.218)

In turn, Theorem 3.2 (including (3.7)) and (3.218), combined with the upperbound (3.217) on Smax− s−, imply that

Dε(s)∩I 32 L∩

(∪j∈{1,...,�2} [bj− 3

4δ,bj+ 3

4δ])c=∅, ∀s∈ [s−,Smax]. (3.219)

For s= Smax this is stronger than (3.215).We proceed to (3.216). In view of (3.219), for any x0 ∈ IL \

(∪j∈{1,...,�2} [bj−δ,bj+ δ]) we have, for ε sufficiently small,

vε(y,s) ∈ B(σi,μ0) ∀(y,s) ∈ [x0− 1

8δ,x0+ 1

8δ]× [s−,Smax].

The latter is nothing but (3.29) for r= 18δ, s0 = s− and S= Smax, and therefore

the conclusion (3.216) follows from (3.31) of Proposition 3.12, with C(δ,L)=15 Ce(8/δ)

1θ−1 as soon as εω/(Smax− s−)≤ δ2/64.

Appendix A

In this appendix we establish properties concerning the stationary solutions∨u+

,�u,

∨u+ε,r, etc., which we have used in the course of the previous discussion,

mainly in Section 3.5.

A.1 The Operator Lμ

Consider for μ > 0 the non-linear operator Lμ, defined for a smooth functionU on R by

Lμ(U)=− d2

dx2U+ 2μθ U2θ−1,

and set for simplicity L ≡ L1. Most results in this section are deduced fromthe comparison principle: if u1 and u2 are two functions defined on somenon-empty interval I, such that

Lμ(u1)≥ 0, Lμ(u2)≤ 0, and u1 ≥ u2 on ∂I, (A.1)

then u1(x)≥ u2(x) for x ∈ I. Scaling arguments are also used extensively.Given r > 0 and η > 0 we provide a rescaling of a given smooth function

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162 Fabrice Bethuel and Didier Smets

U as follows:⎧⎪⎨⎪⎩Uη,R = ηU

(U

r

), and we verify that

Lμ(Uη,r)(x)= η

r2Lγ (U)

(x

r

), where γ = μη2(θ−1)r2.

(A.2)

In particular, if Lμ(U)= 0, then we have

Lμ(r− 1

(θ−1) U(·r))= 0 and Lλμ(λ

− 12(θ−1) U)= 0, for any r > 0 and any λ > 0.

Notice also that U∗ defined on (0,+∞) by U∗(x)= [√2(θ − 1)x]− 1θ−1 solves

L(U∗)= 0.

Lemma 3.59 There exists a unique smooth map∨u+r on (−r,r) such that

L(∨u+r )= 0 and

∨u+r (±r)=+∞, and a unique solution

�ur such that L(�ur)= 0

and�ur(±r)=±∞. Moreover,

∨u+r is even,

�ur is odd, and, setting

∨u+ ≡ ∨

u+1 and

�u ≡ �

u1, we have

∨u+r (x)= r−

1θ−1

∨u+(x

r) and

�ur(x)= r−

1θ−1

�u(

x

r). (A.3)

Proof For n ∈ N∗, we construct on (−r,r) a unique solution∨u+r,n that solves

L(∨u+r,n) = 0 and

∨u+r,n(±r) = n, minimizing the corresponding convex energy.

By the comparison principle,∨u+r,n is non-negative, increasing with n and

uniformly bounded on compact subsets of (−r,r) in view of (A.5). Hence

a unique limit∨u+r exists, solution to L(∨u

+r ) = 0. We observe that

∨u+r,n(·) ≥

U∗(rn−·), where rn = r+[√2(θ−1)]−1n−(θ−1), so that we obtain the required

boundary conditions for∨u+r . Uniqueness may again be established thanks to the

comparison principle. We construct similarly a unique solution�ur,n that solves

L(�ur,n) = 0 and∨u+r,n(±r) = ±n. We notice that

�ur,n is odd, its restriction on

(0,r) non-negative and increasing with n. Moreover, on some interval (a,r),

where 0 < a < r does not depend on n, we have∨u+r,n(·) ≥ U∗(rn − ·), where

rn = r+[(θ − 1)]−1n−(θ−1), and we conclude as for the first assertion.

Remark 3.60 Given r > 0 and λ > 0 we notice that the functions∨

Uλr and

�Uλ

r

defined by

∨Uλ

r (x)= λ− 1

2(θ−1)∨u+r (x) and

�Uλ

r (x)= λ− 1

2(θ−1)�ur(x) (A.4)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 163

solve Lλ(∨

Uλr ) = 0 and Lλ(

�Uλ

r ) = 0 with the same boundary conditions as∨u+r

and�ur.

Lemma 3.61 (i) Assume that L(u) ≤ 0 on (−r,r). Then, we have, for x ∈(−r,r),

u(x)≤(√

2(θ − 1))− 1

θ−1[(x+ r)−

1θ−1 + (x− r)−

1θ−1

]. (A.5)

(ii) Assume that L(u) ≥ 0 on (−r,r) and that u(−r) = u(r) = +∞. Then wehave

u(x)≥(√

2(θ − 1))− 1

θ−1max{(x+ r)−

1θ−1 ,(r− x)−

1θ−1 }. (A.6)

Proof Set U=U∗(·+r)+U∗(r−·). By subaddivity and translation invariance,we have L(U)≥ 0 on (−r,r) with U(±r)=+∞, so that (A.5) follows from thecomparison principle (A.1) with u1 = U and u2 = u. Similarly, (A.6) followsfrom (A.1) with u1 = u and u2 =U∗(·+ r) or u2 =U∗(r−·).

Combining estimate (ii) of Lemma 3.61 with the scaling law of Lemma 3.59,we are led to

| d

dr

∨u+r (x)|+ |

d

dr

�ur(x)| ≤ Cr−

θθ−1 , for x ∈

(−7

8r,

7

8r

). (A.7)

A.2 The Discrepancy for Lμ

The discrepancy 'μ for Lμ relates to a given function u the function 'μ(u)defined by

'μ(u)= u2

2−μu2θ . (A.8)

This function is constant if u solves Lμ(u)= 0. We set '='1,

Aθ ≡'(∨u+)=−(

∨u+(0))2θ < 0 and Bθ ≡'(

�u)= (

�u(0))2

x

2> 0. (A.9)

In view of the scaling relations (A.3) and Remark 3.60, we are led to theidentities ⎧⎪⎨⎪⎩'λ(

∨Uλ

r )= λ− 1

θ−1) r−2θθ−1 Aθ = λ

− 1θ−1) r−(ω+1)Aθ ,

'λ(�

Uλr )= λ

− 1θ−1 r−

2θθ−1 Bθ = λ

− 1θ−1 r−(ω+1)Bθ .

(A.10)

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164 Fabrice Bethuel and Didier Smets

A.3 The Operator Lε

In this subsection, we consider more generally, for given λ > 0, the operatorLε given by

Lε(U)=− d2

dx2U+ 2λfε(U),

with fε defined in (3.88), and the solutions∨u+ε,r,

∨u−ε,r, and

�uε,r of Lε(U)= 0 on

(−r,r) with corresponding infinite boundary conditions, whose existence anduniqueness are proved as for Lemma 3.59.

Lemma 3.62 We have the estimates

| ∨u+ε,r(x)|+ |�uε,r(x)| ≤ C

(λ2(θ − 1)

)− 1θ−1

[(x+ r)−

1θ−1 + (x− r)−

1θ−1

].

Proof It follows from (3.89) that L 34 λ(∨u+ε,r) ≤ 0, so that, invoking the com-

parison principle as well as the scaling law (A.2) we deduce that∨u+ε,r ≤

(3λ/4)−2(θ−1)∨u+r . A similar estimate holds for

�uε,r and the conclusion follows

from Lemma 3.61.

We complete this appendix by comparing the solutions∨u+r and

∨u+ε,r, as well

as�ur and

�uε,r.

Proposition 3.63 In the interval (− 78 r, 7

8 r) we have the estimate

|∨u+ε,r −∨u+r | ≤ Cε

1θ r−

2θ−1θ(θ−1) .

Proof Let ε < δ < r/16 to be fixed. It follows from Lemma 3.62 that, forx ∈ (−r+ δ,r− δ), we have

0≤ ε1

θ−1∨u+ε,r(x)≤ C

(εδ

) 1θ−1

,

and therefore also∣∣∣∣ε 1θ−1

∨u+ε,r(x)g

1θ−1

∨u+ε,r(x)

)∣∣∣∣≤ C(εδ

) 1θ−1

. (A.11)

It follows from (A.11) and the fact that Lε(∨u+ε,r)= 0, that Lλ−ε (

∨u+ε,r)≤ 0, where

λ±ε = λ(1±C( εδ)

1θ−1 ). On the other hand, by the scaling law (A.2), we have

Lλ−ε

((λ−ελ

)− 1

2(θ−1)∨u+r−δ

)= 0.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 165

It follows from the comparison principle, since the second function is infiniteon the boundary of the interval [−r+ δ,r− δ], that

∨u+ε,r ≤ (

λ−ελ

)− 1

2(θ−1)∨u+r−δ on [−r+ δ,r− δ].

Integrating the inequality (A.7) between r− δ and r, we deduce that for x ∈(− 7

8 r, 78 r), we have the inequality

|∨u+r−δ(x)−∨u+r (x)| ≤ Cδr−

θθ−1 . (A.12)

On the other hand, it follows from estimate (A.5) of Lemma 3.61 that for x ∈(− 7

8 r, 78 r),

|(λ−ε

λ)− 1

2(θ−1)∨u+r−δ(x)−

∨u+r−δ(x)| ≤ C

(εδ

) 1θ−1

r−1

θ−1 . (A.13)

We optimize the outcome of (A.12) and (A.13) by choosing δ := ε1θ r

θ−1θ and

we therefore obtain

∨u+ε,r ≤

∨u+r +Cε

1θ r−

2θ−1θ(θ−1) on (− 7

8 r, 78 r).

The lower bound for∨u+ε,r is obtained in a similar way but reversing the role of

super- and subsolutions: the function (λ+ελ)− 1

2(θ−1)∨u+r+δ yields a subsolution for

Lε on [−r,r] whereas∨u+ε,r is a solution. The conclusion then follows.

Similarly, we have the following.

Proposition 3.64 In the interval (− 78 r, 7

8 r) we have the estimate

|�uε,r −�ur| ≤ Cε

1θ r−

2θ−1θ(θ−1) .

Proof We only sketch the necessary adaptations since the argument is closelyparallel to the proof of Proposition 3.63. First, by the maximum principle�uε,r can only have negative maximae and positive minimae, so that actually�uε,r has no critical point and a single zero, which we call aε. Arguing as inProposition 3.63, one first obtains(

λ−ε/λ)− 1

2(θ−1)�ur−δ−aε (·− aε)≥ �

uε,r ≥(λ+ε

/λ)− 1

2(θ−1)�ur+δ−aε (·− aε)

on [aε,r− δ],and

−(λ−ε

/λ)− 1

2(θ−1)�ur+aε−δ(·− aε)≥−�

uε,r ≥−(λ+ε

/λ)− 1

2(θ−1)�ur+aε+δ(·− aε)

on [−r+ δ,aε].

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166 Fabrice Bethuel and Didier Smets

Since�uε,r is continuously differentiable at the point aε (indeed it solves

Lε(�uε,r)= 0), and since the derivative at zero of the function

�ur is a decreasing

function of r, it first follows from the last two sets of inequalities that |aε| ≤ δ,and the conclusion then follows as in Proposition 3.63.

Appendix B

B.1 Some Properties of the OrdinaryDifferential Equation (S)

This appendix is devoted to properties of the system of ordinary differentialequations (S), the result being somewhat parallel to the results in Section 2 of[5]. We assume that we are given � ∈ N∗, and a solution, for k ∈ J = {1, . . . ,�}t �→ a(t)= (a1(t), . . . ,a�(t)) to the system

qkd

dsak(s)=−B(k− 1

2 )[(ak(s)−ak−1(s)]−(ω+1)+B

(k+ 12 )[(ak+1(s)−ak(s)]−(ω+1),

(B.1)where the numbers qk are supposed to be positive, and are actually taken in(S) equal to Si(k), whereas the numbers Bk+1/2, which may have positive ornegative signs, are taken in (S) to be equal to �i(k−1/2). We also define qmin =min{qi} and qmax =max{qi}. We consider a solution on its maximal interval ofexistence, which we call [0,Tmax). An important feature of the equation (B.1)is its gradient flow structure. The behavior of this system is indeed related tothe function F defined on R� by⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

F(U)=�−1∑k=0

Fk+1/2(U), where, for k= 0, . . . ,�− 1, and U = (u1, . . . ,u�),

we set

Fk+ 12(U)=−ω−1Bk+1/2 (uk+1− uk)

−ω with the convention that u0 =−∞.

If u1 < u2 < · · ·< u�, for k= 1, . . . ,�, then we have

∂F

∂uk(U)= Bk−1/2 (uk− uk−1)

−(ω+1)−Bk+1/2 (uk+1− uk)−(ω+1) , (B.2)

so that (S) becomesd

dsak(s)=−q−1

k

∂F

∂uk(a(s)). Hence, we have

d

dtF(a(t))=

�∑k=1

∂F

∂uk(a(t))

dak

dt(t)=−

�∑k=1

q−1k

(∂F

∂uk(a(t))

)2

≤−q−1max|∇F(a(t))|2, (B.3)

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 167

hence F decreases along the flow. We also consider the positive functionalsdefined by

Frep(U)=∑k∈J+

Fk+1/2(U), Fatt =−∑k∈J−

Fk+1/2(U), for U = (u1, . . . ,u�),

where J± = {k ∈ {0,�− 1} such that εk+1/2 ≡ sgn(Bk+1/2)=±1}.Proposition 3.65 Let a = (a1, . . . ,a�) be a solution to (B.1) on its maximalinterval of existence [0,Tmax]. Then, we have, for any time t ∈ [0,Tmax),⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

(Frep(a(t))

)− ω+2ω ≥ (

Frep(a(0)))− ω+2

ω +S0t,

d+a (t)≥(S1t+S2d

+a (0)

ω+2) 1ω+2 ,

(Fatt(a(t)))− ω+2

ω ≤ (Fatt(a(0)))− ω+2

ω −S0t,

d−a (t)≤(S3d

−a (0)

ω+2−S4t) 1ω+2 ,

(B.4)

where S0 > 0, S1 > 0, S2 > 0, S3 > 0 and S4 > 0 depend on only the coefficientsof (B.1).

Since d−a (s)≥ 0, an immediate consequence of (B.4) is that

Tmax ≤ S3

S4d−a (0). (B.5)

Since the system (B.1) involves both attractive and repulsive forces,for the proof of Proposition 3.65 it is convenient to divide the collection{a1(t),a2(t), . . . ,a�(t)} into repulsive and attractive chains. We say that a subsetA of J is a chain if A = {k,k+ 1,k+ 2, . . . ,k+m} is an ordered subset of mconsecutive elements in J, with m≥ 1.

Definition 3.66 A chain A is said to be repulsive (resp. attractive) if and onlyif εj+1/2 = −1 (resp. +1) for j = k, . . . ,k + m. It is said to be a maximalrepulsive chain (resp. maximal attractive chain), if there exists any repulsive(resp. attractive) chain which contains A strictly.

It follows from our definition that a repulsive or attractive chain contains atleast two elements. We may decompose J, in increasing order, as

J = B0 ∪A1 ∪B1 ∪A2 ∪B2 ∪ . . .∪Bp−1 ∪Ap ∪Bp, (B.6)

where the chains Ai are maximal repulsive chains, the chains Bi are maximalattractive for i = 1, . . . ,p− 1, and the sets B0 and Bp being possibly emptyor maximal attractive chains. Moreover, for i = 1 . . . ,p the sets Ai ∩ Bi andBi ∩Ai+1 contain one element.

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168 Fabrice Bethuel and Didier Smets

B.2 Maximal Repulsive Chains

In this section, we restrict ourselves to the study of the behavior of a maximalrepulsive chain A= {j, j+ 1, . . . , j+m} of m+ 1 consecutive points, m≤ �− 2within the general system (B.1). Setting, for k = 0, . . . ,m, uk(·) = ak+j(·), weare led to study U(·)= (u0(·),u1(·), . . . ,um(·)). Since Bk+1/2 < 0 in the repulsivecase the chain U is moved through a system of m− 1 ODEs,

qkd

dsuk(s)=−|B(k−1/2)|[(uk(s)− uk−1(s)]−(ω+1)

+|B(k+1/2)|[(uk+1(s)− uk(s)]−(ω+1) (B.7)

for k= 1, . . . ,m−1, whereas the end points satisfy two differential inequalities

d

dsum(s)≥−q−1

m

∂Frep

∂um(um(s)),

d

dsu0(s)≤−q−1

0

∂Frep

∂u0(a(s)), (B.8)

where we have set Frep(U)=m−1∑k=0

Fk+1/2(U). We assume that at initial time we

have

u0(0) < u1(0) < · · ·< um(0). (B.9)

Set du(t)=min{uk+1(t)− uk(t), k= 0, . . . ,m− 1}. We prove the following.

Proposition 3.67 Assume that the function U satisfies the system (B.7) and(B.8) on [0,Tmax], and that (B.9) holds. Then, we have, for any t ∈ [0,Tmax),(

Frep(U(t)))− ω+2

ω −(

Frep(U(0)))− ω+2

ω ≥ ω+ 1

4ωq−1

max (ωBmax)− 2(ω+1)

ω t, so that

(B.10)

du(t)≥(S1t+S2du(0)

ω+2) 1ω+2 , (B.11)

where S1 > 0 and S2 > 0 depend on only the coefficients of the equation (B.1).

The proof relies on several elementary observations.

Lemma 3.68 Let U be a solution to (B.7), (B.8) and (B.9). Then, we have

d

dtFrep(U(t))≤−q−1

max

∣∣∣∇Frep(U(t))∣∣∣2 , for every t ∈ [0,Tmax]. (B.12)

The proof is similar to (B.3) and we omit it. For U = (u0,u1, . . . ,um) ∈Rm+1 with u0 < · · · < um set ρmin(U)= inf{|uk+1− uk|, k= 0, . . . ,m− 1} andBmin = inf{|Bk+1/2|}, Bmax = sup{|Bk+1/2|}.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 169

Lemma 3.69 Let U = (u0, . . . ,um) be such that u0 < u1 < · · ·< um. We have

Bminω−1ρmin(U)−ω ≤ Frep(U)≤ Bmax(m+ 1)(ω)−1ρmin(U)−ω. (B.13)

|∇Frep(U)| ≤ (m+ 2)Bmax((ω− 1)Bmin)− ω+1

ω (Frep(U))ω+1ω , (B.14)

|∂Frep(U)

∂uk| ≥ 1

2(ωBmax)

− ω+1ω

(Frep(U)

) ω+1ω

, for every k= 0, . . . ,m.

(B.15)

Proof Inequalities (B.13) and (B.14) are direct consequences of the definitionof Frep. We turn therefore to estimate (B.15). In view of formula (B.2), thecases k = 0 and k = m+ 1 are straightforward. Next, let k = 1, . . . ,m and setTk+1/2 = Bk+1/2 (uk+1− uk)

−(ω+1). We distinguish two cases.

Case 1: Tk− 12≤ 1

2 Tk+ 12. Then, we have, in view of (B.2), Tk+1/2 ≤ 2

| ∂F

∂uk(U)| ≤ 2 |∇F(U)|, and we are done.

Case 2: Tk−1/2 ≥ 12 Tk+1/2. In that case, we repeat the argument with k

replaced by k − 1. Then either Tk−3/2 ≤ 12 Tk−1/2, which yields as above

Tk−1/2 ≤ 2|∇F(U)|, so that we are done, or Tk−3/2 ≥ 12 Tk+1/2, and we go on.

Since we have to stop at k= 0, this leads to the desired inequality (B.15).

Proof of Proposition 3.67 Combining (B.12) with (B.15), we are led to

d

dtFrep(U(t))≤−1

4q−1

max (ωBmax)− 2(ω+1)

ω

(Frep(u(t))

) 2(ω+1)ω

.

Integrating this differential equation, we obtain (B.10). Combining the lastinequality of Lemma 3.68 with inequality (B.13), inequality (B.11) fol-lows.

B.3 Maximal Attractive Chains

Maximal attractive chains B = {j, j+ 1, . . . j+m}, with m ≤ �− 1 within thegeneral system (B.1) are handled similarly. Defining U as above, the functionU still satisfies (B.7), but the inequalities (B.8) are now replaced by

d

dsum(s)≤−q−1

m

∂Fatt

∂um(um(s)),

d

dsu0(s)≥−q−1

0

∂Fatt

∂u0(a(s)). (B.16)

Fatt(U) is defined by Fatt=−Frep, so that we have in the attractive case Fatt≥ 0.Up to a change of sign, the function Fatt verifies the properties (B.13), (B.14)and (B.15) stated in Proposition 3.67. However, the differential inequality

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170 Fabrice Bethuel and Didier Smets

(B.12) is now turned into

d

dtFatt(U(t))≥ q−1

max

∣∣∣∇Fatt(U(t))∣∣∣2 ≥ CFatt(U(t))

2(ω+1)ω , (B.17)

where the last inequality follows from (B.15) and where C is some constantdepending on only the coefficients in (B.1). Integrating (B.17) and invoking(B.13) once more, we obtain the following lemma.

Lemma 3.70 Assume that U satisfies the system (B.7) and (B.16) on [0,Tmax]with (B.9). Then for constants S3 > 0 and S4 > 0 depending on only thecoefficients of (B.1), we have

du(t)≤(S3du(0)

ω+2−S4t) 1ω+2 .

B.4 Proof of Proposition 3.65 Completed

Inequalities (B.4) of Proposition 3.65 follow immediately from Proposi-tion 3.67 and Lemma 3.70 applied to each separate maximal chain providedby the decomposition (B.6).

References

[1] F. Bethuel, H. Brezis and F. Helein Ginzburg-Landau vortices, Birkhauser (1994).[2] F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in

dissipative Ginzburg-Landau dynamics, Duke Math. J. 130 (2005) 523–614.[3] F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-

Landau vortices, Comm. Math. Phys. 272 (2007) 229–261.[4] F. Bethuel, G. Orlandi and D. Smets, Slow motion for gradient systems with equal

depth multiple-well potentials, J. Diff. Equations 250 (2011), 53–94.[5] F. Bethuel and D. Smets, Slow motion for equal depth multiple-well gradient

systems: the degenerate case, Discrete Contin. Dyn. Syst. 33 (2013), 67–87.[6] F. Bethuel and D. Smets, On the motion law of fronts for scalar reaction-diffusion

equations with equal depth multiple-well potentials, Chin. Ann. Math. 38 B(1)(2007), 83–148.

[7] L. Bronsard and R.V. Kohn, On the slowness of phase boundary motion in onespace dimension, Comm. Pure Appl. Math. 43 (1990), 983–997.

[8] X. Chen, Generation, propagation, and annihilation of metastable patterns.J. Diff. Equations 206 (2004), no. 2, 399–437.

[9] J. Carr and R. L. Pego, Metastable patterns in solutions of ut = ε2uxx − f (u).Comm. Pure Appl. Math. 42 (1989), 523–576.

[10] G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singularperturbations J. Dynam. Differential Equations 1 (1989), 75–94.

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The Motion Law of Fronts for Scalar Reaction-diffusion Equations 171

[11] R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch.Rational Mech. Anal. 142 (1998), 99–125.

[12] J. B. Keller, On solutions of �u = f (u), Comm. Pure Appl. Math. 10 (1957),503–510.

[13] F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. PureAppl. Math. 49 (1996), 323–359.

[14] R. Osserman, On the inequality �u≥ f (u), Pacific J. Math. 7 (1957), 1641–1647.[15] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applica-

tions to Ginzburg-Landau, Comm. Pure Appl. Math. 57 (2004), 1627–1672.

∗,†Sorbonne Universites, UPMC Universite Paris 06, UMR #7598, Laboratoire Jacques-LouisLions, F-75005, Paris, France

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4

Finite-time Blowup for some NonlinearComplex Ginzburg–Landau Equations

Thierry Cazenave∗ and Seifeddine Snoussi†

In this article, we review finite-time blowup criteria for the family of complexGinzburg–Landau equations ut = eiθ [�u+|u|αu]+ γ u on RN , where 0≤ θ ≤ π

2 ,α > 0 and γ ∈R. We study in particular the effect of the parameters θ and γ , andthe dependence of the blowup time on these parameters.

In memory of our friend Abbas Bahri

4.1 Introduction

In this paper, we review certain known results, and present some new ones, onthe problem of finite-time blowup for the family of complex Ginzburg–Landauequations on RN {

ut = eiθ [�u+|u|αu]+ γ u,

u(0)= u0,(4.1)

where 0 ≤ θ ≤ π2

1, α > 0 and γ ∈ R2. The case θ = 0 of equation (4.1) is thewell-known nonlinear heat equation{

ut =�u+|u|αu+ γ u,

u(0)= u0,(4.2)

2010 Mathematics Subject Classification. Primary 35Q56; secondary 35B44, 35K91, 35Q55Key words and phrases. Complex Ginzburg–Landau equation, finite-time blowup, energy, variance∗ Universite Pierre et Marie Curie & CNRS, Laboratoire Jacques-Louis Lions† UR Analyse Non-Lineaire et Geometrie, UR13ES32, Department of Mathematics, Faculty of

Sciences of Tunis, University of Tunis El-Manar1 One might consider −π

2 ≤ θ ≤ 0, which is equivalent, by changing u to u.2 For a general γ ∈C, the imaginary part is eliminated by changing u(t,x) to e−it$γ u(t,x).

172

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Complex Ginzburg–Landau Equations 173

which arises in particular in chemistry and biology. See e.g. [10]. The caseθ = π

2 of (4.1) is the equally well-known nonlinear Schrodinger equation{ut = i[�u+|u|αu]+ γ u,

u(0)= u0,(4.3)

which is an ubiquitous model for weakly nonlinear dispersive waves andnonlinear optics. See e.g. [43]. Therefore, equation (4.1) can be consideredas “intermediate” between the nonlinear heat and Schrodinger equations.Equation (4.1) is itself a particular case of the more general complexGinzburg–Landau equations on RN :{

ut = eiθ�u+ eiφ |u|αu+ γ u,

u(0)= u0,(4.4)

where 0 ≤ θ ≤ π2 , φ ∈ R, α > 0 and γ ∈ R, which is a generic modula-

tion equation describing the nonlinear evolution of patterns at near-criticalconditions. See e.g. [42, 8, 27].

Two strategies have been developed for studying finite-time blowup. Thefirst one consists of deriving conditions on the initial value, as general as pos-sible, which ensure that the corresponding solution of (4.1) blows up in finitetime. The proofs often use a differential inequality which is satisfied by somequantity related to the solution, and one shows that this differential inequalitycan hold only on a finite-time interval. The major difficulty is to guessthe appropriate quantity to calculate. However, when such a method can beapplied, it usually provides a simple proof of blowup, under explicit conditionson the initial value. On the other hand, this strategy does not give any infor-mation on how the blowup occurs, nor on the mechanism that leads to blowup.Concerning the family (4.1), this is the type of approach used in [16, 20, 2]for the nonlinear heat equation; in [48, 13, 18, 44, 31, 32] for the nonlinearSchrodinger equation; and in [41, 6, 5] for the intermediate case of (4.1).

Another strategy consists of looking for an ansatz of an approximateblowing-up solution, and then showing that the remainder remains bounded,or becomes small with respect to the approximate solution, as time tends tothe blowup time of the approximate solution. The first difficulty is to findthe appropriate ansatz. Then, proving the boundedness of the remainder isoften quite involved technically. When this method is successful, it providesa precise description of how the corresponding solutions blow up. It mayalso explain the mechanism that makes these solutions blow up. For thefamily (4.1), this is the strategy employed in particular in [26] for the nonlinearheat equation; in [22, 23, 24, 38, 39, 25] for the nonlinear Schrodinger

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174 T. Cazenave and S. Snoussi

equation; and in [37, 47, 36, 21] for the intermediate case of (4.1) (andeven (4.4)).

Equation (4.1) enjoys certain properties which the general equation (4.4)does not. in particular its solutions satisfy certain energy identities (seeSection 4.2), which make it possible to study blowup by the first approachdescribed above. We review sufficient conditions for finite-time blowup(obtained using this approach), and we study the influence of the parametersθ and γ . In Sections 4.3 and 4.4, we recall the standard results for the heatequation (4.2) and the Schrodinger equation (4.3), respectively. We are notaware of any previous reference for Theorem 4.10, nor for the case γ > 0 ofTheorem 4.13, although the proofs use standard arguments. The case γ > 0of Theorem 4.14 seems to be new. Section 4.5 is devoted to the complexGinzburg–Landau equation (4.1). In Subsection 4.5.1, we review sufficientconditions for blowup, and the case γ < 0 of Theorem 4.17 is partially new.Finally, we study in Section 4.5.2 the behavior of the blowup time as theparameter θ approaches π

2 , i.e. as the equation gets close to the nonlinearSchrodinger equation (4.3). We consider separately the cases α < 4

N (Subsec-tion 4.5.2) and α ≥ 4

N (Subsection 4.5.2). The case γ > 0 of Theorem 4.22,and Theorem 4.25, are new. A few open questions are collected in Section 4.6.

Notation. We denote by Lp(RN), for 1 ≤ p ≤∞, the usual (complex valued)Lebesgue spaces. H1(RN) and H−1(RN) are the usual (complex valued)Sobolev spaces. (See e.g. [1] for the definitions and properties of these spaces.)We denote by C∞

c (RN) the set of (complex valued) functions that have compactsupport and are of class C∞. We denote by C0(R

N) the closure of C∞c (RN) in

L∞(RN). In particular, C0(RN) is the space of functions u that are continuous

RN →C and such that u(x)→ 0 as |x|→∞. C0(RN) is endowed with the sup

norm.

4.2 The Cauchy Problem and Energy Identities

For the study of the local well-posedness of (4.1), it is convenient to considerthe equivalent integral formulation, given by Duhamel’s formula,

u(t)= T θ (t)u0+∫ t

0T θ (t− s)[eiθ |u(s)|αu(s)+ γ u(s)]ds, (4.5)

where (T θ (t))t≥0 is the semigroup of contractions on L2(RN) generated by theoperator eiθ� with domain H2(RN). Moreover, T θ (t)ψ =Gθ (t) ψ , where thekernel Gθ (t) is defined by

Gθ (t)(x)≡ (4π teiθ )−N2 e

− |x|24teiθ .

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Complex Ginzburg–Landau Equations 175

If 0≤ θ < π2 , it is not difficult to show that (T θ (t))t≥0 is an analytic semigroup

on L2(RN), and a bounded C0 semigroup on Lp(RN) for 1 ≤ p <∞, and onC0(RN). In particular (see [6, formula (2.2)])

‖T θ (t)u0‖L∞ ≤ (cosθ)−N2 ‖u0‖L∞ . (4.6)

It is immediate by a contraction mapping argument (see [40, Theorem 1]) thatthe Cauchy problem (4.1) is locally well-posed in C0(R

N). Moreover, it iseasy to see using the analyticity of the semigroup that C0(R

N) ∩H1(RN) ispreserved under the action of (4.1). The following result is established in [6,Proposition 2.1 and Remark 2.2] in the case γ = 0, and the argument equallyapplies when γ = 0.

Proposition 4.1 Suppose 0≤ θ < π2 . Given any u0 ∈C0(R

N), there exist T > 0and a unique solution u∈C([0,T],C0(R

N)) of (4.5) on (0,T). Moreover, u canbe extended to a maximal interval [0,Tmax), and if Tmax <∞, then ‖u(t)‖L∞ →∞ as t ↑ Tmax. If, in addition, u0 ∈ H1(RN), then u ∈ C([0,T],H1(RN)) ∩C((0,T),H2(RN))∩C1((0,T),L2(RN)) and u satisfies (4.1) in L2(RN) for allt∈ (0,T). Furthermore, if α< 4

N and Tmax <∞, then ‖u(t)‖L2 →∞ as t↑Tmax.

Remark 4.2 Whether the solution given by Proposition 4.1 is global ornot is discussed throughout this paper, but we can observe that, given 0 ≤θ < π

2 and u0 ∈ C0(RN), the corresponding solution of (4.1) is global if γ

is sufficiently negative. More precisely, if γ < − 1α[2(cosθ)−

N2 ‖u0‖L∞]α+1,

then the corresponding solution u of (4.1) is global and satisfies ‖u(t)‖L∞ ≤2(cosθ)−

N2 eγ t‖u0‖L∞ for all t ≥ 0. Indeed, v(t) = e−γ tu(t) satisfies vt =

eiθ (�v+ eγαt|v|αv), so that

v(t)= T θ (t)u0+∫ t

0eγαsT θ (t− s)[|v(s)|αv(s)]ds.

Setting φ(t) = sup{‖v(s)‖L∞ ; 0 ≤ s ≤ t}, it follows from (4.6) thatφ(t) ≤ c‖u0‖L∞ + c

−γαφ(t)α+1 with c = (cosθ)−

N2 . Therefore, if γ <

− 1α(2c)α+1‖u0‖αL∞ , then φ(t)≤ 2c‖u0‖L∞ for all 0≤ t < Tmax, and the desired

conclusion follows.

If θ = π2 , then (4.1) is the nonlinear Schrodinger equation, and (T θ (t))t≥0

is a group of isometries (which is not analytic). More restrictive conditions areneeded for the local solvability of (4.1), and the proofs make use of Strichartz’sestimates. The following result is proved in [17, Theorem I] (except for theblowup alternatives, which follow from [4, Theorems 4.4.1 and 4.6.1]).

Proposition 4.3 Suppose θ = π2 and (N − 2)α < 4. Given any u0 ∈ H1(RN),

there exist T > 0 and a unique u ∈ C([0,T],H1(RN)) ∩ C1((0,T),H−1(RN))

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176 T. Cazenave and S. Snoussi

which satisfies (4.1) for all t ∈ [0,T] and such that u(0)= u0. Moreover, u canbe extended to a maximal interval [0,Tmax), and if Tmax <∞, then ‖u(t)‖H1 →∞ as t ↑ Tmax. In addition, if α < 4

N and Tmax <∞, then ‖u(t)‖L2 →∞ ast ↑ Tmax.

As observed above, an essential feature of equation (4.1) is the energyidentities satisfied by its solutions. Set

I(w)=∫RN|∇w|2−

∫RN|w|α+2, (4.7)

E(w)= 1

2

∫RN|∇w|2− 1

α+ 2

∫RN|w|α+2. (4.8)

The functionals I and E are well defined on C0(RN) ∩ H1(RN); and if

(N− 2)α ≤ 4, they are well defined on H1(RN).Suppose 0 ≤ θ < π

2 , let u0 ∈ C0(RN) ∩ H1(RN) and let u be the corre-

sponding solution of (4.1) defined on the maximal interval [0,Tmax), given byProposition 4.1. Multiplying the equation by u, we obtain∫

RNuut = γ

∫RN|u|2− eiθ I(u) (4.9)

and in particular, taking the real part,

d

dt

∫RN|u|2 = 2γ

∫RN|u|2− 2cosθ I(u) (4.10)

for all 0< t < Tmax. Multiplying the equation by e−iθut, taking the real part andusing (4.9) yields

d

dtE(u(t))=−cosθ

∫RN|ut|2+ γ 2 cosθ

∫RN|u|2− γ cos(2θ)I(u) (4.11)

for all 0 < t < Tmax. Applying (4.10), we see that this is equivalent to

d

dt

[E(u(t))− γ

2cosθ

∫RN|u|2

]=−cosθ

∫RN|ut|2+ γ sin2 θ I(u). (4.12)

Suppose now θ = π2 and (N − 2)α < 4, let u0 ∈ H1(RN) and let u be the

corresponding solution of (4.1) defined on the maximal interval [0,Tmax), givenby Proposition 4.3. Identities corresponding to (4.9), (4.10) and (4.11) hold.More precisely, the functions t �→ ‖u(t)‖2

L2 and t �→ E(u(t)) are C1 on [0,Tmax)

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Complex Ginzburg–Landau Equations 177

and ∫RN

uut = γ

∫RN|u|2− iI(u), (4.13)

d

dt

∫RN|u|2 = 2γ

∫RN|u|2, (4.14)

d

dtE(u(t))= γ I(u) (4.15)

for all 0≤ t < Tmax. Identity (4.13) is obtained by taking the H−1−H1 dualityproduct of the equation with u (the term

∫RN uut is understood as the duality

bracket 〈ut,u〉H1,H−1 ). (4.14) follows, by taking the real part. Identity (4.15)is formally obtained by multiplying the equation by e−iθut and taking the realpart. However, the solution is not smooth enough to do so, thus a regularizationprocess is necessary. See [34] for a simple justification. Still in the case of theSchrodinger equation θ = π

2 , an essential tool in the blowup arguments is thevariance identity. It concerns the variance

V(w)=∫RN|x|2|w|2, (4.16)

which is not defined on L2(RN), but on the weighted space L2(RN , |x|2dx). Itcan be proved that if u0 ∈ H1(RN) ∩ L2(RN , |x|2dx), then the correspondingsolution u of (4.1) satisfies u∈C([0,Tmax),L2(RN , |x|2dx)). Moreover, the mapt �→ V(u(t)) is C2 on [0,Tmax) and

d

dtV(u(t))=−4J(u(t))+ 2γV(u(t)), (4.17)

d

dtJ(u(t))=−2

∫RN|∇u|2+ Nα

α+ 2

∫RN|u|α+2+ 2γ J(u(t)) (4.18)

for all 0≤ t < Tmax, where the functional J is defined by

J(w)=$∫RN

(x · ∇w)w (4.19)

for w ∈ H1(RN) ∩ L2(RN , |x|2dx). The proofs of these properties requireappropriate regularizations and multiplications, see [4, Section 6.5].

4.3 The Nonlinear Heat Equation

In this section, we consider the nonlinear heat equation (4.2). The first blowupresult was obtained by Kaplan [16, Theorem 8]. Its argument applies topositive solutions of the equation set on a bounded domain, and is based ona differential inequality satisfied by the scalar product of the solution with the

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178 T. Cazenave and S. Snoussi

first eigenfunction. It is easy to extend the argument to the equation set on

RN . Let w(x) ≡ e−√

N2+|x|2 , so that �w ≥ −w by elementary calculations.If ψ = ‖w‖−1

L1 w and ψλ(x) = λNψ(λx) for λ > 0, then ‖ψλ‖L1 = 1 and�ψλ ≥ −λ2ψλ. Let now u0 ∈ C0(R

N)∩H1(RN), u0 ≥ 0, u0 ≡ 0, and let u bethe corresponding solution of (4.2) defined on the maximal interval [0,Tmax).The maximum principle implies that u(t)≥ 0 for all 0≤ t < Tmax. Multiplyingthe equation by ψλ, integrating by parts on RN and using Jensen’s inequality,we obtain

d

dt

∫RN

uψλ =∫RN

u�ψλ+∫RN

uα+1ψλ+ γ

∫RN

uψλ

≥ (γ −λ2)

∫RN

uψλ+(∫

RNuψλ

)α+1.

It follows that f (t)= ∫RN uψλ satisfies

df

dt≥ (γ −λ2+ f α) f (4.20)

on [0,Tmax). It is not difficult to show that if f (0)α > λ2 − γ , then (4.20) canhold on only a finite interval, so that Tmax <∞. Therefore, we can distinguishtwo cases. If γ ≤ 0, we choose for instance λ = 1 and we see that if u0 issufficiently “large” so that

∫RN u0ψ1 > (1−γ )

1α , then the solution blows up in

finite time. If γ > 0, then we let λ=√γ , so that the condition f (0)α >λ2−γ isalways satisfied if u0 ≡ 0. In this case, we see that every nonnegative, nonzeroinitial value produces a solution of (4.2) which blows up in finite time.

Levine [20] established blowup by a different argument. It is based on adifferential inequality satisfied by the L2 norm of the solution, derived fromthe energy identities. This argument applies to sign-changing solutions and,more generally, to complex valued solutions, and to the equation set on anydomain, bounded or not. Strangely enough, even though Kaplan’s argumentseems to indicate that blowup is more likely to happen if γ > 0, it turns outthat Levine’s result only applies to the case γ ≤ 0, which we consider first.

4.3.1 The Case γ ≤ 0

It is convenient to set

Iγ (w)=∫RN|∇w|2−

∫RN|w|α+2− γ

∫RN|u|2, (4.21)

Eγ (w)= 1

2

∫RN|∇w|2− 1

α+ 2

∫RN|w|α+2− γ

2

∫RN|u|2 (4.22)

for w ∈ C0(RN)∩H1(RN).

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Complex Ginzburg–Landau Equations 179

Theorem 4.4 ([20], Theorem I) Let u0 ∈ C0(RN)∩H1(RN) and let u be the

corresponding solution of (4.2) defined on the maximal interval [0,Tmax), givenby Proposition 4.1. If γ ≤ 0 and Eγ (u0)< 0, where Eγ is defined by (4.22), thenu blows up in finite time, i.e. Tmax <∞.

Proof We obtain a differential inequality on the quantity

M(t)= 1

2

∫ t

0‖u(s)‖2

L2 ds. (4.23)

Formulas (4.9) and (4.12) (with θ = 0) yield∫RN

uut =−Iγ (u), (4.24)

d

dtEγ (u(t))=−

∫RN|ut|2 ≤ 0. (4.25)

Identity (4.25) implies

Eγ (u(t))+∫ t

0‖ut‖2

L2 = Eγ (u0). (4.26)

Moreover, it follows from (4.21) and (4.22) that

Iγ (u(t))≤ (α+ 2)Eγ (u(t))− (−γ )α

2‖u‖2

L2 (4.27)

so that by (4.26),

Iγ (u(t))≤ (α+ 2)Eγ (u0)− (−γ )α

2‖u‖2

L2 − (α+ 2)∫ t

0‖ut‖2

L2 < 0. (4.28)

We deduce from (4.23), (4.24) and (4.28) that

M′′(t)=(∫RN

uut =−Iγ (u)≥−(α+ 2)Eγ (u0)+ (α+ 2)∫ t

0‖ut‖2

L2 > 0.

(4.29)All the above formulas hold for 0 ≤ t < Tmax. Assume now by contradictionthat Tmax =∞. We deduce in particular from (4.29) that

M′(t)−→t→∞∞, M(t)−→

t→∞∞. (4.30)

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180 T. Cazenave and S. Snoussi

It follows from (4.23), (4.29), and Cauchy–Schwarz’s inequality (in time andspace) that

M(t)M′′(t)≥ α+ 2

2

(∫ t

0‖u‖2

L2

)(∫ t

0‖ut‖2

L2

)≥ α+ 2

2

(∫ t

0

∣∣∣∫RN

uut

∣∣∣)2 ≥ α+ 2

2

(∫ t

0

∣∣∣(∫RN

uut

∣∣∣)2

= α+ 2

2

(∫ t

0M′′(s)

)2 = α+ 2

2(M′(t)−M′(0))2.

(4.31)

We deduce from (4.30) that α+22 (M′(t)−M′(0))2 ≥ α+4

4 M′(t)2 for t sufficientlylarge. Therefore (4.31) yields

M(t)M′′(t)≥ α+ 4

4M′(t)2

which means that M(t)−α4 is concave for t large. Since M(t)−

α4 → 0 as t→∞

by (4.30), we obtain a contradiction.

The proof of Theorem 4.4 does not immediately provide an estimate ofTmax in terms of u0. It turns out that a variant of that proof, given in [14,Proposition 5.1] yields such an estimate.

Theorem 4.5 Under the assumptions of Theorem 4.4, we have

Tmax ≤

⎧⎪⎪⎪⎨⎪⎪⎪⎩‖u0‖2

L2

α(α+ 2)(−Eγ (u0)), γ = 0,

1

−γαlog

(1+ −2γ ‖u0‖2

L2

2(α+ 2)(−Eγ (u0))− γα‖u0‖2L2

), γ < 0.

(4.32)

Proof Setf (t)= ‖u(t)‖2

L2 , e(t)= Eγ (u(t)). (4.33)

We first obtain an upper bound on e in terms of f , then a differential inequalityon f . It follows from (4.24) and (4.27) that

df

dt≥ 2(α+ 2)(−e)+ (−γ )αf . (4.34)

Since dfdt > 0 by (4.29), we deduce from (4.25), Cauchy–Schwarz’s inequality,

(4.24) and (4.34) that

−fde

dt=∫|u|2

∫|ut|2 ≥

∣∣∣∫ uut

∣∣∣2 = 1

4

(df

dt

)2

≥ 1

2

(−(α+ 2)e+ (−γ )α

2f)df

dt.

(4.35)

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Complex Ginzburg–Landau Equations 181

This means thatd

dt

(−ef−

α+22 + −γ

2f−

α2

)≥ 0 (4.36)

so that

− e+ −γ

2f ≥ ηf

α+22 (4.37)

with

η=−Eγ (u0)‖u0‖−(α+2)L2 + −γ

2‖u0‖−α

L2 > 0. (4.38)

It follows from (4.34) and (4.37) that

df

dt≥−2(−γ )f + 2(α+ 2)ηf

α+22 .

Therefore,d

dt

[(e−2γ tf )−

α2

]+α(α+ 2)ηeγαt ≤ 0. (4.39)

After integration, then letting t ↑ Tmax, we deduce that

α(α+ 2)η∫ Tmax

0eγαtdt≤ f (0)−

α2 . (4.40)

Expressing η and f (0) in terms of u0, estimate (4.32) easily follows in both thecases γ = 0 and γ < 0.

Remark 4.6 Here are some comments on Theorems 4.4 and 4.5.

1. Suppose u0 = 0 and Eγ (u0) = 0. In particular, Iγ (u0) < 0. Therefore,u0 is not a stationary solution of (4.2), and it follows from (4.25) thatEγ (u(t)) < 0 for all 0 < t < Tmax. Thus we can apply Theorems 4.4 and 4.5with u0 replaced by u(ε), and let ε ↓ 0. In particular, we see that Tmax <∞.Moreover, estimate (4.32) holds if γ < 0.

2. Given any nonzero ϕ ∈ C0(RN)∩H1(RN), we have Eγ (λϕ)) < 0 if |λ| is

sufficiently large. Thus we see that the sufficient condition Eγ (u0) < 0 canindeed be achieved by certain initial values, for any α > 0 and γ ≤ 0.

3. Let α > 0 and fix u0 ∈ C0(RN)∩H1(RN). It is clear that if γ is sufficiently

negative, then Eγ (u0) ≥ 0, so that one cannot apply Theorem 4.4. This isnot surprising, since the corresponding solution of (4.2) is global if γ issufficiently negative. (See Remark 4.2.)

4. Suppose γ < 0. It follows from (4.32) and the assumption Eγ (u0) < 0 thatTmax ≤ 1

−γαlog(1+ 2

α). In particular, we see that for u0 as in Theorem 4.5,

the blowup time is bounded in terms of α and γ only, independently of u0.

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182 T. Cazenave and S. Snoussi

As observed in Remark 4.6 (4), in the case γ < 0, Theorem 4.5 does notapply to solutions for which the blowup time would be arbitrarily large. When

(N− 2)α < 4 (4.41)

this can be improved by using the potential well argument of Payne andSattinger [35]. To this end, we introduce some notation. Assuming (4.41) andγ < 0, we denote by Qγ the unique positive, radially symmetric, H1 solutionof the equation

−�Q− γQ= |Q|αQ, (4.42)

and we recall below the following well-known properties of Qγ .

Proposition 4.7 Assume (4.41) and γ < 0, and let Qγ ∈H1(RN) be the uniquepositive, radially symmetric solution of (4.42).

1. Eγ (Qγ ) > 0 and Iγ (Qγ )= 0.

2. Eγ (Qγ )= inf{

Eγ (v); v ∈H1(RN),v = 0, Iγ (v)= 0}

.

3. If u ∈ H1(RN), Eγ (u) < Eγ (Qγ ) and Iγ (u) < 0, then Iγ (u)≤−(Eγ (Qγ )−Eγ (u)).

Proof The first two properties are classical, see for instance [46, Chapter 3].Next, let u ∈H1(RN) with Iγ (u) < 0, and set h(t)= Eγ (tu) for t > 0. It followseasily that h′(t) = 1

t Iγ (tu), so that there exists a unique t∗ > 0 such that his increasing on [0, t∗] and decreasing and concave on [t∗,∞). In particular,Iγ (t∗u) = 0, thus h(t∗) = Eγ (t∗u) ≥ Eγ (Qγ ). Moreover, since Iγ (u) < 0 wehave t∗ < 1 and by the concavity of h on [t∗,1],

Eγ (u)= h(1)≥ h(t∗)+ (1− t∗)h′(1)= h(t∗)+ (1− t∗)Iγ (u)≥ Eγ (Qγ )+ (1− t∗)Iγ (u)≥ Eγ (Qγ )+ Iγ (u)

from which (3) follows.

We have the following result.

Theorem 4.8 Assume (4.41), γ < 0, and let Qγ ∈ H1(RN) be the uniquepositive, radially symmetric solution of (4.42). Let u0 ∈ C0(R

N)∩H1(RN) andlet u be the corresponding solution of (4.2) defined on the maximal interval[0,Tmax). If Eγ (u0) < Eγ (Qγ ) and Iγ (u0) < 0, then u blows up in finite time,and

Tmax ≤ 1

−γα

[(α+ 4)[Eγ (u0)]+Eγ (Qγ )−Eγ (u0)

+ log(2(α+ 2)

α

)]. (4.43)

Proof The key observation is that

Iγ (u(t))≤−(Eγ (Qγ )−Eγ (u0)) < 0 (4.44)

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Complex Ginzburg–Landau Equations 183

for 0 ≤ t < Tmax. Indeed, it follows from (4.25) that Eγ (u(t)) ≤ Eγ (u0) <

Eγ (Qγ ). Therefore, Proposition 4.7 (3) implies that (4.44) holds as long asIγ (u(t)) < 0. Since the right-hand side of (4.44) is a negative constant, we seeby continuity and Proposition 4.7 (2) that Iγ (u(t)) must remain negative; andso (4.44) holds for all 0≤ t < Tmax.

If Eγ (u0) ≤ 0, then the result follows from Remark 4.6 (2) and (4), so wesuppose

0 < Eγ (u0) < Eγ (Qγ ). (4.45)

We use the notation (4.33) introduced in the proof of Theorem 4.5. It followsfrom (4.24) and (4.44) that df

dt ≥ 2(Eγ (Qγ )−Eγ (u0)), so that

f (t)≥ 2(Eγ (Qγ )−Eγ (u0))t. (4.46)

In particular, we see that

σ(t)def=−e(t)+ −γ

2f (t)≥−Eγ (u0)− γ (Eγ (Qγ )−Eγ (u0))t. (4.47)

We set

τ = (α+ 4)Eγ (u0)

−γα(Eγ (Qγ )−Eγ (u0))>

Eγ (u0)

−γ (Eγ (Qγ )−Eγ (u0)). (4.48)

If Tmax ≤ τ then (4.43) follows from (4.48). We now suppose

Tmax > τ . (4.49)

Setting v0 = u(τ ), we see that the solution v of (4.2) with the initial conditionv(0)= v0 is v(t)= u(t+ τ) and that its maximal existence time Smax is Smax =Tmax− τ . Since σ(τ) > 0 by (4.47) and (4.48) we can argue as in the proof ofTheorem 4.5 (note that η > 0, where η is given by (4.38) with u0 replaced byv0), and we deduce (see (4.40)) that

2(α+ 2)(−Eγ (v0))− γα‖v0‖2L2

2(α+ 2)(−Eγ (v0))− γ (α+ 2)‖v0‖2L2

≤ eγαSmax . (4.50)

(Observe that σ(τ) > 0, so that the denominator on the left-hand side of (4.50)is positive.) Note that by (4.46), (4.48), and the fact that e(t) is nonincreasing

‖v0‖2L2 = ‖u(τ )‖2

L2 ≥ 2(Eγ (Qγ )−Eγ (u0))τ

≥ 2(α+ 4)

−γαEγ (u0)≥ 2(α+ 4)

−γαEγ (v0)

from which it follows that

2(α+ 2)(−Eγ (v0))− γα‖v0‖2L2

2(α+ 2)(−Eγ (v0))− γ (α+ 2)‖v0‖2L2

≥ α

2(α+ 2). (4.51)

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184 T. Cazenave and S. Snoussi

(4.50) and (4.51) yield Smax ≤ 1α

log(

2(α+2)α

). Since Tmax = τ +Smax, the result

follows by applying (4.48).

Remark 4.9 Theorem 4.8 applies to solutions for which the maximal existencetime is arbitrary large. Indeed, given ε > 0, let uε

0 = (1+ ε)Qγ and uε thecorresponding solution of (4.2). It is straightforward to verify that for all ε > 0,Eγ (uε

0) < Eγ (Qγ ) and Iγ (uε0) < 0. Indeed, the function ε �→ Eγ ((1+ ε)Qγ ) is

decreasing on [1,+∞), Iγ (uε0) < (1+ ε)2Iγ (Qγ ) and Iγ (Qγ ) = 0. Hence uε

0

satisfies the assumptions of Theorem 4.8. On the other hand, Qγ is a stationary(hence global) solution of (4.2), so that the blowup time of vε goes to infinityas ε ↓ 0, by continuous dependence.

4.3.2 The Case γ > 0

Levine’s method (Subsection 4.3.1) does not immediately apply when γ > 0,but it can easily be adapted, after a suitable change of variable.

Theorem 4.10 Suppose γ > 0. Let u0 ∈ C0(RN) ∩H1(RN) and let u be the

corresponding solution of (4.2) defined on the maximal interval [0,Tmax). IfE(u0) < 0, where E is defined by (4.8), then u blows up in finite time, i.e.,Tmax <∞. Moreover,

Tmax ≤ 1

αγlog

(1+ γ ‖u0‖2

L2

(α+ 2)(−E(u0))

)<∞. (4.52)

Proof We set v(t)= e−γ tu(t), so that{vt =�v+ eαγ t|v|αv,

v(0)= u0,(4.53)

and we use the arguments in the proof of Theorem 4.5. Setting

f = ‖v‖2L2 , j = ‖∇v‖2

L2 − eαγ t‖v‖α+2Lα+2 , e= 1

2‖∇v‖2

L2 − eαγ t

α+ 2‖v‖α+2

Lα+2 ,

(4.54)it follows from (4.53) that ∫

RNvvt =−j (t) (4.55)

and dedt =−

∫RN |vt|2+αγ e− αγ

2

∫RN |∇u|2, so that

de

dt−αγ e≤−

∫RN|vt|2. (4.56)

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Complex Ginzburg–Landau Equations 185

In particular,e(t)≤ eαγ t e(0)= eαγ tE(u0) < 0. (4.57)

Applying (4.56), Cauchy–Schwarz’s inequality and (4.55), we obtain

− f( de

dt−αγ e

)≥∫|v|2

∫|vt|2 ≥

∣∣∣∫ vvt

∣∣∣2 = j 2 = 1

2(−j )

df

dt. (4.58)

Note that

j = (α+ 2)e− α

2

∫RN|∇v|2 ≤ (α+ 2)e. (4.59)

Since dfdt > 0 by (4.55), (4.59) and (4.57), we deduce from (4.58) and (4.59)

that −f ( dedt −αγ e)≥−α+2

2 e dfdt . Therefore d

dt [e−αγ t(−e)f−α+2

2 ] ≥ 0, and so

e−αγ t(−e(t))≥ [−e(0)]f (0)− α+22 f (t)

α+22 = (−E(u0))‖u0‖−(α+2)

L2 f (t)α+2

2 .

Thus we see that

df

dt=−2j ≥−2(α+ 2)e≥ 2(α+ 2)(−E(u0))‖u0‖−(α+2)

L2 fα+2

2 eαγ t.

This shows that α(α + 2)(−E(u0))‖u0‖−(α+2)L2 eαγ t + d

dt (f− α

2 ) ≤ 0, and (4.52)easily follows.

Remark 4.11 Below are a few comments on Theorem 4.10.

1. Kaplan’s calculations at the beginning of Section 4.3 show that if γ >

0, every nonnegative, nonzero initial value produces finite-time blowup.On the other hand, in dimension N ≥ 2, there exist nontrivial stationarysolutions in C0(RN), which are global solutions of (4.2). Indeed, it isnot difficult to prove that for every η > 0, the solution u of the ODEu′′ + N−1

r u′ + γ u + |u|αu = 0 with the initial conditions u(0) = η andu′(0)= 0 oscillates indefinitely and converges to 0 as r→∞. This yields asolution u ∈C2(RN)∩C0(R

N) of the equation �u+γ u+|u|αu= 0, hencea stationary solution of (4.2). Note that if (N−2)α≤ 4, Pohozaev’s identityimplies that there is no nontrivial stationary solution in H1(RN)∩C0(R

N).When N ≥ 3 and (N − 2)α > 4, whether or not there exist nontrivialstationary solutions in H1(RN) ∩ C0(R

N) seems to be an open problem.Note also that in dimension N = 1, there is no stationary solution inC0(R

N), this can be easily deduced from the resulting ODE.2. In the case γ = 0, α = 2

N is the Fujita critical exponent. If α > 2N , then

small initial values in an appropriate sense give rise to global solutionsof (4.2). On the other hand, if α ≤ 2

N , then every nonnegative, nonzeroinitial value produces finite-time blowup. (See [11, Theorem 1], [15], [19,

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186 T. Cazenave and S. Snoussi

Theorem 2.1], [45, Theorem 1], [18, Corollary 1.12].) However, givenany α ≤ 2

N , there exist nonzero initial values producing global solutions.In the one-dimensional case, they can be initial values that change signsufficiently many times and are sufficiently small [29, Theorem 1.1], [30,Theorem 1.2]. In any dimension, they can be self-similar solutions [14,Theorem 3]. If γ > 0, then equation (4.2) is not scaling-invariant, so thatone cannot expect self-similar solutions.

4.4 The Nonlinear Schrodinger Equation

In this section, we consider the nonlinear Schrodinger equation (4.3). Weassume α < 4

N−2 , and it follows from Proposition 4.3 that the Cauchy problemis locally well-posed in H1(RN). In contrast with the nonlinear heat equation,for which blowup may occur no matter how small α is, blowup for (4.3) cannotoccur if α is too small.

Proposition 4.12 ([12], Theorem 3.1) Suppose 0 < α < 4N and let γ ∈ R.

It follows that for every u0 ∈ H1(RN), the corresponding solution of (4.3) isglobal, i.e. Tmax =∞.

Proof Let u0 ∈ H1(RN) and u be the corresponding solution of (4.3) definedon the maximal interval [0,Tmax). Formula (4.14) yields

‖u(t)‖L2 = eγ t‖u0‖L2 (4.60)

for all 0 ≤ t < Tmax. Applying the blowup alternative on the L2 norm ofProposition 4.3, we conclude that Tmax =∞.

When α ≥ 4N , finite-time blowup may occur. This was proved in [48, p. 911]

in the three-dimensional cubic, radial case with γ = 0, then in [13, p. 1795]in the general case (still with γ = 0). Note that all solutions have locallybounded L2-norm by (4.60), so that Levine’s method used in Section 4.3cannot be applied. Instead, the proof in [48, 13] is based on the varianceidentity (4.17)–(4.18). This argument can easily be applied to the case γ ≥ 0,which we consider first.

4.4.1 The Case γ ≥ 0

The following result is proved in [48, 13] when γ = 0.

Theorem 4.13 Suppose 4N ≤ α < 4

N−2 and γ ≥ 0. Let u0 ∈ H1(RN) and u bethe corresponding solution of (4.3) defined on the maximal interval [0,Tmax).

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Complex Ginzburg–Landau Equations 187

If E(u0) < 0 and u0 ∈ L2(RN , |x|2dx), where E is defined by (4.8), then u blowsup in finite time, i.e., Tmax <∞.

Proof The proof is based on a differential inequality for the variance. Moreprecisely, it follows from (4.17) that

d

dt(e−2γ tV(u(t)))=−4e−2γ tJ(u) (4.61)

and from (4.18) that

d

dt(e−2γ tJ(u(t)))= e−2γ t

[−4E(u(t))+ Nα− 4

α+ 2‖u‖α+2

Lα+2

]≥−4e−2γ tE(u(t)),

(4.62)where we used the assumption Nα ≥ 4 in the last inequality. (4.61) and (4.62)yield

d2

dt2(e−2γ tV(u(t)))=−4

d

dt(e−2γ tJ(u(t)))≤ 16e−2γ tE(u(t)). (4.63)

Since

I(w)= (α+ 2)E(w)− α

2

∫RN|∇u|2 ≤ (α+ 2)E(w)

and γ ≥ 0, it follows from (4.15) that

d

dtE(u(t))≤ γ (α+ 2)E(u(t)) (4.64)

so that

E(u(t))≤ eγ (α+2)tE(u0) < 0. (4.65)

Applying (4.63) and (4.65) we obtain

d2

dt2(e−2γ tV(u(t)))≤ 16eαγ tE(u0)≤ 16E(u0). (4.66)

Note that by (4.61)

d

dt(e−2γ tV(u(t)))|t=0 =−4J(u0). (4.67)

Integrating (4.66) twice and applying (4.67) yields

e−2γ tV(u(t))≤ V(u0)− 4tJ(u0)+ 16E(u0)

∫ t

0

∫ s

0eαγσ dσds (4.68)

for all 0≤ t < Tmax. The right-hand side of (4.68), considered as a function oft ≥ 0, is negative for t large (because E(u0) < 0). Since e−2γ tV(u(t)) ≥ 0, weconclude that Tmax <∞.

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188 T. Cazenave and S. Snoussi

The “natural” condition in Theorem 4.13 is E(u0) < 0. However, we requirethat u0 ∈ L2(RN , |x|2dx) because we calculate the variance V(u). Whether thefinite variance assumption is necessary or not in Theorem 4.13 seems to bean open question. A partial answer is known in the case α = 4

N and γ = 0: ifE(u0) < 0 and ‖u0‖L2 is not too large, then Tmax <∞. (See [24, Theorem 1.1].)Another partial answer is given by Ogawa and Tsutsumi [31, p. 318] forradially symmetric solutions in the case γ = 0 and N ≥ 2. The proof can beadapted to the case γ ≥ 0, under the additional restriction N ≥ 3. (The caseN = 1, γ = 0 and α = 4 is considered in [32, p. 488], but we do not study itsextension to γ > 0 here.)

Theorem 4.14 Suppose 4N ≤ α < 4

N−2 and γ ≥ 0. Assume further N ≥ 2 andα≤ 4 if γ = 0, and N ≥ 3 if γ > 0. Let u0 ∈H1(RN) and u be the correspondingsolution of (4.3) defined on the maximal interval [0,Tmax). If E(u0) < 0, whereE is defined by (4.8), and if u0 is radially symmetric, then u blows up in finitetime, i.e., Tmax <∞.

Proof The proof uses calculations similar to those in the proof of Theo-rem 4.13, but for a truncated variance. It is convenient to set v(t) = e−γ tu(t),so that v satisfies the equation vt = i(�v+ eαγ t|v|αv). Moreover,

‖v(t)‖L2 = ‖u0‖L2 (4.69)

by formula (4.60). Let % ∈C∞(RN)∩W4,∞(RN) be spherically symmetric andset

ζ(t)=∫RN

%|v|2dx. (4.70)

It follows from (4.150) and (4.152) that

1

2ζ ′(t)=$

∫RN

v(∇v · ∇%) (4.71)

and

1

2ζ ′′ =

∫RN

(−1

2|v|2�2%− αeαγ t

α+ 2|v|α+2�%+ 2|∇v|2% ′′

). (4.72)

Note that the calculations in Proposition 4.31 are formal in the case θ = π2 .

However, they are easily justified for H2 solutions, and then by continuousdependence for H1 solutions. We observe that∫RN

(−αeαγ t

α+ 2|v|α+2�%+ 2|∇v|2% ′′

)= 2Nαe−2γ tE(u)− (Nα− 4)

∫RN|∇v|2

+ 2∫RN

(% ′′ − 2)|∇v|2+ αeαγ t

α+ 2

∫RN

(2N−�%)|v|α+2.

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Complex Ginzburg–Landau Equations 189

Since Nα ≥ 4 and E(u(t))≤ eγ (α+2)tE(u0) by (4.65), we deduce that∫RN

(−αeαγ t

α+ 2|v|α+2�%+ 2|∇v|2% ′′

)≤ 2Nαeαγ tE(u0)

+ 2∫RN

(% ′′ − 2)|∇v|2+ αeαγ t

α+ 2

∫RN

(2N−�%)|v|α+2

so that (4.72) yields

1

2ζ ′′ ≤ 2Nαeαγ tE(u0)

+∫RN

(−1

2|v|2�2%+ αeαγ t

α+ 2|v|α+2(2N−�%)+ 2|∇v|2(% ′′ − 2)

).

(4.73)

We first consider the case γ = 0. We apply Lemma 4.32 with A = ‖u0‖L2 ,μ = α

α+2 and ε > 0 sufficiently small so that χμε2(N−1) < 1 and κ(μ,ε) ≤−NαE(u0). With % =%ε given by Lemma 4.32, it follows from (4.69), (4.73)and (4.160) that ζ ′′ ≤ 2NαE(u0) < 0. Since ζ(t) ≥ 0 for all 0 ≤ t < Tmax, weconclude as in the proof of Theorem 4.13 that Tmax <∞.

We next consider the case γ > 0 and N ≥ 3. Let 0 < τ < Tmax. We set

μτ = eαγ τ ≥ 1, (4.74)

we fix1

2> λ>

1

2(N− 1)(4.75)

(here we use N ≥ 3) and we set

ετ = aμ−λτ ≤ a. (4.76)

Here, the constant 0 < a≤ 1 is chosen sufficiently small so that χa2(N−1) < 1,where χ is the constant in Lemma 4.32. Since μτ ≥ 1 and 1− 2(N− 1)λ < 0by (4.75), it follows in particular that χμτε

2(N−1)τ = χμ1−2(N−1)λ

τ a2(N−1) ≤χa2(N−1) < 1. Moreover, we deduce from (4.75) that κ defined by (4.161)satisfies κ(μ,ε)≤ Cμ1−δ

τ , where C is independent of τ , and

δ = αmin{λN

2,2(N− 1)λ− 1

4−α

}> 0. (4.77)

We now let % =%ετ where %ε is given by Lemma 4.32 for this choice of ε. Itfollows from (4.159), (4.160) and the inequality κ(μ,ε)≤ Cμ1−δ

τ that

−2∫RN

(2−% ′′ετ)|∇v|2+ α

α+ 2eαγ t

∫RN

(2N−�%ετ )|v|α+2

− 1

2

∫RN|v|2�2%ετ ≤ Cμ1−δ

τ .(4.78)

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190 T. Cazenave and S. Snoussi

Estimates (4.73) and (4.78) yield

1

2ζ ′′ ≤ 2Nαeαγ tE(u0)+Cμ1−δ

τ (4.79)

for all 0≤ t≤ τ . Integrating (4.79) twice and applying (4.71), we deduce that

1

2ζ(τ )≤ 1

2‖%ετ ‖L∞‖u0‖2

L2 + τ‖∇%ετ ‖L∞‖u0‖2H1

+ 2N

αγ 2E(u0)(e

αγ τ − τ −αγ τ)+Cμ1−δτ τ .

(4.80)

Using (4.158) to estimate % in the above inequality, applying (4.74) and (4.76)to express μτ and ετ in terms of τ , and since ζ(τ )≥ 0, we obtain

0≤ Ce2λαγ τ +Cτeλαγ τ + 2N

αγ 2E(u0)(e

αγ τ − τ −αγ τ)+Cτe(1−δ)αγ τ . (4.81)

Since E(u0) < 0 and max{2λ,1− δ}< 1, the right-hand side of (4.81) is nega-tive for τ large. Since τ < Tmax is arbitrary, we conclude that Tmax <∞.

4.4.2 The Case γ < 0

If γ < 0, the argument used in the proof of Theorem 4.13 breaks downbecause (4.64) does not hold. Yet blowup occurs when α > 4

N , as is shownby the following result of Tsutsumi [44, Theorem 1].

Theorem 4.15 Suppose 4N < α < 4

N−2 and γ < 0. Let u0 ∈ H1(RN) andlet u be the corresponding solution of (4.3) defined on the maximal interval[0,Tmax). If

V(u0)+ Nα− 4

γαJ(u0)+ (Nα− 4)2

γ 2α2E(u0) < 0, (4.82)

where the functionals E, V and J are defined by (4.8), (4.16) and (4.19),respectively, then u blows up in finite time, i.e., Tmax <∞.

Proof We follow the simplified argument given in [33]. We define

W(w)= 1

2

∫RN|∇w|2− Nα

4(α+ 2)

∫RN|w|α+2 (4.83)

and we set e(t)= E(u(t)), v(t)= V(u(t)), ı(t)= I(u(t)), j (t)= J(u(t)), w(t)=W(u(t)), where the functionals E, V , I, J and W are defined by (4.8), (4.16),(4.7), (4.19) and (4.83), respectively. It follows from (4.15), (4.17) and (4.18)that

de

dt= γ ı(t),

dv

dt= 2γ v(t)− 4j (t),

dj

dt= 2γ j (t)− 4w(t). (4.84)

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Complex Ginzburg–Landau Equations 191

It is convenient to define

b=−2γ4− (N− 2)α

Nα− 4> 0, η=−2γ + b= −4γα

Nα− 4> 0.

Using the identity γ ı − be=−ηw, we deduce from (4.84) that

d

dt(e−bte(t))= e−bt(γ ı − be)=−ηe−btw(t), (4.85)

d

dt(e−btv(t))=−ηe−btv(t)− 4e−btj (t), (4.86)

d

dt(e−btj (t))=−ηe−btj (t)− 4e−btw(t). (4.87)

Integrating (4.85) on (0, t), we obtain

e−bte(t)+η

∫ t

0e−bsw(t)= E(u0).

Since α ≥ 4N , we have e≥w so that

e−btw(t)+η

∫ t

0e−bsw(s)ds≤ E(u0). (4.88)

We now set

w(t)=∫ t

0e−bsw(s)ds, j (t)=

∫ t

0e−bsj (s)ds

so that (4.88) becomes dwdt + ηw ≤ E(u0). Therefore, eηtw(t) ≤ eηt−1

ηE(u0),

which implies ∫ t

0eηsw(s)ds≤ eηt− 1−ηt

η2E(u0). (4.89)

Integrating now (4.87) on (0, t), we obtain djdt +ηj = J(u0)− 4w, so that

eηtj (t)=∫ t

0eηs[J(u0)− 4w(s)]ds. (4.90)

We deduce from (4.90) and (4.89) that

eηtj (t)≥ eηt − 1

ηJ(u0)− 4

eηt − 1−ηt

η2E(u0). (4.91)

Finally, since v(t) ≥ 0 we deduce from (4.86) that ddt (e

−btv(t)) ≤ −4e−btj (t),so that

e−btv(t)≤ V(u0)− 4j (t). (4.92)

It now follows from (4.92) and (4.91) that

e−btv(t)≤ V(u0)− 41− e−ηt

ηJ(u0)+ 16

1− (1−ηt)e−ηt

η2E(u0). (4.93)

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192 T. Cazenave and S. Snoussi

Assumption (4.82) means that V(u0)− 4ηJ(u0)+ 16

η2 E(u0) < 0. Therefore, theright-hand side of (4.93) becomes negative for t large, which implies thatTmax <∞.

Remark 4.16 Here are a few comments on Theorem 4.15.

1. The condition (4.82) is satisfied if u0 = cϕ with ϕ ∈ H1(RN), ϕ = 0 and cis large.

2. The condition α > 4N is essential in the proof, for the definition of η and b.

If α= 4/N, then finite-time blowup occurs for some initial data [28, 7], butthe proof follows a very different argument.

3. We are not aware of a result similar to Theorem 4.15 for initial values ofinfinite variance (in the spirit of Theorem 4.14).

4.5 The Complex Ginzburg–Landau Equation

4.5.1 Sufficient Condition for Finite-time Blowup

In this section, we derive sufficient conditions for finite-time blowup inequation (4.1), and upper estimates of the blowup time. Such conditions areobtained in [41, Theorem 1.2] in the case γ ≤ 0, in [6, Theorem 1.1] in thecase γ = 0, and in [41, Theorem 1.3] and [5, Theorem 1.1] in the case γ > 0.The upper bound is established in [6, Theorem 1.1] in the case γ = 0.

Theorem 4.17 Let γ ∈ R, α > 0, 0 ≤ θ < π2 , u0 ∈ C0(R

N) ∩ H1(RN), andlet u be the corresponding solution of (4.1) defined on the maximal interval[0,Tmax). If {

E(u0) < 0, γ ≥ 0,

E γcosθ

(u0) < 0, γ ≤ 0(4.94)

(with the definitions (4.8) and (4.22)) then u blows up in finite time, i.e., Tmax <

∞. Moreover,

Tmax ≤

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1

γαlog

(1+ γ ‖u0‖2

L2

(α+ 2)(−E(u0))cosθ

), γ > 0,

‖u0‖2L2

α(α+ 2)(−E(u0))cosθ, γ = 0,

1

−γαlog

(1+ −2γ ‖u0‖2

L2

2(α+ 2)(−E γcosθ

(u0))cosθ − γα‖u0‖2L2

), γ < 0.

(4.95)

Proof We consider separately the cases γ ≥ 0 and γ < 0.

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Complex Ginzburg–Landau Equations 193

The case γ ≥ 0 We follow the argument of the proof of Theorem 4.10, and inparticular we use the same notation (4.54). We indicate only the minor changesthat are necessary. The function v(t)= e−γ tu(t) now satisfies the equation{

vt = eiθ [�v+ eαγ t|v|αv],v(0)= u0.

(4.96)

Identity (4.55) becomes ∫RN

vvt =−eiθ j (t), (4.97)

so thatdf

dt= 2(

∫RN

vvt =−2j (t)cosθ . (4.98)

Moreover, dedt =−cosθ

∫RN |vt|2+αγ e− αγ

2

∫RN |∇u|2, so that inequality (4.56)

becomesde

dt−αγ e≤−cosθ

∫RN|vt|2. (4.99)

Applying (4.99), Cauchy–Schwarz, (4.97) and (4.98), we obtain

−f( de

dt−αγ e

)≥ cosθ

∫|v|2

∫|vt|2 ≥ cosθ

∣∣∣∫ vvt

∣∣∣2 = cosθ j 2 = 1

2(−j )

df

dt.

(4.100)

The crux is that the factor cosθ in the first inequalities in (4.100) has beencancelled in the last one by using (4.98). In particular, the left-hand and theright-hand terms in (4.100) are the same as in (4.58). Therefore, we may nowcontinue the argument as in the proof of Theorem 4.10. Using (4.98) insteadof (4.55), we arrive at the inequality

α(α+ 2)(−E(u0))‖u0‖−(α+2)L2 eαγ t cosθ + d

dt(f−

α2 )≤ 0

and estimate (4.95) easily follows in both the cases γ > 0 and γ = 0.

The case γ < 0 Since the result in the case γ ≥ 0 is obtained by the argumentof the proof of Theorem 4.10, one could try now to follow the proof ofTheorem 4.5. It turns out that this strategy leads to intricate calculations andunnecessary conditions. (See [5].) Instead, we follow the strategy of [41] andwe set

μ= (−γ )−12 (cosθ)

12 , (4.101)

v(t,x)= e−it sinθμ2α u(μ2t,μx), (4.102)

v0(x)=μ2α u0(μx), (4.103)

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194 T. Cazenave and S. Snoussi

so that {vt = eiθ [�v+|v|αv− v],v(0)= v0.

(4.104)

Since u is defined on [0,Tmax), v is defined on [0,Smax) with

Smax = −γTmax

cosθ. (4.105)

We introduce the notation

f = ‖v‖2L2 , j = I−1(v(t)), e= E−1(v(t)), (4.106)

where I−1 and E−1 are defined by (4.21) and (4.22), and we observe that

‖v0‖2L2 =μ

4α−N‖u0‖2

L2 , E−1(v0)=μ2+ 4α−NE γ

cosθ(u0). (4.107)

We now follow the proof of Theorem 4.5. Equation (4.104) yields∫RN

vvt =−eiθ j (t), (4.108)

df

dt=−2j (t)cosθ , (4.109)

de

dt=−cosθ

∫RN|vt|2. (4.110)

Since E γcosθ

(u0) < 0, we deduce from (4.107) that e(0) < 0. Therefore, e(t) <

0 by (4.110) (hence j (t) < 0) and dfdt > 0 by (4.109). Applying (4.110),

Cauchy–Schwarz, (4.108) and (4.109), we obtain

− fde

dt= cosθ

∫|v|2

∫|vt|2 ≥ cosθ

∣∣∣∫ vvt

∣∣∣2 = cosθ j 2 = 1

2(−j )

df

dt. (4.111)

At this point, we use the property

j(t)= (α+2)e(t)− α

2f (t)− α

2

∫RN|∇v(t)|2≤ (α+2)e(t)− α

2f (t)< 0, (4.112)

so that (4.111) yields −f dedt ≤ 1

2 (−(α+ 2)e+ α2 f ) df

dt . Therefore, ddt (−ef−

α+22 +

12 f−

α2 )≥ 0, and so

− e+ 1

2f ≥ ηf

α+22 , (4.113)

with

η= (−e(0))f (0)−α+2

2 + 1

2f (0)−

α2 > 0. (4.114)

It follows from (4.109), (4.112) and (4.113) that

df

dt≥ [2(α+ 2)(−e)+ αf ]cosθ ≥ (−2f + 2(α+ 2)ηf

α+22 )cosθ .

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Complex Ginzburg–Landau Equations 195

Therefore, ddt [(e2t cosθ f )−

α2 − (α + 2)ηe−αt cosθ ] ≤ 0, so that (α + 2)η(1 −

e−αt cosθ )≤ f (0)−α2 for all 0≤ t < Smax. It follows easily that

Smax ≤ 1

α cosθlog

(1+ 2‖v0‖2

L2

2(α+ 2)(−E−1(v0)+α‖v0‖2L2)

). (4.115)

Applying (4.105) (4.107), and (4.101), estimate (4.95) follows.

Remark 4.18 One can study equation (4.104) for its own sake. The proofof Theorem 4.17 shows that if v0 ∈ C0(RN) ∩ H1(RN) satisfies E−1(v0) ≤0 and v0 ≡ 0, then the corresponding solution of (4.104) defined on themaximal interval [0,Smax) blows up in finite time, and (4.115) holds. It followsfrom (4.115) that

Smax ≤ 1

α cosθlog

(α+ 2

α

). (4.116)

In particular, the bound in (4.116) is independent of v0, so that this result doesnot apply to solutions for which the blowup time would be large. When α <

4N−2 , this restriction can be improved by the potential well argument we usedin Theorem 4.8 for the heat equation. More precisely, if v0 ∈C0(R

N)∩H1(RN)

satisfies E−1(v0)<E−1(Q−1) and I−1(v0)< 0, where Q−1 is as in Theorem 4.8,then the corresponding solution of (4.104) defined on the maximal interval[0,Smax). blows up in finite time, i.e., Smax <∞, and

Smax ≤ 1

α cosθ

[(α+ 4)[E−1(v0)]+

E−1(Q−1)−E−1(v0)+ log

(2(α+ 2)

α

)]. (4.117)

The proof is easily adapted from the proof of Theorem 4.8, in the sameway as the proof of Theorem 4.17 (case γ ≤ 0) is adapted from the proofof Theorem 4.5. Note that this last result applies to solutions for which themaximal existence time is arbitrary large. Indeed, given ε > 0, vε0 = (1+ε)Q−1

satisfies E−1(vε0) < E−1(Q−1) and I−1(v

ε0) < 0, while the blowup time of the

corresponding solution of (4.104) goes to infinity as ε ↓ 0. (See Remark 4.9.)

Remark 4.19 Here are some comments on Theorem 4.17.

1. If γ ≤ 0, one can replace assumption (4.94) by the slightly weakerassumption E γ

cosθ(u0)≤ 0 and u0 = 0. See Remark 4.6 (1).

2. Let α > 0, 0≤ θ < π2 , and fix u0 ∈ C0(R

N)∩H1(RN) such that E(u0) < 0.It follows from (4.95) that Tmax → 0 as γ →∞. On the other hand, itis clear that if γ is sufficiently negative, then E γ

cosθ(u0) ≥ 0, so that one

cannot apply Theorem 4.17. This is not surprising, since the correspondingsolution of (4.1) is global if γ is sufficiently negative. (See Remark 4.2.)

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196 T. Cazenave and S. Snoussi

3. Let α > 0 and γ < 0. Given u0 ∈C0(RN)∩H1(RN) such that E γ

cosθ(u0) < 0,

it follows from (4.95) that Tmax <1

−γαlog( α+2

α). In particular, we see that

Theorem 4.17 does not apply to solutions for which the blowup time wouldbe large. However, one can use the result presented in Remark 4.18. Usingthe transformation (4.101)–(4.103), and formulas (4.105) and (4.107), wededuce from (4.117) that if E γ

cosθ(u0)<μ−2− 4

α+NE−1(Q−1) and I γcosθ

(u0)<

0, then the corresponding solution of (4.1) blows up in finite time and

Tmax ≤ 1

−γα

⎡⎣ (α+ 4)[E γcosθ

(u0)]+μ−2− 4

α+NE−1(Q−1)−E γcosθ

(u0)+ log

(2(α+ 2)

α

)⎤⎦ .

Moreover, this property applies to solutions for which the maximal exis-tence time is arbitrary large. (The stationary solution Q−1 of (4.104) corre-sponds to the standing wave u(t,x)=μ−

2α eitμ−2 sinθQ−1(μ

−1x) of (4.1).)

4.5.2 Behavior of the Blowup Time as a Function of θ

Fix α > 0 and γ ∈R. Given u0 ∈ C0(RN)∩H1(RN), we let uθ , for 0≤ θ < π2 ,

be the solution of (4.1) defined on the maximal interval [0,Tθmax). If γ ≥ 0

and E(u0) < 0, or if γ < 0 and E γcosθ

(u0) < 0, then we know that Tθmax <∞.

(See Theorem 4.17.) We now study the behavior of Tθmax as θ → π

2 , i.e. asthe equation (4.1) approaches the nonlinear Schrodinger equation (4.3). Weconsider separately the cases α < 4

N and α ≥ 4N .

The Case α < 4N

If α < 4N , then all solutions of the limiting equation (4.3) are global by

Proposition 4.12, and so we may expect that Tθmax → ∞ as θ → π

2 . Thisis indeed what happens. Indeed, one possible proof of global existence forthe nonlinear Schrodinger equation (4.3) is based on the Gagliardo–Nirenberginequality

‖w‖α+2Lα+2 ≤ ‖∇w‖2

L2 +A‖w‖2+ 4α4−Nα

L2 , (4.118)

where A is a constant independent of w ∈ H1(RN). (See e.g. [1].) Similarly,using (4.118) one can prove the following result. (For γ = 0, this is [6,Theorem 1.2].)

Theorem 4.20 Let 0 < α < 4N and γ ∈ R. Given u0 ∈ C0(R

N)∩H1(RN), letuθ , for 0 ≤ θ < π

2 , be the solution of (4.1) defined on the maximal interval

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Complex Ginzburg–Landau Equations 197

[0,Tθmax). If γ ≥ 0, then

Tθmax ≥

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩4−Nα

4αγlog

(1+ γ

A‖u0‖4α

4−Nα

L2 cosθ

), γ > 0,

4−Nα

4αA‖u0‖4α

4−Nα

L2 cosθ, γ = 0,

(4.119)

and if γ < 0, then

Tθmax ≥

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∞, cosθ ≤ −γ

A‖u0‖4α

4−Nα

L2

,

4−Nα

4αγlog

(1+ γ

A‖u0‖4α

4−Nα

L2 cosθ

), cosθ >

−γ

A‖u0‖4α

4−Nα

L2

,(4.120)

where A is the constant in (4.118).

Proof We combine (4.10) and (4.118) to obtain the desired conclusion. Settingf (t)= ‖uθ (t)‖2

L2 , we deduce from (4.10) and (4.118) that

df

dt= 2γ f + cosθ(−2‖∇u‖2

L2 + 2‖w‖α+2Lα+2)≤ 2γ f + 2Af 1+ 2α

4−Nα cosθ

so thatd

dt(e−2γ tf )≤ 2Ae

4αγ t4−Nα (e−2γ tf )1+ 2α

4−Nα cosθ .

This means that ddt (−(e−2γ tf )−

2α4−Nα ) ≤ 4Aα cosθ

4−Nαe

4αγ t4−Nα , from which we deduce

that

(e−2γ tf )−2α

4−Nα ≥ ‖u0‖− 4α4−Nα − 4Aα cosθ

4−Nα

∫ t

0e

4αγ s4−Nα ds.

The above inequality yields a control on ‖u(t)‖L2 for all 0≤ t < Tθmax such that

4Aα cosθ4−Nα

∫ t0 e

4αγ s4−Nα ds < ‖u0‖− 4α

4−Nα . Therefore, if we set

τ = sup{

0≤ t <∞;4Aα cosθ

4−Nα

∫ t

0e

4αγ s4−Nα ds < ‖u0‖− 4α

4−Nα

}(4.121)

then it follows from the blowup alternative on the L2-norm in Proposition 4.1that Tθ

max ≥ τ . The result follows by calculating the integral in (4.121) in thevarious cases γ > 0, γ = 0 and γ < 0.

Remark 4.21 Here are some comments on Theorem 4.20.

1. Suppose γ ≥ 0. It follows from (4.120) that the upper estimate (4.95) ofTθ

max established in Theorem 4.17 is optimal with respect to the dependence

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198 T. Cazenave and S. Snoussi

in θ . Indeed, if E(u0) < 0, then it follows from (4.120) and (4.95) that

0 < liminfθ→ π

2

φ(θ)Tθmax ≤ limsup

θ→ π2

φ(θ)Tθmax <∞,

with φ(θ)= cosθ for γ = 0 and φ(θ)= [log((cosθ)−1)]−1 for γ > 0.2. Suppose γ < 0. It follows from (4.120) that, given any initial value u0 ∈

C0(RN) ∩H1(RN), the corresponding solution of (4.1) is global for all θ

sufficiently close to π2 .

3. We can apply Theorem 4.20 to equation (4.104). The upper esti-mate (4.120), together with formulas (4.107) and (4.105), shows that ifv0 ∈ C0(R

N) ∩ H1(RN) and v is the corresponding solution of (4.104)

defined on the maximal interval [0,Smax), then Smax =∞ if A‖v0‖4α

4−Nα

L2 ≤ 1,and

Smax ≥− 4−Nα

4α cosθlog

(1− 1

A‖v0‖4α

4−Nα

L2

)(4.122)

if A‖v0‖4α

4−Nα

L2 > 1. Suppose now that either E−1(v0)≤ 0 and v0 = 0 or else0 < E−1(v0) < E−1(Q−1) and I−1(v0) < 0. In both cases I−1(v0) < 0, and itfollows from (4.118) that

‖∇v0‖2L2 +‖v0‖2

L2 < ‖v0‖α+2Lα+2 ≤ ‖∇v0‖2

L2 +A‖v0‖2+ 4α4−Nα

L2 .

In particular, A‖v0‖4α

4−Nα

L2 > 1 so that

0 < liminfθ→ π

2

(cosθ)Sθmax ≤ limsup

θ→ π2

(cosθ)Sθmax <∞

by (4.122) and either (4.115) or (4.117).

The Case α > 4N

If 4N ≤ α < 4

N−2 , then the solution of the limiting nonlinear Schrodingerequation (4.3) blows up in finite time, under appropriate assumptions on theinitial value u0. See Theorems 4.13, 4.14 and 4.15. Under these assumptions,one might expect that Tθ

max(u0), which is finite (under suitable assumptions) byTheorem 4.17, remains bounded as θ→ π

2 . It appears that no complete answeris known to this problem.

We first consider the case γ ≥ 0, for which one can give a partial answer. IfE(u0) < 0, then Tθ

max <∞ for all 0 ≤ θ < π2 by Theorem 4.17. However, the

bound in (4.95) blows up as θ → π2 . The proof of (4.95) is based on Levine’s

argument for blowup in the nonlinear heat equation (4.2). Since, as observed

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Complex Ginzburg–Landau Equations 199

before, Levine’s argument does not apply to the limiting nonlinear Schrodin-ger equation, it is not surprising that the bound in (4.95) becomes inaccurateas θ → π

2 . This observation suggests we should adapt the proof of blowupfor (4.3) to equation (4.1), in order to obtain a bound on Tθ

max as θ → π2 . This

strategy proved to be successful in [6, Theorem 1.5] for γ = 0, and it can beextended to the case γ ≥ 0. More precisely, we have the following result.

Theorem 4.22 Let 4N ≤ α ≤ 4, γ ≥ 0 and N ≥ 2. Assume that N ≥ 3 if γ > 0.

Let u0 ∈ H1(RN)∩C0(RN) be radially symmetric and, given any 0 ≤ θ < π

2 ,let uθ be the corresponding solution of (4.1) defined on the maximal interval[0,Tθ

max). If E(u0) < 0, then sup0≤θ< π2

Tθmax <∞.

We prove Theorem 4.22 by following the strategy of [6]. We adapt the proofof Theorem 4.14, and in particular we consider v(t) = e−γ tu(t), which satis-fies equation (4.96). The corresponding identities for the truncated varianceare given by Proposition 4.31; and the Caffarelli–Kohn–Nirenberg estimateby Lemma 4.32. The terms involving cosθ in (4.152) can be controlledusing (4.98). Yet there is a major difference between equations (4.3) and (4.1)that must be taken care of. For (4.3), the L2-norm of the solutions is controlledby formula (4.14). The resulting estimate for v is essential when applyingLemma 4.32. For (4.1), there is no such a priori estimate. However, one canestimate v on an interval [0,T], where T is proportional to Tθ

max. More precisely,we have the following result, similar to [6, Lemma 5.2].

Lemma 4.23 Fix 0 ≤ θ < π2 . Let u0 ∈ C0(RN) ∩H1(RN), and consider the

corresponding solution v of (4.96) defined on the maximal interval [0,Tmax).Set

τ = sup{t ∈ [0,Tmax); ‖v(s)‖2L2 ≤ K‖u0‖2

L2 for 0≤ s≤ t}, (4.123)

where

K =[1−

( α+ 4

2α+ 4

) 12]−1

> 1 (4.124)

so that 0 < τ ≤ Tmax. If E(u0) < 0, then Tmax ≤ α+4α

τ .

Proof We use the notation of the proof of Theorem 4.17, and in partic-ular (4.54). The proof is based Levine’s argument used in the proof ofTheorem 4.4, which shows that if ‖v‖2

L2 achieves the value K‖u0‖2L2 at a certain

time t, then v must blow up within a lapse of time which is controlled byt. More precisely, let τ be given by (4.123). If τ = Tmax, there is nothing toprove, so we assume τ < Tmax. It follows that ‖v(τ )‖2

L2 = K‖u0‖2L2 , so that

f (t)≤ f (τ )= Kf (0), 0≤ t≤ τ . (4.125)

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200 T. Cazenave and S. Snoussi

Since E(u0) < 0, it follows (see the proof of Theorem 4.17) that f isnondecreasing on [0,Tmax); and so, using (4.125),

f (t)≥ Kf (0), τ ≤ t < Tmax. (4.126)

We deduce from (4.99) that

e−αγ t e(t)≤ E(u0)− cosθ∫ t

0e−αγ s

∫RN|vt(s)|2,

so that

e(t)≤−cosθ∫ t

0

∫RN|vt|2. (4.127)

Since dfdt ≥ −2cosθ j ≥ −2(α+ 2)cosθ e by (4.98), we deduce from (4.127)

thatdf

dt≥ 2(α+ 2)cos2 θ

∫ t

0

∫RN|vt|2. (4.128)

Set

M(t)= 1

2

∫ t

0f (s)ds. (4.129)

It follows from (4.128) and Cauchy–Schwarz’s inequality that (see the proofof Theorem 4.4)

MM′′ ≥ α+ 2

2cos2 θ

(∫ t

0

∣∣∣∫RN

vtv∣∣∣)2

. (4.130)

Since j ≤ (α+ 2)e≤ 0, identities (4.97) and (4.98) yield∣∣∣∫RN

vtv∣∣∣=−j = 1

2cosθ

df

dt= 1

cosθM′′(t)

so that (4.130) becomes

MM′′ ≥ α+ 2

2(M′(t)− M′(0))2 = α+ 2

8(f (t)− f (0))2. (4.131)

It follows from (4.131), (4.126) and (4.124) that

MM′′ ≥ α+ 2

8

(K− 1

K

)2f (t)2 = α+ 4

16f (t)2 = α+ 4

4[M′(t)]2 (4.132)

for all τ ≤ t < Tmax. This means that (M− α4 )′′ ≤ 0 on [τ ,Tmax); and so

M(t)−α4 ≤ M(τ )−

α4 +(t−τ)(M− α

4 )′(τ )= M(τ )−α4

[1− α

4(t−τ)M(τ )−1M′(τ )

]

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Complex Ginzburg–Landau Equations 201

for τ ≤ t < Tmax. Since M(t)−α4 > 0, we deduce that for every τ ≤ t < Tmax,

α4 (t− τ)M(τ )−1M′(τ )≤ 1, i.e.,

(t− τ )f (τ )≤ 4

α

∫ τ

0f (s)ds≤ 4

ατ f (τ ), (4.133)

where we used (4.125) in the last inequality. Thus t≤ α+4α

τ for all τ ≤ t <Tmax,which proves the desired inequality.

Proof of Theorem 4.22 We set vθ (t) = e−γ tuθ (t), thus vθ is the solutionof (4.96) on [0,Tθ

max). We let K be defined by (4.124) and we set

τθ = sup{t ∈ [0,Tθmax); ‖vθ (s)‖2

L2 ≤ K‖u0‖2L2 for 0≤ s≤ t}. (4.134)

Therefore,

sup0≤θ< π

2

sup0≤t<τθ

‖vθ (t)‖2L2 ≤ K‖u0‖2

L2 (4.135)

and, by Lemma 4.23,

Tθmax ≤

α+ 4

ατθ (4.136)

so that we only need a bound on τθ . We first derive an inequality (4.141) bycalculating a truncated variance. Let % ∈C∞(RN)∩W4,∞(RN) be real-valued,nonnegative, and radially symmetric. We set

ζθ (t)=∫RN

%(x)|vθ (t,x)|2dx,

Hθ (t)=∫RN

{−2(2−% ′′)|vθr |2+

α

α+ 2eαγ t(2N−�%)|vθ |α+2−1

2|vθ |2�2%

},

Kθ (t)=∫RN

{−2%|vθr |2+

α+ 4

α+ 2eαγ t%|vθ |α+2+|vθ |2�%

}and we observe that

−Kθ (0)≤ C(‖%‖L∞ +‖�%‖L∞)‖u0‖2H1 . (4.137)

We apply Proposition 4.31 with f (t)≡ eαγ t. It follows from (4.150) that

1

2ζ ′θ (0)≤ C(‖%‖L∞ +‖∇%‖L∞ +‖�%‖L∞)(1+‖u0‖α+2

H1 ) (4.138)

and from (4.152) that

1

2ζ ′′θ ≤−

1

2

∫RN|vθ |2�2%− αeαγ t

α+ 2

∫RN|vθ |α+2�%+ 2

∫RN

% ′′|vθr |2

+ cosθd

dtKθ .

(4.139)

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202 T. Cazenave and S. Snoussi

Using the identity

− 1

2

∫RN|vθ |2�2%− αeαγ t

α+ 2

∫RN|vθ |α+2�%+ 2

∫RN

% ′′|vθr |2,

= 2Nαe(t)+Hθ (t)− (Nα− 4)∫RN|vθr |2

where e(t) is defined by (4.54), the estimate e(t)≤ eαγ tE(u0) by (4.99), and theassumption Nα ≥ 4, we deduce from (4.139) that

1

2ζ ′′θ ≤ 2Nαeαγ tE(u0)+Hθ (t)+ cosθ

d

dtKθ . (4.140)

Integrating the above inequality twice, and since ζθ ≥ 0, we obtain

0≤τθ

(1

2ζ ′θ (0)− cosθKθ (0)

)+ 2NαE(u0)

∫ τθ

0

∫ s

0eαγσdσds

+∫ τθ

0

∫ s

0Hθ (σ )dσds+ cosθ

∫ τθ

0Kθ (s)ds.

(4.141)

We derive a bound on τθ from (4.141). In order to do so we show that, for largetime, the dominating term in the right-hand side is the middle one, which isnegative.

We first obtain an estimate of the last term in (4.141), for which the factorcosθ is essential. Indeed, it follows from (4.98) that

d

dt

∫RN|vθ |2 = 2cosθ

(−2e(t)+ α

α+ 2eαγ t

∫RN|vθ |α+2

),

where e(t) is defined by (4.54). Since e(t) ≤ 0 by (4.99), we deduce byintegrating on (0,τθ ) and applying (4.135) that

α+ 2cosθ

∫ t

0eαγ s

∫RN|vθ |α+2 ≤ (K− 1)‖u0‖2

L2

for all 0≤ t≤ τθ . It follows that

cosθ∫ τθ

0Kθ (s)ds≤ C(‖%‖L∞ +‖�%‖L∞)‖u0‖2

L2 , (4.142)

where we used (4.135) again to estimate the factor of �%.We conclude by estimating the term involving Hθ in (4.141) with

Lemma 4.32. We first consider the case γ = 0. We apply Lemma 4.32 withA=‖u0‖L2 , μ= α

α+2 and ε > 0 chosen sufficiently small so that χμε2(N−1) < 1and κ(μ,ε) ≤ −NαE(u0). With % = %ε given by Lemma 4.32, it followsfrom (4.160) that Hθ (t)≤−NαE(u0) for all 0≤ t < τθ . Therefore, we deduce

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Complex Ginzburg–Landau Equations 203

from (4.141) and (4.142) that

0≤ τθ

(1

2ζ ′θ (0)− cosθKθ (0)

)+Nα

(τθ )2

2E(u0).

Since E(u0) < 0, we conclude that sup0≤θ< π2τθ <∞.

We next consider the case γ > 0 (and so N ≥ 3). We apply Lemma 4.32, thistime with with A= ‖u0‖L2 and

μ=μθ = eαγ τθ . (4.143)

The additional difficulty with respect to the case γ = 0 is that μθ may, inprinciple, be large. We fix λ satisfying (4.75) (we use the assumption N ≥ 3)and we set

εθ = aμ−λθ ≤ a. (4.144)

Here, 0 < a ≤ 1 is chosen sufficiently small so that χa2(N−1) < 1, whereχ is the constant in Lemma 4.32. Since μθ ≥ 1 and 1 − 2(N − 1)λ < 0by (4.75), it follows in particular that χμθε

2(N−1)θ = χμ

1−2(N−1)λθ a2(N−1) ≤

χa2(N−1) < 1. Moreover, we deduce from (4.75) that κ defined by (4.161)satisfies κ(μθ ,εθ ) ≤ Cμ1−δ

θ , where C is independent of θ , and δ > 0 is givenby (4.77). We now let % = %εθ where %ε is given by Lemma 4.32 for thischoice of ε, and it follows from (4.159) and (4.160) that

Hθ (t)≤ Ce(1−δ)αγ τθ , (4.145)

and from (4.158), (4.144) and (4.143) that

‖%εθ ‖L∞ +‖∇%εθ ‖L∞ +‖�%εθ ‖L∞ ≤ Ce2λαγ τθ . (4.146)

Finally, we estimate the first term in the right-hand side of (4.141) and wededuce from (4.138), (4.137), (4.146) and (4.143) that

1

2ζ ′θ (0)− cosθKθ (0)≤ Ce2λαγ τθ . (4.147)

Estimates (4.141), (4.147), (4.145), (4.142), (4.146) and (4.143) now yield

0≤C(1+τθ )e2λαγ τθ + 2N

αγ 2E(u0)(e

αγ τθ −1−αγ τθ )+Cτ 2θ e(1−δ)αγ τθ . (4.148)

Since E(u0) < 0, and max{2λ,1− δ} < 1, the right-hand side of the aboveinequality is negative if τθ is large. Thus sup0≤θ< π

2τθ <∞, which completes

the proof.

Remark 4.24 Under the assumptions of Theorem 4.22, we know that Tθmax

remains bounded. On the other hand, we do not know if Tθmax has a limit as

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204 T. Cazenave and S. Snoussi

θ → π2 , and if it does, if this limit is the blowup time of the solution of the

limiting Schrodinger equation.

We end this section by considering the case γ < 0. The condition for blowupin Theorem 4.17 in this case is E γ

cosθ(u0) < 0. Given u0 ∈ C0(RN)∩H1(RN),

u0 = 0, it is clear that E γcosθ

(u0) > 0 for all θ sufficiently close to π2 , and we do

not know if there exists an initial value u0 such that the corresponding solutionof (4.1) blows up in finite time for all θ close to π

2 . (See Open Problem 4.28.)Another point of view concerning the case γ < 0 is to apply the trans-

formation (4.101)–(4.103) and study the resulting equation (4.104). Let v0 ∈C0(R

N) ∩ H1(RN) and, given 0 ≤ θ < π2 , let vθ the corresponding solution

of (4.104) defined on the maximal interval [0,Sθmax). If E−1(v0) < 0, then it

follows from (4.116) that Sθmax <∞ for all 0 ≤ θ < π

2 . Therefore, it makessense to study the behavior of Sθ

max as θ → π2 , and we have the following

result.

Theorem 4.25 Suppose N ≥ 2, 4N ≤ α ≤ 4, and fix a radially symmetric initial

value v0 ∈H1(RN)∩C0(RN). Given any 0≤ θ < π

2 , let vθ be the correspondingsolution of (4.104) defined on the maximal interval [0,Sθ

max). If E−1(v0) < 0,then sup0≤θ< π

2Tθ

max <∞.

The proof of Theorem 4.25 is very similar to the proof of [6, Theorem 1.5],with minor modifications only. More precisely, it is not difficult to adapt theproof of Lemma 4.23 to show that if

τθ = sup{t ∈ [0,Sθmax); ‖vθ (s)‖2

L2 ≤ K‖v0‖2L2 for 0≤ s≤ t},

where K is defined by (4.124), then Sθmax ≤ α+4

ατθ . Moreover, given a

real-valued, radially symmetric function % ∈C∞(RN)∩W4,∞(RN), and setting

ζθ (t)=∫RN

%(x)|vθ (t,x)|2dx,

it is not difficult to deduce from Proposition 4.31 the variance identities

1

2ζ ′θ (t)=cosθ

∫RN

{−%|vθr |2+%|vθ |α+2−%|vθ |2+ 1

2|vθ |2�%

}+ sinθ$

∫RN

vθ (∇vθ · ∇%)

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Complex Ginzburg–Landau Equations 205

and

1

2ζ ′′θ (t)≤ 2NαE−1(v

θ )

+∫RN

{−2(2−% ′′)|vθr |2+

α

α+ 2(2N−�%)|vθ |α+2− 1

2(2Nα+�2%)|vθ |2

}+ cosθ

d

dt

∫RN

{−2%|vθr |2+

α+ 4

α+ 2%|vθ |α+2+ (�%− 2%)|vθ |2

}.

One can then conclude exactly as in the case γ = 0 of the proof of Theo-rem 4.22.

4.6 Some Open Problems

Open problem 4.26 Suppose 4N < α < 4

N−2 . Let u0 ∈H1(RN), u0 = 0 and letu be the corresponding solution of (4.3). It follows from [33, Theorem 1] that,if γ is sufficiently negative, then u is global. Does u blow up in finite time forγ > 0 sufficiently large (this is true if E(u0) < 0 and u0 ∈ L2(RN , |x|2dx), byTheorem 4.13), or does there exist u0 = 0 such that u is global for all γ > 0?

Open problem 4.27 Let 0 ≤ θ < π2 , u0 ∈ C0(R

N)∩H1(RN), u0 = 0, and letu be the corresponding solution of (4.1). If γ is sufficiently negative, then u isglobal, by Remark 4.2. Does u blow up in finite time for all sufficiently largeγ > 0 (this is true if E(u0) < 0, by Theorem 4.17), or does there exist u0 = 0such that u is global for all γ > 0? (The question is open even for the nonlinearheat equation (4.2).)

Open problem 4.28 Let γ < 0. Does there exist an initial value u0 ∈C0(RN)∩

H1(RN), u0 = 0 such that the corresponding solution of (4.1) blows up in finitetime for all θ close to π

2 ? One possible strategy for constructing such initialvalues when 4

N < α < 4N−2 would be to adapt the proof of Theorem 4.15 to

equation (4.1).

Open problem 4.29 Suppose γ ≥ 0 and 0 ≤ θ < π2 . The sufficient condition

for blowup in Theorem 4.17 is E(u0) < 0. Does there exist a constantκ > 2

α+2 such that the (weaker) condition∫RN |∇u0|2 − κ

∫RN |u0|α+2 < 0

implies finite-time blowup? Note that for the equation with γ = 0 set ona bounded domain with Dirichlet boundary conditions, κ = 1 is not admis-sible. Indeed, there exist initial values for which I(u0) < 0 and Tmax = ∞.(See [9, Theorem 1].)

Open problem 4.30 Theorems 4.20 and 4.25 require that α ≤ 4 and thesolution is radially symmetric. Are these assumptions necessary? Note that

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206 T. Cazenave and S. Snoussi

they are necessary in Lemma 4.32 (see Section 6 in [6]) which is an essentialtool in our proof. Could these assumptions be replaced by stronger decayconditions on the initial value, such as u0 ∈ L2(RN , |x|2dx)? In particular, onecould think of adapting the proof of Theorem 4.13 (instead of the proof ofTheorem 4.14), but this does not seem to be simple, see Section 7 in [6].

4.7 A Truncated Variance Identity

We prove the following result, which is a slightly more general form of [6,Lemma 5.1].

Proposition 4.31 Fix α > 0, 0 ≤ θ < π2 , and a real-valued function % ∈

C∞(RN)∩W4,∞(RN). Let u0 ∈ C0(RN)∩H1(RN), f ∈ C1(R,R), and consider

the corresponding solution v of{vt = eiθ [�v+ f (t)|v|αv],v(0)= u0,

(4.149)

defined on the maximal interval [0,Tmax). If ζ is defined by

ζ(t)=∫RN

%(x)|v(t,x)|2dx

then ζ ∈ C2([0,Tmax)),

1

2ζ ′(t)=cosθ

∫RN

{−%|∇v|2+ f (t)%|v|α+2+ 1

2|v|2�%

}+ sinθ$

∫RN

v(∇v · ∇%)

(4.150)

and

1

2ζ ′′(t)=

∫RN

{−1

2|v|2�2%− αf (t)

α+ 2|v|α+2�%+ 2(〈H(%)∇v,∇v〉

}+ cosθ

d

dt

∫RN

{−2%|∇v|2+ α+ 4

α+ 2f (t)%|v|α+2+|v|2�%

}− 2cos2 θ

∫RN

%|vt|2− 2f ′(t)α+ 2

cosθ∫RN

%|v|α+2

(4.151)

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Complex Ginzburg–Landau Equations 207

for all 0≤ t < Tmax, where H(%) is the Hessian matrix (∂2ij%)i,j. In particular,

if both % and u0 (hence, v) are radially symmetric, then

1

2ζ ′′(t)=

∫RN

{−1

2|v|2�2%− αf (t)

α+ 2|v|α+2�%+ 2% ′′|vr|2

}+ cosθ

d

dt

∫RN

{−2%|vr|2+ α+ 4

α+ 2f (t)%|v|α+2+|v|2�%

}− 2cos2 θ

∫RN

%|vt|2− 2f ′(t)α+ 2

cosθ∫RN

%|v|α+2

(4.152)

for all 0≤ t < Tmax.

Proof Multiplying the equation (4.149) by %(x)v, taking the real part, andusing the identity

2([v(∇v · ∇%)] = ∇ · (|v|2∇%)−|v|2�%

we obtain (4.150). Next, the identity

v(∇vt · ∇%)=∇ · (vtv∇%)− (∇% · ∇v)vt− vvt�%

and integration by parts yield

d

dt

(sinθ$

∫RN

v(∇v · ∇%))=−sinθ$

∫RN[v�%+ 2∇% · ∇v]vt.

We rewrite this last identity in the form

d

dt

(sinθ$

∫RN

v(∇v · ∇%))=cosθ(

∫RN[v�%+ 2∇% · ∇v]vt

−(∫RN[v�%+ 2∇% · ∇v]e−iθvt.

(4.153)

Using (4.149) and the identities

([(∇% · ∇v)|v|αv] = 1

α+ 2∇ · (|v|α+2∇%)− 1

α+ 2|v|α+2�%,

([�v(v�%+ 2∇v · ∇%)] = (∇ ·[∇v(v�%+ 2∇v · ∇%)−|∇v|2∇%

− 1

2|v|2∇(�%)

]− 2(〈H(%)∇v,∇v〉

+ 1

2|v|2�2%,

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208 T. Cazenave and S. Snoussi

we see that

−(∫RN[v�%+ 2∇% · ∇v]e−iθvt

=−(∫RN[v�%+ 2∇% · ∇v](�v+ f (t)|v|αv)

=−1

2

∫RN|v|2�2%− αf (t)

α+ 2

∫RN|v|α+2�%+ 2(

∫RN〈H(%)∇v,∇v〉.

(4.154)

Next, we observe that

(∫RN[v�%+ 2∇% · ∇v]vt = 1

2

d

dt

∫RN|v|2�%+ 2(

∫RN

(∇v · ∇%)vt.

(4.155)On the other hand,

f ′

α+ 2

∫RN

%|v|α+2+ d

dt

∫RN

%( |∇v|2

2− f (t)

α+ 2|v|α+2

)=(

∫RN

%(∇v · ∇vt− f |v|αvvt)=−(∫RN[%(�v+ f |v|αv)vt+ (∇% · ∇v)vt]

= −cosθ∫RN

%|vt|2−(∫RN

(∇% · ∇v)vt

so that

2(∫RN

(∇% · ∇v)vt =−2cosθ∫RN

%|vt|2− 2f ′

α+ 2

∫RN

%|v|α+2 (4.156)

− d

dt

∫RN

%(|∇v|2− 2f (t)

α+ 2|v|α+2

).

Applying (4.153), (4.154), (4.155) and (4.156), we deduce that

d

dt

(sinθ$

∫RN

v(∇% · ∇v))

=∫RN

{−1

2|v|2�2%− αf (t)

α+ 2|v|α+2�%+ 2(〈H(%)∇v,∇v〉

}+ cosθ

d

dt

∫RN

(−%|∇v|2+ 2f (t)

α+ 2%|v|α+2+ 1

2|v|2�%

)− 2cos2 θ

∫RN

%|vt|2− 2f ′(t)α+ 2

cosθ∫RN

%|v|α+2.

(4.157)

Taking now the time derivative of (4.150) and applying (4.157), weobtain (4.151). Finally, if both % and u0 (hence v) are radially symmetric,then (〈H(%)∇v,∇v〉 =% ′′|vr|2, so that (4.152) follows from (4.151).

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Complex Ginzburg–Landau Equations 209

4.8 A Caffarelli–Kohn–Nirenberg Inequality

We use the following form of Caffarelli–Kohn–Nirenberg inequality [3]. Itextends an inequality which was established in [31] and generalized in [6,Lemma 5.3].

Lemma 4.32 Suppose N ≥ 2 and α≤ 4 and let A> 0. There exist a constant χand a family (%ε)ε>0 ⊂ C∞(RN)∩W4,∞(RN) of radially symmetric functionssuch that %ε(x) > 0 for x = 0,

supε>0

[ε2‖%ε‖L∞ + ε‖∂r%ε‖L∞ +‖�%ε‖L∞ + ε−2‖�2%ε‖L∞]<∞, (4.158)

2N−�%ε ≥ 0 (4.159)

and

−2∫RN

(2−% ′′ε )|ur|2+μ

∫RN

(2N−�%ε)|u|α+2

− 1

2

∫RN|u|2�2%ε ≤ κ(μ,ε)

(4.160)

for all radially symmetric u ∈ H1(RN) such that ‖u‖L2 ≤ A and all μ,ε > 0such that χμε2(N−1) < 1, where

κ(μ,ε)=⎧⎨⎩χμ

Nα2 +[χμε2(N−1)] α

4−α

)+χε2 if 0 < α < 4,

χμεNα2 +χε2 if α = 4.

(4.161)

Proof We follow the method of [31], and we construct a family (%ε)ε>0 suchthat, given A, the estimate (4.160) holds with % = %ε provided ε > 0 issufficiently small. Fix a function h ∈ C∞([0,∞)) such that

h≥ 0, supph⊂ [1,2],∫ ∞

0h(s)ds= 1

and let

ζ(t)= t−∫ t

0(t− s)h(s)ds= t−

∫ t

0

∫ s

0h(σ )dσds

for t≥ 0. It follows that ζ ∈C∞([0,∞))∩W4,∞((0,∞)), ζ ′ ≥ 0, ζ ′′ ≤ 0, ζ(t)= tfor t≤ 1, and ζ(t)=M for t≥ 2 with M = ∫ 2

0 sh(s)ds. Set

�(x)= ζ(|x|2).It follows in particular that � ∈ C∞(RN)∩W4,∞(RN). Given any ε > 0, set

%ε(x)= ε−2�(εx),

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210 T. Cazenave and S. Snoussi

so that‖Dβ%ε‖L∞ = ε|β|−2‖Dβ�‖L∞ , (4.162)

where β is any multi-index such that 0≤ |β| ≤ 4. Next, set

ξ(t)=√2(1− ζ ′(t))− 4tζ ′′(t)=

√2∫ t

0h(s)ds+ 4th(t). (4.163)

It is not difficult to check that ξ ∈ C1([0,∞))∩W1,∞(0,∞). Let

γ (r)= ξ(r2)

and, given ε > 0, letγε(r)= γ (εr).

It easily follows that γε is supported in [ε−1,∞), so that

‖r−(N−1)γ ′ε‖L∞ ≤ εN−1‖γ ′ε‖L∞ = εN‖γ ′‖L∞ , (4.164)

‖r−(N−1)γεur‖L2 ≤ εN−1‖γεur‖L2 . (4.165)

Set

Iε(u)=−2∫RN

(2−% ′′ε )|ur|2+μ

∫RN

(2N−�%ε)|u|α+2

− 1

2

∫RN|u|2�2%ε.

(4.166)

Elementary but long calculations using in particular (4.163) show that

2−% ′′ε (x)= γε(|x|)2, (4.167)

2N−�%ε(x)= N[γε(|x|)]2+ 4(N− 1)(ε|x|)2ζ ′′(ε2|x|2)≤ N[γε(|x|)]2.(4.168)

We deduce from (4.166), (4.167), (4.168), and (4.162) that

Iε(u)≤−2∫RN

γ 2ε |ur|2+Nμ

∫RN

γ 2ε |u|α+2+ ε2

2‖�2�‖L∞‖u‖2

L2 . (4.169)

We next claim that

‖γ12ε u‖2

L∞ ≤ εN‖γ ′‖L∞‖u‖2L2 + 2εN−1‖u‖L2‖γεur‖L2 . (4.170)

Indeed,

γε(r)|u(r)|2 =−∫ ∞

r

d

ds[γε(s)|u(s)|2] ≤

∫ ∞

0|γ ′ε| |u|2+ 2

∫ ∞

0γε|u| |ur|

≤ ‖r−(N−1)γ ′ε‖L∞‖u‖2L2 + 2‖u‖L2‖r−(N−1)γεur‖L2 .

(4.171)

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Complex Ginzburg–Landau Equations 211

(The above calculation is valid for a smooth function u and is easily justifiedfor a general u by density.) The estimate (4.170) follows from (4.171), (4.164),and (4.165).

In what follows, χ denotes a constant that may depend on N,γ ,� and A andchange from line to line, but is independent of 0 <α≤ 4 and ε > 0. We assume‖u‖L2 ≤ A, and we observe that∫

RNγ 2ε |u|α+2 =

∫RN

γ4−α

2ε [γ

12ε |u|]α|u|2 ≤ ‖γ ‖

4−α2

L∞ ‖γ12ε u‖αL∞A2

≤ χ‖γ12ε u‖αL∞ .

(4.172)

Applying (4.170) and the inequality (x + y)α2 ≤ χ(x

α2 + y

α2 ), we deduce

from (4.172) that ∫RN

γ 2ε |u|α+2 ≤ χε

Nα2 +χε

(N−1)α2 ‖γεur‖

α2L2 . (4.173)

We first consider the case α < 4. Applying the inequality xy≤ xp

pδp + δp′ yp′p′ with

δ > 0 and p= 44−α

, p′ = 4α

, we see that

ε(N−1)α

2 ‖γεur‖α2L2 ≤ χδ

− 44−α ε

2(N−1)α4−α +χδ

4α ‖γεur‖2

L2

so that (4.173) yields∫RN

γ 2ε |u|α+2 ≤ χδ

4α ‖γεur‖2

L2 +χ(ε

Nα2 + δ

− 44−α ε

2(N−1)α4−α

). (4.174)

Estimates (4.169) and (4.174) now yield

Iε(u)≤−(

2−χμδ4α

)‖γεur‖2

L2 +χμ(ε

Nα2 +[δ−4ε2(N−1)α] 1

4−α

)+χε2.

(4.175)We choose δ > 0 so that the first term in the right-hand side of (4.175)vanishes, i.e. χμδ

4α = 2. For this choice of δ, it follows from (4.175) that if

‖u‖L2 ≤ A, then Iε(u) ≤ κ(μ,ε), where κ is defined by (4.161). This provesinequality (4.160) for α < 4. The case α = 4 follows by letting α ↑ 4 andobserving that [χμε2(N−1)] α

4−α → 0 as α ↑ 4 when χμε2(N−1) < 1.

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∗ B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, FranceE-mail address: [email protected]

† UR Analyse Non-Lineaire et Geometrie, UR13ES32, Department of Mathematics, Faculty ofSciences of Tunis, University of Tunis El-Manar

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5

Asymptotic Analysis for the Lane–EmdenProblem in Dimension Two

Francesca De Marchis∗, Isabella Ianni† and Filomena Pacella‡

Introduction

We consider the Lane–Emden Dirichlet problem{ −�u= |u|p−1u in ,u= 0 on ∂,

(5.1)

when p > 1 and ⊂R2 is a smooth bounded domain. The aim of the paper isto survey some recent results on the asymptotic behavior of solutions of (5.1)as the exponent p→∞.

We will start in Section 5.1 with a summary of some basic and well-knownfacts about the solutions of (5.1). We will also describe a recent result aboutthe existence, for p large, of a special class of sign-changing solutions of (5.1)in symmetric domains (see [19]) and we will provide, for p large, the exactcomputation of the Morse index of least energy nodal radial solutions of (5.1)in the ball, as obtained in [23].

The asymptotic behavior as p →∞ will be described in Sections 5.2–5.3.In Section 5.2 a general “profile decomposition” theorem obtained in [20] andholding for both positive and sign-changing solutions will be presented with adetailed proof, together with some additional new results for positive solutionsrecently obtained in [22]. Finally, in Section 5.3 we will describe the preciselimit profile of the symmetric nodal solutions found in [19] and then studied in[20]. In particular, the result of this section will show that, asymptotically, as

2010 Mathematics Subject classification: 35B05, 35B06, 35J91.Keywords: semilinear elliptic equations, superlinear elliptic boundary value problems, asymptoticanalysis, concentration of solutions.Research supported by: PRIN 201274FYK7 005 grant, INDAM - GNAMPA and SapienzaResearch Funds: “Avvio alla ricerca 2015” and “Awards Project 2014”∗ Francesca De Marchis, University of Roma Sapienza† Isabella Ianni, Second University of Napoli‡ Filomena Pacella, University of Roma Sapienza

215

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216 Francesca De Marchis, Isabella Ianni and Filomena Pacella

p→∞, these solutions look like a superposition of two bubbles with differentsign corresponding to radial solutions of the regular and singular Liouvilleproblem in R2.

Acknowledgments

This paper originates from a short course given by F. Pacella at aConference-School held in Hammamet in March 2015 in honor of AbbasBahri. She would like to thank all the organizers for the wonderful and warmhospitality.

5.1 Various Results for Solutions of theLane–Emden Problem

We consider the Lane–Emden Dirichlet problem{ −�u= |u|p−1u in ,u= 0 on ∂,

(5.2)

where p > 1 and ⊂R2 is a smooth bounded domain.Since in two dimensions any exponent p > 1 is subcritical (with respect to

the Sobolev embedding), it is well known, by standard variational methods,that (5.2) has at least one positive solution. Moreover, exploiting the oddnessof the nonlinearity f (u)= |u|p−1u and using topological tools it can be provedthat (5.2) admits infinitely many solutions.

It was first proved in [6], and later in [5] for more general nonlinearities,that there exists at least one solution which changes sign, so it makes senseto study the properties of both positive and sign-changing solutions. Thelatter will be often referred to as nodal solutions. Among these solutions onecan select those which have the least energy, therefore named “least energy”(or “least energy nodal”) solutions. More precisely, considering the energyfunctional:

Ep(u)= 1

2

|∇u|2 dx− 1

p+ 1

|u|p+1 dx, u ∈H10()

and the Nehari manifold

N = {u ∈H10() : 〈E′p(u),u〉 = 0}

or the nodal Nehari set

N± = {u ∈H10() : 〈E′p(u),u±〉 = 0},

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 217

where u± are the positive and negative part of u, it is possible to prove thatthe infN Ep (resp. infN± Ep) is achieved. The corresponding minimizers arethe least energy positive (resp. nodal) solutions (see [46], [5]). Note that anyminimizer on N cannot change sign and we will assume that it is positive(rather than negative).

Let us observe that N is a codimension one manifold in H10() while N±

is a C1-manifold of codimension two in H10() ∩H2() (but not in H1

0(),see [5]).

For the least energy solutions several qualitative properties can be obtained.We start by considering the case of positive solutions.Let us first define the Morse index of a solution of (5.2).

Definition 5.1 The Morse index m(u) of a solution u of (5.2) is the maximaldimension of a subspace of C1

0() on which the quadratic form

Q(ϕ)=∫

|∇ϕ|2 dx− p∫

|u|p−1ϕ2 dx

is negative definite.

In the case when is a bounded domain, m(u) can be equivalently definedas the number of the negative Dirichlet eigenvalue of the linearized operatorat u:

Lu =−�− p|u|p−1

in the domain .It is easy to see, just multiplying the equation by u and integrating, that

Q(u) < 0, so that there is at least one negative direction for Q(u), i.e. m(u)≥ 1.This holds for any solution of (5.2), either positive or sign-changing. For theleast energy solution u, since it minimizes the energy on a codimension onemanifold, one could guess that m(u) = 1. This is what was indeed provedin [44] (see also [46]), for more general nonlinearities. Another importantproperty of a solution, both for theoretical reasons and for applications, is itssymmetry in symmetric domains.

For positive solutions u of (5.2), as a consequence of the famous result byGidas, Ni and Nirenberg [29] it holds that if is symmetric and convex withrespect to a line, then u is invariant by reflection with respect to that line. Inparticular a positive solution of (5.2) in a ball is radial and strictly radiallydecreasing.

This result allows us to prove that if is a ball there exists only one positivesolution of (5.2) (this holds also in higher dimensions, when p is a subcriticalexponent) ([29], [45], [3], [14]).

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218 Francesca De Marchis, Isabella Ianni and Filomena Pacella

The question of the uniqueness of the positive solution in more generalbounded domains is a very difficult one, still open, and will be addressed inthe subsequent paper [16]. Here we recall that it has been conjectured ([29])that the uniqueness should hold in convex domains (also in higher dimensions)and that, so far, in the case of planar domains it has only been proved whenthe domains are symmetric and convex with respect to two orthogonal linespassing through the origin ([14], [38]). If one restricts the question to the leastenergy solutions (or more generally to solutions of Morse index one) then theuniqueness, in convex planar domains, has been proved in [34]. On the otherside it is easy to see that there are nonconvex domains for which multiplepositive solutions exist; examples of such domains are annular domains ordumbbell domains ([15], [38]).

More properties of positive solutions and, actually, a good description oftheir profile, can be obtained, for large exponents p, by the asymptotic analysisof the solutions of (5.2), as p→∞.

This study started in [41] and [42] where the authors considered families(up) of least energy (hence positive) solutions and, for some domains, provedconcentration at a single point, as well as asymptotic estimates, as p →∞.Later, inspired by the paper [2], Adimurthi and Grossi in [1] identified a “limitproblem” by showing that suitable rescalings of up converge, in C2

loc(R2) to a

regular solution U of the Liouville problem{ −�U = eU in R2,∫R2 eUdx= 8π .

(5.3)

They also considered general bounded domains and showed that ‖up‖∞converges to

√e as p→∞, as it had been previously conjectured.

So the asymptotic profile of the least energy solutions is clear, as well astheir energy.

Concerning general positive solutions, a first asymptotic analysis (actuallyholding for general families of solutions, both positive and sign-changing)under the following energy condition:

p∫

|∇up|2 dx≤ C (5.4)

for some positive constant C ≥ 8πe and independent of p, was carried out in[20]. Then, recently in [22], starting from this, a complete description of theasymptotic profile of any family of positive solutions up satisfying (5.4) hasbeen obtained, showing that (up) concentrates at a finite number of distinctpoints in , having the limit profile of the solution U of (5.3) when a suitablerescaling around each of the concentration points is made. These results have

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 219

been recently improved in [17] where it has been proved that ‖up‖∞ convergesto√

e as p→∞, generalizing the result in [1], and that a quantization of theenergy holds (see the end of Section 5.2 for the complete statement). Positivesolutions with the described profile have been found in [27].

Now let us analyze the case of sign-changing solutions of (5.2).Since any such solution u has at least two nodal regions (i.e. connected

components of the set where u does not vanish), multiplying the equation byu and integrating on each nodal domain, we get that the Morse index m(u) isat least two. For the least energy nodal solution, since it minimizes the energyfunctional Ep on N±, it is proved in [5] that its Morse index is exactly two.

Concerning symmetry properties of sign-changing solutions, a general resultas the one of Gidas, Ni and Nirenberg for positive solutions cannot hold. Thisis easily understood by just thinking of the Dirichlet eigenfunctions of theLaplacian in a ball.

Nevertheless, by using maximum principles, properties of the linearizedoperator and bounds on the Morse index, partial symmetry results can beobtained also for nodal solutions. This direction of research started in [37]and continued in [39] and [30]. In particular in these papers, semilinear ellipticequations with nonlinear terms f (u) either convex or with a convex derivativewere studied in rotationally symmetric domains, showing the foliated Schwarzsymmetry of solutions (of any sign) having Morse index m(u) ≤ N, where Nis the dimension of the domain. We recall the definition of foliated Schwarzsymmetry.

Definition 5.2 Let B ⊆ RN,N ≥ 2, be a ball or an annulus. A continuousfunction v in B is said to be foliated Schwarz symmetric if there exists a unitvector p ∈RN such that v(x) depends only on |x| and ϑ = arccos( x

|x| ·p) and isnonincreasing in ϑ .

In other words a foliated Schwarz symmetric function is axially symmetricand monotone with respect to the angular coordinate.

In particular, in dimension two, the results of [39] allow us to claim that,in a ball or in an annulus, any solution u of (5.2) with Morse index m(u) ≤ 2is foliated Schwarz symmetric. Thus, in such domains, the least energy nodalsolutions are foliated Schwarz symmetric.

Since radial functions are, obviously, foliated Schwarz symmetric, onemay ask whether the least energy nodal solutions are radial or not. Theanswer to this question was provided by [4] where it was proved that anysign-changing solution u of a semilinear elliptic equation with a generalautonomous nonlinearity f (u) in a ball or an annulus must have Morse indexm(u)≥ N+ 2 (again N denotes the dimension of the domain).

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220 Francesca De Marchis, Isabella Ianni and Filomena Pacella

An immediate corollary of this theorem is that, since a least energy nodalsolution of (5.2) has Morse index two, it cannot be radial.

Another interesting consequence of the result of [4] is that the nodal set ofa least energy nodal solution of (5.2) in a ball or an annulus must intersect theboundary of . We recall that the nodal set N(u) of a function u defined in thedomain is:

N(u)= {x ∈ : u(x)= 0}.To understand the property of the nodal line is important while studyingsign-changing functions. It is an old question related to the study of the nodaleigenfunctions of the Laplacian, in particular of the second eigenfunction. In[36] it has been proved that in convex planar domains the nodal set of a secondeigenfunction touches the boundary, but the question is still open in higherdimension, except for the case of some symmetric domains ( [33], [13]).

Coming back to nodal solutions of (5.2) we observe that if is a ball or anannulus, it is easy to see that there exist both nodal solutions with an interiornodal line and solutions whose nodal line intersects the boundary. Examplesof solutions of the first type are the radial ones while those of the second typeare antisymmetric with respect to a line passing through the center. It is naturalto ask whether both kinds of solution exist in more general domains. Whileit is not difficult to provide examples of symmetric domains where there arenodal solutions whose nodal line intersects the boundary (rectangles, regularpolygons etc.) it is not obvious at all that solutions with an interior nodal lineexist. In the paper [19] we have succeeded in proving the existence of thistype of solution in some symmetric planar domains, for large exponents p. Theprecise statement is the following.

Theorem 5.3 Assume that is simply connected, invariant under the actionof a finite group G of orthogonal transformations of R2. If |G| ≥ 4 (|G| is theorder of the group) then, for p sufficiently large (5.2) admits a sign-changingG-symmetric solution up, with two nodal domains, whose nodal line neithertouches ∂, nor passes through the origin. Moreover

p∫

|∇up|2dx≤ α 8πe for some α < 5 and p large.

Let us now come back to the question of the Morse index of nodal solutionsof (5.2). As recalled before, the result of [4] allows us to give an estimate frombelow in the radial case:

m(u)≥ 4

for any radial sign-changing solution u of (5.2) in a ball or an annulus. In therecent paper [23] we have been able to compute the Morse index exactly for

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 221

these solutions when the exponent p is large and u has the least energy amongthe radial nodal solutions. The result is the following.

Theorem 5.4 Let up be the least energy sign-changing radial solution of (5.2).Then

m(up) = 12

for p sufficiently large.

The proof of this theorem is based on a decomposition of the spectrum ofthe linearized operator at up, as well as on fine estimates of the radial solutionup obtained in [32].

As in the case of positive solutions, a better description of nodal solutionsand of their profile can be obtained, for large exponents p, by performing anasymptotic analysis, as p →∞. This study started in [31] by considering afamily (up) of solutions of (5.2) satisfying the condition

p∫

|∇u|2 dx→ 16πe as p→+∞,

where 16πe is the “least-asymptotic” energy for nodal solutions. Under someadditional conditions it was proved in [31] that these low-energy solutionsconcentrate at two distinct points of and suitable scalings of u+p and u−pconverge to a regular solution U of (5.3).

Next, the case of least energy radial nodal solutions was considered in [32]where the new phenomenon of u+p and u−p concentrating at the same point butwith different profiles was shown. The precise result is the following.

Theorem 5.5 Let (up) be a family of least energy radial nodal solutions of(5.2) in the ball with up positive at the center. Then

(i) a suitable scaling of u+p converges in C2loc(R

2) to a regular solution Uof (5.3),

(ii) a suitable scaling and translation of u−p converges in C2loc(R

2 \ {O}) to asingular radial solution V of{ −�V = eV +Hδ0 in R2,∫

R2 eVdx <+∞,(5.5)

where H is a suitable negative constant and δ0 is the Dirac measure centeredat O.

Moreover:

p∫

|∇up|2 dx→ C > 16πe as p→+∞.

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222 Francesca De Marchis, Isabella Ianni and Filomena Pacella

So the theorem shows the existence of solutions which asymptoticallylook like a tower of two bubbles corresponding to solutions of two differentLiouville problems in R2, namely (5.3) and (5.5).

In Section 5.3 of this paper we show that the same phenomenon appears inother symmetric domains different from the balls. We obtain this through theasymptotic analysis of the sign-changing solutions found in Theorem 5.3.

The starting point for this result is an asymptotic analysis of a generalfamily (up) of solutions of (5.2) satisfying the condition (5.4). This first result,inspired by the paper [25] (see also [26]) can be viewed as a first step towardsthe complete classification of the asymptotic behavior of general sign-changingsolutions of (5.2).

The hardest part of the proof of this result relies on showing that therescaling about the minimum point x−p converges to a radial singular solutionof a singular Liouville problem in R2. Indeed, while the rescaling of up aboutthe maximum point x+p can be studied in a “canonical” way, the analysisof the rescaling about x−p requires additional arguments. In particular thepresence of the nodal line, with an unknown geometry, gives difficultieswhich, obviously, are not present when dealing with positive solutions orwith radial sign-changing solutions. Also the proofs of the results for nodalradial solutions of [32] cannot be of any help since they depend strongly onone-dimensional estimates.

We would like to point out that the analysis carried out in [20] also allowsus to get the same asymptotic result by substituting the bound on the energywith a bound on the Morse index of the solutions (see [21]).

Finally, we observe that the bubble-tower solutions of (5.1) are also inter-esting in the study of the associated heat flow because they induce a peculiarblow-up phenomenon (see [7, 24, 35] and in particular [18]).

We conclude by remarking that the phenomenon of nodal solutions of (5.2)with positive and negative parts concentrating at the same point and havingdifferent asymptotic profiles does not seem to appear in higher dimensions asp approaches the critical Sobolev exponent.

Finally, nodal solutions to (5.2) concentrating at a finite number of pointwithout exhibiting the bubble tower phenomenon, i.e. only simple concentra-tion points, also exist (see [28]).

5.2 General Asymptotic Analysis

This section is mostly devoted to the study of the asymptotic behavior of ageneral family (up)p>1 of nontrivial solutions of (5.2) satisfying the uniform

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 223

upper bound

p∫

|∇up|2dx≤ C, for some C > 0 independent of p. (5.6)

At the end of the section we also exhibit some recent results related to familiesof positive solutions. The material presented is based mainly on some of theresults contained in [20] plus smaller additions or minor improvements. Wealso refer to [22] (see also [16]) for the complete analysis in the case of positivesolutions.

Recall that in [41] it has been proved that for any family (up)p>1 of nontrivialsolutions of (5.1) the following lower bound holds:

liminfp→+∞ p

|∇up|2dx≥ 8πe, (5.7)

so the constant C in (5.6) is intended to satisfy C ≥ 8πe. Moreover, if up issign-changing then we also know that (see again [41])

liminfp→+∞ p

|∇u±p |2dx≥ 8πe. (5.8)

If we denote by Ep the energy functional associated with (5.2), i.e.

Ep(u) := 1

2‖∇u‖2

2−1

p+ 1‖u‖p+1

p+1, u ∈H10(),

since for a solution u of (5.2)

Ep(u)=(

1

2− 1

p+ 1

)‖∇u‖2

2 =(

1

2− 1

p+ 1

)‖u‖p+1

p+1, (5.9)

then (5.6), (5.7) and (5.8) are equivalent to uniform upper and lower boundsfor the energy Ep or for the Lp+1-norm, indeed

limsupp→+∞

2pEp(up)= limsupp→+∞

p∫

|up|p+1 dx= limsupp→+∞

p∫

|∇up|2 dx≤ C,

liminfp→+∞ 2pEp(up)= liminf

p→+∞ p∫

|up|p+1 dx= liminfp→+∞ p

|∇up|2 dx≥ 8πe,

and if up is sign-changing, also

liminfp→+∞ 2pEp(u

±p )= liminf

p→+∞ p∫

|u±p |p+1 dx= liminfp→+∞ p

|∇u±p |2 dx≥ 8πe.

We will use all these equivalent formulations throughout the paper.Observe that by the assumption in (5.6) we have that

Ep(up)→ 0, ‖∇up‖2 → 0, as p→+∞,

Ep(u±p )→ 0, ‖∇u±p ‖2 → 0, as p→+∞ (if up is sign-changing),

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224 Francesca De Marchis, Isabella Ianni and Filomena Pacella

so in particular u±p → 0 a.e. as p→+∞.In this section we will show that the solutions up do not vanish as p →

+∞ (both u±p do not vanish if up is sign-changing) and that moreover,differently from what happens in higher dimension, they do not blow up(see Theorem 5.6). Moreover, we will show that they concentrate at a finitenumber of points and we will also describe the asymptotic behavior of suitablerescalings of up (“bubbles”) around suitable “concentrating” sequences ofpoints (see Theorem 5.8).

Our first result is the following.

Theorem 5.6 Let (up) be a family of solutions to (5.2) satisfying (5.6). Thenthe following held.

(i) (No vanishing)

‖up‖p−1∞ ≥ λ1,

where λ1 = λ1()(> 0) is the first eigenvalue of the operator −� inH1

0().If up is sign-changing then also ‖u±p ‖p−1

∞ ≥ λ1.(ii) (Existence of the first bubble) Let (x+p )p ⊂ such that |up(x+p )| = ‖up‖∞.

Let us set

μ+p := (p|up(x

+p )|p−1)− 1

2 (5.10)

and for x ∈ +p := {x ∈R2 : x+p +μ+p x ∈}

v+p (x) := p

up(x+p )(up(x

+p +μ+p x)− up(x1,p)). (5.11)

Then μ+p → 0 as p→+∞ and

v+p −→U in C2loc(R

2) as p→+∞,

where

U(x)= log

(1

1+ 18 |x|2

)2

(5.12)

is the solution of −�U = eU in R2, U ≤ 0, U(0)= 0 and∫R2 eU = 8π .

Moreover

liminfp→+∞ ‖up‖∞ ≥ 1. (5.13)

(iii) (No blow-up) There exists C > 0 such that

‖up‖∞ ≤ C, for p large. (5.14)

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 225

(iv) There exist constants c,C > 0, such that for all p sufficiently large we have

c≤ p∫

|up|pdx≤ C. (5.15)

(v)√

pup ⇀ 0 in H10() as p→+∞.

Proof Point (i) was first proved for positive solutions in [41], here we followsthe proof in [31, Proposition 2.5]. If up is sign-changing, just observe that u±p ∈H1

0(), where we know that

0 < 8πe− ε(5.7)/(5.8)≤

|∇u±p |2 dx(5.6)≤ C <+∞

and that also by the Poincare inequality∫

|∇u±p |2 dx=∫

|u±p |p+1 dx≤‖u±p ‖p−1∞

|u±p |2 dx≤ ‖u±p ‖p−1L∞()

λ1()

|∇u±p |2 dx.

If up is not sign-changing just observe that either up = u+p or up = u−p and thesame proof as before applies.

The proof of (ii) follows the same ideas as in [1] where the same result hasbeen proved for least energy (positive) solutions. We let x+p be a point in

where |up| achieves its maximum. Without loss of generality we can assumethat

up(x+p )=max

up > 0. (5.16)

By (i) we have that pup(x+p )p−1 → +∞ as p → +∞, so (5.13) holds andmoreover μ1,p → 0, where μ+p is defined in (5.10). Let +

p and v+p be definedas in (5.11), then by (5.16) we have

v+p (0)= 0 and v+p ≤ 0 in +p . (5.17)

Moreover, v+p solves

−�v+p =∣∣∣∣∣1+ v+p

p

∣∣∣∣∣p(

1+ v+pp

)in +

p , (5.18)

with ∣∣∣∣∣1+ v+pp

∣∣∣∣∣≤ 1 and v+p =−p on ∂+p .

Then|−�v+p | ≤ 1 in +

p . (5.19)

Using (5.17) and (5.19) we prove that

+p →R2 as p→+∞. (5.20)

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226 Francesca De Marchis, Isabella Ianni and Filomena Pacella

Indeed, since μ+p → 0 as p→+∞, either +p →R2 or +

p →R×]−∞,R[ asp→+∞ for some R≥ 0 (up to a rotation). In the second case we let

v+p = ϕp+ψp in +p ∩B2R+1(0)

with −�ϕp =−�v+p in +p ∩B2R+1(0) and ψp = v+p in ∂

(+

p ∩B2R+1(0)).

Thanks to (5.19) we have, by standard elliptic theory, that ϕp is uniformlybounded in +

1 ∩B2R+1(0). So the function ψp is harmonic in +p ∩B2R+1(0),

bounded from above by (5.17) and satisfies ψp = −p → −∞ on ∂+p ∩

B2R+1(0). Since ∂+p ∩ B2R+1(0) → (R× {R}) ∩ B2R+1(0) as p → +∞ one

easily gets that ψp(0)→−∞ as p →+∞ (if R = 0 this is trivial, if R > 0it follows by the Harnack inequality). This is a contradiction since ψp(0) =−ϕp(0) and ϕp is bounded, hence (5.20) is proved.

Then for any R> 0, BR(0)⊂ 1,p for p sufficiently large. So (v+p ) is a familyof nonpositive functions with uniformly bounded Laplacian in BR(0) and withvp+(0)= 0.

Thus, arguing as before, we write v+p = ϕp + ψp, where ϕp is uniformlybounded in BR(0) and ψp is an harmonic function which is uniformly boundedfrom above. By the Harnack inequality, either ψp is uniformly bounded inBR(0) or it tends to −∞ on each compact set of BR(0). The second alternativecannot happen because, by definition, ψp(0)= v+p (0)−ϕp(0)=−ϕp(0)≥−C.Hence we obtain that v+p is uniformly bounded in BR(0), for all R > 0. By

standard elliptic regularity theory one has that v+p is bounded in C2,αloc (R

2).Thus, by the Arzela–Ascoli theorem and a diagonal process on R → +∞,after passing to a subsequence

v+p →U in C2loc(R

2) as p→+∞, (5.21)

with U ∈ C2(R2), U ≤ 0 and U(0) = 0. Thanks to (5.18) (on each ball also

1+ v+pp > 0 for p large) and (5.21) we get that U is a solution of −�U = eU in

R2. Moreover for any R > 0, by (5.24), we have∫BR(0)

eU(x)dx(5.21)+Fatou≤

∫BR(0)

|up(x+p +μ+p x)|p+1

up(x+p )p+1dx+ op(1)

= p

‖up‖2∞

∫BRμ1,p (x1,p)

|up(y)|p+1dy+ op(1)

(5.13)≤ p

(1− ε)2

|up(y)|p+1dy+ op(1)(5.6)≤ C <+∞,

so that eU ∈ L1(R2). Thus, since U(0) = 0, by virtue of the classification dueto Chen and Li [8] we obtain (5.12). Lastly an easy computation shows that∫R2 eU = 8π .

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 227

Point (iii) was first proved in [41], here we write a simpler proof whichfollows directly from (ii) by applying Fatou’s lemma. An analogous argumentcan be found in [1, Lemma 3.1]. Indeed,

C(5.6)≥ p

|up(y)|p+1dy= ‖up‖2∞

∫+p

∣∣∣∣∣1+ v+p (x)p

∣∣∣∣∣p+1

dx

(ii)-Fatou≥ ‖up‖2∞

∫R2

eU(x)dx= ‖up‖2∞8π .

(iv) follows directly from (iii). Indeed, on the one hand

0 < C(5.7)−(5.9)≤ p

|up|p+1 dx≤ ‖up‖∞p∫

|up|p dx(iii)≤ Cp

|up|p dx.

On the other hand, by the Holder inequality

p∫

|up|p dx≤ || 1p+1 p

(∫

|up|p+1 dx

) pp+1 (5.6)≤ C.

To prove (v) we need (iv). Indeed, let us note that, since (5.6) holds, thereexists w ∈ H1

0() such that, up to a subsequence,√

pup ⇀ w in H10(). We

want to show that w= 0 a.e. in .Using the equation (5.2), for any test function ϕ ∈ C∞

0 (), we have∫

∇(√

pup)∇ϕ dx=√p∫

|up|p−1upϕ dx≤ ‖ϕ‖∞√p

p∫

|up|p dx(iv)≤ ‖ϕ‖∞√

pC

for p large. Hence ∫

∇w∇ϕ dx= 0 ∀ϕ ∈ C∞0 (),

which implies that w= 0 a.e. in .

In order to show our next results we need to introduce some notation. Givena family (up) of solutions of (5.2) and assuming that there exists n ∈ N \ {0}families of points (xi,p), i= 1, . . . ,n in such that

p|up(xi,p)|p−1 →+∞ as p→+∞, (5.22)

we define the parameters μi,p by

μ−2i,p = p|up(xi,p)|p−1, for all i= 1, . . . ,n. (5.23)

By (5.22) it is clear that μi,p → 0 as p→+∞ and that

liminfp→+∞ |up(xi,p)| ≥ 1. (5.24)

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228 Francesca De Marchis, Isabella Ianni and Filomena Pacella

Then we define the concentration set

S ={

limp→+∞xi,p, i= 1, . . . ,n

}⊂ (5.25)

and the function

Rn,p(x)= mini=1,...,n

|x− xi,p|, ∀x ∈. (5.26)

Finally, we introduce the following properties.

(Pn1 ) For any i, j ∈ {1, . . . ,n}, i = j,

limp→+∞

|xi,p− xj,p|μi,p

=+∞.

(Pn2 ) For any i= 1, . . . ,n, for x ∈ i,p := {x ∈R2 : xi,p+μi,px ∈},

vi,p(x) := p

up(xi,p)(up(xi,p+μi,px)− up(xi,p))−→U(x) (5.27)

in C2loc(R

2) as p→+∞, where U is the same function as in (5.12).(Pn

3 ) There exists C > 0 such that

pRn,p(x)2|up(x)|p−1 ≤ C

for all p > 1 and all x ∈.(Pn

4 ) There exists C > 0 such that

pRn,p(x)|∇up(x)| ≤ C

for all p > 1 and all x ∈.

Lemma 5.7 If there exists n ∈ N \ {0} such that the properties (Pn1 ) and (Pn

2 )

hold for families (xi,p)i=1,...,n of points satisfying (5.22), then

p∫

|∇up|2 dx≥ 8πn∑

i=1

α2i + op(1) as p→+∞,

where αi := liminfp→+∞ |up(xi,p)| ((5.24)≥ 1).

Proof Let us write, for any R > 0

p∫

BRμi,p (xi,p)

|up|p+1 dx=∫

BR(0)

|up(xi,p+μi,py)|p+1

|up(xi,p)|p−1dy

= u2p(xi,p)

∫BR(0)

∣∣∣∣1+ vi,p(y)

p

∣∣∣∣p+1

dy. (5.28)

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 229

Thanks to (Pn2 ), we have

‖vi,p−U‖L∞(BR(0)) = op(1) as p→+∞. (5.29)

Thus by (5.28), (5.29) and Fatou’s lemma

liminfp→+∞

(p∫

BRμi,p (xi,p)

|up|p+1 dx

)≥ α2

i

∫BR(0)

eU dx. (5.30)

Moreover, by virtue of (Pn1 ) it is not hard to see that BRμi,p(xi,p) ∩

BRμj,p(xj,p)= ∅ for all i = j. Hence, in particular, thanks to (5.30),

liminfp→+∞

(p∫

|up|p+1 dx

)≥

n∑i=1

(α2

i

∫BR(0)

eU dx

).

At last, since this holds for any R > 0, we get

p∫

|∇up|2 dx= p∫

|up|p+1 dx≥n∑

i=1

α2i

∫R2

eU dx+ o(1)

= 8πn∑

i=1

α2i + o(1) as p→+∞.

The next result shows that the solutions concentrate at a finite number ofpoints and also establishes the existence of a maximal number of “bubbles”.

Theorem 5.8 Let (up) be a family of solutions to (5.2) and assume that (5.6)holds. Then there exist k∈N\{0} and k families of points (xi,p) in i= 1, . . . ,ksuch that, after passing to a sequence, (Pk

1), (Pk2) and (Pk

3) hold. Moreover,x1,p = x+p and, given any family of points xk+1,p, it is impossible to extract a

new sequence from the previous one such that (Pk+11 ), (Pk+1

2 ) and (Pk+13 ) hold

with the sequences (xi,p), i= 1, . . . ,k+ 1. Lastly, we have√

pup → 0 in C2loc( \S) as p→+∞. (5.31)

Moreover, there exists v ∈ C2( \S) such that

pup → v in C2loc( \S) as p→+∞, (5.32)

and (Pk4) holds.

Proof This result is mainly contained in [20]. The proof is inspired by theone in [25, Proposition 1] (see also [43]), but we have to deal with an extradifficulty because we allow the solutions up to be sign-changing. We divide theproof into several steps and all the claims are up to a subsequence.

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230 Francesca De Marchis, Isabella Ianni and Filomena Pacella

STEP 1. We show that there exists a family (x1,p) of points in such that,after passing to a sequence (P1

2 ) holds.We let x+p be a point in where |up| achieves its maximum. The proof then

follows by taking x1,p := x+p and using the results in Theorem 5.6-(ii).STEP 2. We assume that (Pn

1 ) and (Pn2 ) hold for some n ∈ N \ {0}. Then

we show that either (Pn+11 ) and (Pn+1

2 ) hold or (Pn3 ) holds, specifically there

exists C > 0 such thatpRn,p(x)

2|up(x)|p−1 ≤ C

for all x ∈, with Rn,p defined as in (5.26).Let n ∈N \ {0} and assume that (Pn

1 ) and (Pn2 ) hold while

supx∈

(pRn,p(x)

2|up(x)|p−1)→+∞ as p→+∞. (5.33)

We will prove that (Pn+11 ) and (Pn+1

2 ) hold.We let xn+1,p ∈ be such that

pRn,p(xn+1,p)2|up(xn+1,p)|p−1 = sup

x∈

(pRn,p(x)

2|up(x)|p−1)

. (5.34)

Clearly xn+1,p ∈ because up = 0 on ∂. By (5.34) and since is bounded itis clear that

p|up(xn+1,p)|p−1 →+∞ as p→+∞.

We claim that |xi,p− xn+1,p|μi,p

→+∞ as p→+∞ (5.35)

for all i= 1, . . . ,n and μi,p as in (5.23). In fact, assuming by contradiction thatthere exists i∈ {1, . . . ,n} such that |xi,p−xn+1,p|/μi,p →R as p→+∞ for someR≥ 0, thanks to (Pn

2 ), we get

limp→+∞p|xi,p− xn+1,p|2|up(xn+1,p)|p−1 = R2

(1

1+ 18 R2

)2

<+∞,

against (5.34). Setting

(μn+1,p)−2 := p|up(xn+1,p)|p−1, (5.36)

by (5.33) and (5.34) we deduce that

Rn,p(xn+1,p)

μn+1,p→+∞ as p→+∞. (5.37)

Then (5.36), (5.37) and (Pn1 ) imply that (Pn+1

1 ) holds with the added sequence(xn+1,p).

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 231

Next we show that also (Pn+12 ) holds with the added sequence (xn+1,p). Let

us define the scaled domain

n+1,p = {x ∈R2 : xn+1,p+μn+1,px ∈},and, for x ∈ n+1,p, the rescaled functions

vn+1,p(x)= p

up(xn+1,p)(up(xn+1,p+μn+1,px)− up(xn+1,p)), (5.38)

which, by (5.2), satisfy

−�vn+1,p(x)= |up(xn+1,p+μn+1,px)|p−1up(xn+1,p+μn+1,px)

|up(xn+1,p)|p−1up(xn+1,p)in n+1,p,

(5.39)or equivalently

−�vn+1,p(x)=∣∣∣∣1+ vn+1,p(x)

p

∣∣∣∣p−1(1+ vn+1,p(x)

p

)in n+1,p. (5.40)

Fix R > 0 and let (zp) be any point in n+1,p ∩ BR(0), whose correspondingpoints in are

xp = xn+1,p+μn+1,pzp.

Thanks to the definition of xn+1,p we have

pRn,p(xp)2|up(xp)|p−1 ≤ pRn,p(xn+1,p)

2|up(xn+1,p)|p−1. (5.41)

Since |xp− xn+1,p| ≤ Rμn+1,p we have

Rn,p(xp)≥ mini=1,...,n

|xn+1,p− xi,p|− |xp− xn+1,p|≥ Rn,p(xn+1,p)−Rμn+1,p

and, analogously,

Rn,p(xp)≤ Rn,p(xn+1,p)+Rμn+1,p.

Thus, by (5.37) we get

Rn,p(xp)= (1+ o(1))Rn,p(xn+1,p)

and in turn from (5.41)

|up(xp)|p−1 ≤ (1+ o(1))|up(xn+1,p)|p−1. (5.42)

In the following we show that for any compact subset K of R2

− 1+ o(1)≤−�vn+1,p ≤ 1+ o(1) in n+1,p ∩K (5.43)

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232 Francesca De Marchis, Isabella Ianni and Filomena Pacella

and

limsupp→+∞

supn+1,p∩K

vn+1,p ≤ 0. (5.44)

In order to prove (5.43) and (5.44) we will distinguish two cases.

(i) Assume thatup(xp)

up(xn+1,p)= |up(xp)||up(xn+1,p)| . (5.45)

Then by (5.42) we get |up(xp)|p ≤ (1+o(1))|up(xn+1,p)|p and so by (5.39)

(0≤)−�vn+1,p(zp)(5.45)= |up(xp)|p

|up(xn+1,p)|p ≤ 1+ o(1). (5.46)

Moreover, since (5.40) implies −�vn+1,p(zp)= evn+1,p(zp)+ o(1), we get

limsupp→+∞

vn+1,p(zp)≤ 0. (5.47)

(ii) Assume thatup(xp)

up(xn+1,p)=− |up(xp)|

|up(xn+1,p)| . (5.48)

Then, by the expression of vn+1,p necessarily

vn+1,p(zp)≤ 0, (5.49)

and moreover by (5.42)

0≥−�vn+1,p(zp)=− |up(xp)|p|up(xn+1,p)|p ≥−1+ o(1). (5.50)

(5.46) and (5.50) imply (5.43), while (5.49), (5.47) and the arbitrariness of zp

give (5.44).Using (5.43) and (5.44) we can prove, similarly to the proof of

Theorem 5.6-(ii), that

n+1,p →R2 as p→+∞. (5.51)

Then for any R > 0, BR(0)⊂ n+1,p for p large enough and vn+1,p are functionswith uniformly bounded Laplacian in BR(0) and with vn+1,p(0)= 0. So, by theHarnack inequality, vn+1,p is uniformly bounded in BR(0) for all R > 0 andthen by standard elliptic regularity vn+1,p → U in C2

loc(R2) as p →+∞ with

U ∈ C2(R2), U(0)= 0 and, by (5.44), U ≤ 0. Passing to the limit in (5.40) weget that U is a solution of −�U = eU in R2. Then by Fatou’s lemma, as in theproof of Theorem 5.6-(ii), we get that eU ∈ L1(R2) and so by the classificationresult in [8] we have the explicit expression of U.

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 233

This proves that (Pn+12 ) holds with the added points (xn+1,p), thus STEP 2 is

proved.STEP 3. We complete the proof of Theorem 5.8.From STEP 1 we have that (P1

1 ) and (P12 ) hold. Then, by STEP 2, either

(P21 ) and (P2

2 ) hold or (P13 ) holds. In the last case the assertion is proved with

k = 1. In the first case we go on and proceed with the same alternative untilwe reach a number k ∈ N \ {0} for which (Pk

1), (Pk2) and (Pk

3) hold up to asequence. Note that this is possible because the solutions up satisfy (5.6) andLemma 5.7 holds and hence the maximal number k of families of points forwhich (Pk

1), (Pk2) hold must be finite.

Moreover, given any other family of points xk+1,p, it is impossible to extracta new sequence from it such that (Pk+1

1 ), (Pk+12 ) and (Pk+1

3 ) hold together withthe points (xi,p)i=1,..,k+1. Indeed, if (Pk+1

1 ) was verified then

|xk+1,p− xi,p|μk+1,p

→+∞ as p→+∞, for any i ∈ {1, . . . ,k},

but this would contradict (Pk3).

Finally, the proofs of (5.31) and (5.32) are a direct consequence of (Pk3).

Indeed, if K is a compact subset of \ S by (Pk3) we have that there exists

CK > 0 such that

p|up(x)|p−1 ≤ CK for all x ∈ K.

Then by (5.2) ‖�(√

pup)‖L∞(K) ≤ CK‖up‖L∞(K)√

p → 0, as p → +∞. Hence

standard elliptic theory shows that√

pup → w in C2(K), for some w.But by Theorem 5.6 we know that

√pup ⇀ 0, so w = 0 and (5.31) is

proved. Iterating, we then have ‖�(pup)‖L∞(K) ≤ CK‖up‖L∞(K) → 0, as p →+∞ by (5.31). And so by standard elliptic theory we have that pup → v

in C2(K), for some v. The arbitrariness of the compact set K ends theproof of (5.32).

It remains to prove (Pk4). Green’s representation gives

p|∇up(x)| = p

∣∣∣∣∫

∇G(x,y)up(y)pdy

∣∣∣∣≤ p∫

|∇G(x,y)||up(y)|pdy

≤ Cp∫

|up(y)|p|x− y| dy, (5.52)

where G is the Green’s function of −� in with Dirichlet boundaryconditions, and in the last estimate we have used that |∇xG(x,y)| ≤

C|x−y| ∀x,y∈, x = y (see for instance [12]). Let Rk,p(x)=mini=1,...,k |x−xi,p|

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234 Francesca De Marchis, Isabella Ianni and Filomena Pacella

and i,p = {x ∈ : |x− xi,p| = Ri,p(x)}, i= 1, . . . ,k. We then have

p∫i,p

|x− y|−1|up(y)|p dy= p∫i,p∩B |x−xi,p|

2

(xi,p)

|x− y|−1|up(y)|p dy

+ p∫i,p\B |x−xi,p|

2

(xi,p)

|x− y|−1|up(y)|p dy.

Note that by (5.14) and (Pk3) for y ∈i,p \B |x−xi,p|

2(xi,p) we have

p |up(y)|p|x− y| ≤ C

p |up(y)|p−1

|x− y| ≤ C

|x− y||y− xi,p|2 ≤C

|x− y||x− xi,p|2 ,

and hence∫i,p\B |x−xi,p|

2

(xi,p)

p |up(y)|p|x− y| dy ≤ 1

|x− xi,p|2∫|x−y|≤|x−xi,p|

C

|x− y| dy

+ 1

|x− xi,p|∫i,p

p |up(y)|pdy

(5.15)≤ C

|x− xi,p| .

For y ∈ i,p ∩ B |x−xi,p|2

(xi,p), |x− y| ≥ |x− xi,p| − |y− xi,p| ≥ |x− xi,p|/2, and

hence by (5.14) and (5.15) we get

p∫i,p∩B |x−xi,p|

2

(xi,p)

|up(y)|p|x− y| dy≤ C

|x− xi,p| , for i= 1, . . . ,k

so that (Pk4) is proved.

In the rest of this section we derive some consequences of Theorem 5.8.

Remark 5.9 Under the assumptions of Theorem 5.8 we have

dist(xi,p,∂)

μi,p−→

p→+∞+∞ for all i ∈ {1, . . . ,k}.

Corollary 5.10 Under the assumptions of Theorem 5.8, if up is sign-changingit follows that

dist(xi,p,NLp)

μi,p−→

p→+∞+∞ for all i ∈ {1, . . . ,k},

where NLp denotes the nodal line of up.As a consequence, for any i ∈ {1, . . . ,k}, letting Ni,p ⊂ be the nodal domain

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 235

of up containing xi,p and setting uip := upχNi,p (χA is the characteristic function

of the set A), then the scaling of uip around xi,p:

zi,p(x) := p

up(xi,p)(ui

p(xi,p+μi,px)− up(xi,p)),

defined on Ni,p := Ni,p−xi,pμi,p

, converges to U in C2loc(R

2), where U is the same

function defined in (Pk2).

Proof Let us suppose by contradiction that

dist(xi,p,NLp)

μi,p−→

p→+∞ �≥ 0,

then there exist yp ∈ NLp such that|xi,p−yp|

μi,p→ � as p→+∞. Setting

vi,p(x) := p

up(xi,p)(up(xi,p+μi,px)− up(xi,p)),

on the one handvi,p(

yp− xi,p

μi,p)=−p −→

p→+∞−∞,

on the other hand by (Pk2) and up to subsequences

vi,p

(yp− xi,p

μi,p

)−→

p→+∞ U(x∞) >−∞,

where x∞ = limp→+∞yp−xi,pμi,p

∈ R2 and so |x∞| = �. Thus we have obtained acontradiction which proves the assertion.

For a family of points (xp)p ⊂ we denote by μ(xp) the numbersdefined by [

μ(xp)]−2

:= p|up(xp)|p−1. (5.53)

Proposition 5.11 Let (xp)p ⊂ be a family of points such that p|up(xp)|p−1 →+∞ and let μ(xp) be as in (5.53). By (Pk

3), up to a sequence, Rk,p(xp) =|xi,p− xp|, for a certain i ∈ {1, . . . ,k}. Then

limsupp→+∞

μi,p

μ(xp)≤ 1.

Proof To shorten the notation let us denote μ(xp) simply by μp. Let us start byproving that

μi,pμp

is bounded. So we assume by contradiction that there exists asequence pn →+∞, as n→+∞, such that

μi,pn

μpn

→+∞ as n→+∞. (5.54)

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236 Francesca De Marchis, Isabella Ianni and Filomena Pacella

By (Pk3) and (5.54) we then have

|xpn − xi,pn |μi,pn

= |xpn − xi,pn |μpn

μpn

μi,pn

→ 0 as n→+∞,

so that by (Pk2)

vi,pn

(xpn − xi,pn

μi,pn

)→U(0)= 0 as n→+∞.

As a consequence

μi,pn

μpn

=(

upn(xpn)

upn(xi,pn)

)pn−1

=⎛⎝1+

vi,pn

(xpn−xi,pn

μi,pn

)pn

⎞⎠pn−1

→ eU(0) = 1

as n→+∞,

which contradicts with (5.54). Hence we have proved thatμi,pμp

is bounded.

Next we show thatμi,pμp≤ 1. Assume by contradiction that there exist � > 1

and a sequence pn →+∞ as n→+∞ such that

μi,pn

μpn

→ � as n→+∞. (5.55)

By (Pk3) and (5.55) we then have

|xpn − xi,pn |μi,pn

= |xpn − xi,pn |μpn

μpn

μi,pn

≤ 2√

C

for n large, so that by (Pk2) there exists x∞ ∈R2, |x∞| ≤ 2

√C

�such that, up to a

subsequence,

vi,pn

(xpn − xi,pn

μi,pn

)→U(x∞)≤ 0 as n→+∞.

As a consequence

μi,pn

μpn

=(

upn(xpn)

upn(xi,pn)

)pn−1

=⎛⎝1+

vi,pn

(xpn−xi,pn

μi,pn

)pn

⎞⎠pn−1

→ eU(x∞)

as n→+∞.

By (5.55) and the assumption � > 1 we deduce

U(x∞)= log�+ on(1) > 0,

reaching a contradiction.

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 237

Proposition 5.12 Let xp and xi,p be as in the statement of Proposition 5.11. If

|xp−xi,p|μi,p

→+∞ as p→+∞, (5.56)

thenμi,pμ(xp)

→ 0 as p→+∞,

where μ(xp) is defined in (5.53).

Proof By Proposition 5.11 we know that

μi,p

μ(xp)≤ 1+ o(1).

Assume by contradiction that (5.56) holds but there exist 0 < � ≤ 1 and asequence pn →+∞ such that

μi,pn

μ(xpn)→ �, as n→+∞. (5.57)

Then (5.57) and (Pk3) imply

|xpn − xi,pn |μi,pn

= |xpn − xi,pn |� μ(xpn)

+ on(1)≤ C

�+ on(1) as n→+∞,

which contradicts (5.56).

Remark 5.13 If up(xp) and up(xi,p) have opposite sign, i.e.

up(xp)up(xi,p) < 0,

then, by Corollary 5.10, necessarily (5.56) holds. Hence in this case

μi,pμ(xp)

→ 0 as p→+∞.

The next result characterizes in different ways the concentration set S .

Proposition 5.14 (Characterizations of S) Let (up) be a family of solutionsto (5.2) satisfying (5.6). Then the following hold:

S ={

x ∈ : ∀r0 > 0, ∀p0 > 1, ∃p > p0 s.t. p∫

Br0 (x)∩|up(x)|p+1 dx≥ 1

};

(5.58)

S = {x ∈ : ∃a subsequence of (up) and a sequencexp →p x

s.t. p|up(xp)|→p +∞}

. (5.59)

Proof Proof of (5.58): by (Pk3) it is immediate to see that if x /∈ S then

p∫

Br(x)∩ |up(x)|p+1 dx → 0 as r → 0+, uniformly in p. On the other hand

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238 Francesca De Marchis, Isabella Ianni and Filomena Pacella

if x ∈ S , i.e. x = limp→+∞ xi,p for some i = 1, . . . ,k, we fix R > 0 such that∫BR(0)

eU dx > 1 (where U is defined in (5.12)) and then for p large, reasoningas in the proof of Lemma 5.7, we get:

p∫

Br0 (x)∩|up(x)|p+1dx≥ p

∫BRμi,p (xi,p)

|up(x)|p+1 dx

= |up(xi,p)|2∫

BR(0)

(1+ vj,p(y)

p

)p+1

dy,

where by Fatou’s lemma

liminfp→+∞ |up(xi,p)|2

∫BR(0)

(1+ vj,p(y)

p

)p+1

dy ≥ liminfp→+∞ |up(xi,p)|2

∫BR(0)

eU(y) dy

(5.24)≥∫

BR(0)eU(y) dy > 1.

Proof of (5.59): if x /∈ S , then by (Pk4), which holds by Theorem 5.8, p|up|

is uniformly bounded in L∞(K) for some compact set K containing x and thenthere can not exist a sequence xp → x such that p|up(xp)| →+∞. Conversely,if x ∈ S , i.e. x= limp→+∞ xi,p for some i= 1, . . . ,k, and by (5.22) we know that|up(xi,p)| ≥ 1

2 for p large, therefore p|up(xi,p)|→+∞. This proves (5.59).

We conclude this section with a result for positive solutions that we haverecently obtained (see [22] and the latest improvement in [17]) carrying on theasymptotic analysis started in Theorem 5.8.

Theorem 5.15 Let (up) be a family of positive solutions to (5.2) and assumethat (5.6) holds. Let k ∈ N \ {0} and (xi,p), i = 1, . . . ,k, be the integer and thefamilies of points of introduced in Theorem 5.8. One has:

limp→+∞‖up‖∞ =

√e.

Moreover, there exist k distinct points xi ∈, i= 1, . . . ,k such that:

(i) up to a subsequence limp→+∞xi,p = xi, for any i = 1, . . . ,k; therefore the

concentration set S , introduced in (5.25), consists of the k points

S = {x1, . . . ,xk} ⊂;

(ii)

pup(x)→ 8π√

ek∑

i=1

G(x,xi) as p→+∞, in C2loc( \S),

where G is the Green’s function of −� in under Dirichlet boundaryconditions;

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 239

(iii)

p∫

|∇up(x)|2 dx→ 8πe · k, as p→+∞;

(iv) the concentration points xi, i= 1, . . . ,k satisfy

∇xH(xi,xi)+∑i=�

∇xG(xi,x�)= 0, (5.60)

where

H(x,y)=G(x,y)+ log(|x− y|)2π

(5.61)

is the regular part of the Green’s function G;(v)

limp→+∞‖up‖L∞(Bδ(xi))

=√e, i= 1, . . . ,k

for any δ > 0 such that Bδ(xi) does not contain any other xj, j = i.

Remark 5.16 Observe that k= 1 for the least energy solutions and that in thiscase the results in the previous theorem were already known (see [41], [42]and [1]).

5.3 The G-symmetric Case

In this section we focus on sign-changing solutions. Of course all theresults in Section 5.2 hold true if assumption (5.6) is satisfied, in particularTheorems 5.6 and 5.8.

Hence letting x±p be the family of points where |up(x±p )| = ‖u±p ‖∞, then fromTheorem 5.6-(i) we know that for x±p the analogs of (5.22) and (5.24) hold andso we have

μ±p := (p|up(x

±p )|p−1)− 1

2 → 0 as p→+∞. (5.62)

From now on w.l.g. we assume that the L∞-norm of up is assumed at amaximum point, namely that up(x+p ) = ‖u+p ‖∞ = ‖up‖∞ and that −up(x−p ) =‖u−p ‖∞.

So by Theorem 5.6-(ii) we already know that scaling up about the maximumpoint x+p as in (5.11) gives a first “bubble” converging to the function U definedin (5.12).

In general, for sign-changing solutions, one would like to investigate thebehavior of up when scaling about the minima x−p and understand whether x−pcoincides with one of the k sequences in Theorem 5.8 or not. Moreover, onewould like to describe the set of concentration S .

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240 Francesca De Marchis, Isabella Ianni and Filomena Pacella

Recall that if x−p is one of the sequences of Theorem 5.8 then by Corol-lary 5.10 one has

dist(x−p ,NLp)

μ−p→+∞, as p→+∞, (5.63)

where NLp denotes the nodal line of up. On the contrary, it is easy to seethat if (5.63) is satisfied, then one can use the same ideas as in the proof ofTheorem 5.6-(ii) also for the scaling about the minimum, which we define inthe natural way as

v−p (x) := pup(x−p +μ−p x)− up(x−p )

up(x−p ), x ∈ −

p := {x ∈R2 : x−p +μ−p x ∈},(5.64)

and obtain that v−p → U in C2loc(R

2) (this has been done for instance in [31]for the case of low-energy sign-changing solutions under some additionalassumptions).

Here we analyze the case when up belongs to a family of G-symmetricsign-changing solutions satisfying the same properties as the ones inTheorem 5.3 recalled in the Introduction and show that a different phenomenonappears.

All the results of this section are mainly based on the work [20], theexistence result (Theorem 5.3) is instead contained in [19].

We prove the following.

Theorem 5.17 Let ⊂R2 be a connected bounded smooth domain, invariantunder the action of a cyclic group G of rotations about the origin, with |G| ≥ 4e(|G| is the order of G) and such that the origin O ∈. Let (up) be a family ofsign-changing G-symmetric solutions of (5.2) with two nodal regions, NLp ∩∂= ∅, O ∈ NLp and

p∫

|∇up|2dx≤ α 8πe (5.65)

for some α < 5 and p large. Then, assuming w.l.g. that ‖up‖∞ = ‖u+p ‖∞, wehave:

(i) S = {O} and k= 1 where S and k are the ones in Theorem 5.8;(ii) |x+p |→O as p→+∞;

(iii) |x−p |→O as p→+∞;(iv) NLp shrinks to the origin as p→+∞;

(v) There exists x∞ ∈ R2 \ {0} such that, up to a subsequence,x−pμ−p→−x∞

and

v−p (x)−→ V(x− x∞) in C2loc(R

2 \ {x∞}) as p→+∞,

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 241

where

V(x) := log

(2α2βα|x|α−2

(βα+|x|α)2

), (5.66)

with α = α(|x∞|) =√

2|x∞|2+ 4, β = β(|x∞|) = |x∞|(α+2α−2

) 1α , is a

singular radial solution of{ −�V = eV +Hδ0 in R2,∫R2 eVdx <∞,

(5.67)

where H=H(|x∞|) < 0 is a suitable constant and δ0 is the Dirac measurecentered at 0.

Observe that the existence of families of solutions up having all theproperties as in the assumptions of Theorem 5.17 has been proved in [19] when is simply connected and when |G|> 4 (see Theorem 5.3 in Section 5.1).

We also recall that in [32] the case of least energy sign-changing radialsolutions in a ball has been studied, proving a result similar to that inTheorem 5.17 with precise estimates of α,β and H.

5.3.1 Proofs of (i)− (ii) of Theorem 5.17

Let us introduce the following notation:

• N±p ⊂ denotes the positive/negative nodal domain of up,

• N±p are the rescaled nodal domains about the points x±p by the parameters

μ±p defined in the introduction, i.e.

N±p := N±

p − x±pμ±p

= {x ∈R2 : x±p +μ±p x ∈N±p }.

Let k, (xi,p), i= 1, . . . ,k and S be as in Theorem 5.8 then, defining μi,p as in(5.23), we get the following.

Proposition 5.18 Under the assumptions of Theorem 5.17,

|xi,p|μi,p

is bounded, ∀i= 1, . . . ,k.

In particular |xi,p|→ 0, ∀i= 1, . . . ,k, as p→+∞, so that S = {O}.Proof W.l.g. we can assume that for each i = 1, . . . ,k either (xi,p)p ⊂ N+

p or(xi,p)p ⊂N−

p . We prove the result in the case (xi,p)p ⊂N+p , the other case being

similar. Moreover, in order to simplify the notation we drop the dependence oni, that is we set xp := xi,p and μp :=μi,p.

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242 Francesca De Marchis, Isabella Ianni and Filomena Pacella

Let h := |G| and let us denote by gj, j = 0, . . . ,h− 1, the elements of G. Weconsider the rescaled nodal domains

Npj,+

:= {x ∈R2 : μpx+ gjxp ∈N+p }, j= 0, . . . ,h− 1,

and the rescaled functions zj,+p (x) : Np

j,+ →R defined by

zj,+p (x) := p

u+p (xp)

(u+p (μpx+ gjxp)− u+p (xp)

), j= 0, . . . ,h− 1. (5.68)

Hence it’s not difficult to see (as in Corollary 5.10) that each zj,+p converges to U

in C2loc(R

2) as p→∞, where U is the function in (5.12). Assume by contradic-

tion that there exists a sequence pn →+∞ as n→+∞ such that |xpn |μpn

→+∞.

Then, since the h distinct points gjxpn , j= 0, . . . ,h−1, are the vertex of a regularpolygon centered in O, dn := |gjxpn −gj+1xpn | = 2dn sin π

h , where dn := |gjxpn |,j= 0, ..,h− 1, and so we also have that dn

μpn→+∞ as n→+∞. Let

Rn :=min

{dn

3,d(xpn ,∂)

2,d(xpn ,NLpn)

2

}, (5.69)

then by construction BRn(gjxpn)⊆N+

pnfor j= 0, . . . ,h− 1,

BRn(gjxpn)∩BRn(g

lxpn)= ∅, for j = l (5.70)

andRn

μpn

→+∞ as n→+∞. (5.71)

Using (5.71), the convergence of zj,+pn to U, (5.24) and Fatou’s lemma, we have

8π =∫R2

eU dx≤ limn

∫B Rnμpn

(0)

∣∣∣∣∣1+ zj,+pn

pn

∣∣∣∣∣(pn+1)

dx

= limn

pn∣∣u+pn(xpn)

∣∣2∫

BRn (gjxpn )

∣∣u+pn

∣∣(pn+1)dx

(5.24)≤ limn

pn

∫BRn (g

jxpn )

∣∣u+pn

∣∣(pn+1)dx. (5.72)

Summing on j= 0, . . . ,h− 1, using (5.70), (5.65), (5.8) and (5.9) we get:

h · 8π ≤ limn

pn

h−1∑j=0

∫BRn (g

jxpn )

∣∣u+pn

∣∣(pn+1)dx

(5.70)≤ limn

pn

∫N+

pn

∣∣u+pn

∣∣(pn+1)dx

= limn

(pn

∣∣upn

∣∣(pn+1)dx− pn

∫N−

pn

∣∣u−pn

∣∣(pn+1)dx

)

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 243

(5.65)+(5.8)≤ (α− 1) · 8πeα<5< 4 · 8πe,

which contradicts the assumption |G| ≥ 4e.

Remark 5.19 If we knew that ‖up‖∞ ≥ √e, then we would obtain a betterestimate in (5.72), and so Proposition 5.18 would hold under the weakersymmetry assumption |G| ≥ 4 (recall that |G| ≥ 4 is the assumption underwhich one can prove Theorem 5.3).

It is also possible to prove the following (see [20, Corollary 3.5] for moredetails).

Corollary 5.20 Under the assumptions of Theorem 5.17

(i) O ∈N+p for p large;

(ii) let i ∈ {1, . . . ,k}, then xi,p ∈N+p for p large.

Proposition 5.21 Under the assumptions of Theorem 5.17, the maximalnumber k of families of points (xi,p), i = 1, . . . ,k, for which (Pk

1), (Pk2) and

(Pk3) hold is 1.

Proof Let us assume by contradiction that k > 1 and set x+p = x1,p. For a family(xj,p) j ∈ {2, . . . ,k} by Proposition 5.18, there exists C > 0 such that

|x1,p|μ1,p

≤ C and|xj,p|μj,p

≤ C.

Thus, since by definition μ+p =μ1,p ≤μj,p, also

|x1,p|μj,p

≤ C.

Hence |x1,p− xj,p|μj,p

≤ |x1,p|+ |xj,p|μj,p

≤ C,

which contradicts (Pk1) when p→+∞.

Then we easily get the following.

Proposition 5.22 Under the assumptions of Theorem 5.17 there exists C > 0such that |xp|

μ(xp)≤ C (5.73)

for any family (xp)p ⊂ , where μ(xp) is defined as in (5.53). In particular,since by (5.62) μ−p → 0, then |x−p |→ 0.

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244 Francesca De Marchis, Isabella Ianni and Filomena Pacella

Proof (5.73) holds for x+p by Proposition 5.18. Moreover, k = 1 by Proposi-tion 5.21, so applying (P1

3 ) to the points (xp), for xp = x+p , we have

|xp− x+p |μ(xp)

≤ C.

By definition, μ+p ≤μ(xp), hence we get

|xp|μ(xp)

≤ |xp− x+p |μ(xp)

+ |x+p |μ(xp)

≤ |xp− x+p |μ(xp)

+ |x+p |μ+p

≤ C.

Lemma 5.23 Let the assumptions of Theorem 5.17 be satisfied and let (xp)⊂

be such that p|up(xp)|p−1 →+∞ and μ(xp) be as in (5.53). Assume also thatthe rescaled functions vp(x) := p

up(xp)(up(xp+μ(xp)x)− up(xp)) converge to U

in C2loc(R

2 \ {− limpxp

μ(xp)}) as p→+∞ (U as in (5.12)). Then

|xp|μ(xp)

→ 0 as p→+∞. (5.74)

As a byproduct we deduce that vp →U in C2loc(R

2 \ {0}), as p→+∞.

Proof By Proposition 5.22 we know that |xp|μ(xp)

≤ C. Assume by contradiction

that |xp|μ(xp)

→ �> 0. Let g∈G such that |xp−gxp| =Cg|xp| with constant Cg > 1(such a g exists because G is a group of rotation about the origin). Hence

|xp− gxp|μ(xp)

= Cg|xp|μ(xp)

→ Cg� > �.

Then x0 := limp→+∞gxp−xpμ(xp)

∈R2\{− limpxp

μ(xp)} and so by the C2

loc convergencewe get

vp

(gxp− xp

μ(xp)

)→U(x0) < 0 as p→+∞.

On the other hand, for any g ∈G, one also has

vp

(gxp− xp

μ(xp)

)= p

up(xp)(up(gxp)− up(xp))= 0,

by the symmetry of up and this gives a contradiction.

5.3.2 Asymptotic Analysis about the Minimum Points x−pand Study of NLp

Proposition 5.21 implies that (P13 ) holds, from which

|x+p − x−p |μ−p

≤ C, (5.75)

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 245

with μ−p as in (5.62). Moreover, since we already know thatd(x+p ,NLp)

μ+p→+∞

as p→+∞, we deduce that|x+p −x−p |

μ+p→+∞ as p→+∞, and in turn by (5.75)

we getμ+pμ−p

→ 0 as p→+∞. (5.76)

Note that (5.75) and (5.76) hold more generally for any family of points (xp)

such that up(xp) < 0 and p|up(xp)|p−1 →+∞.By Proposition 5.22 we have

|x−p |μ−p

≤ C, (5.77)

so there are two possibilities: either|x−p |μ−p

→ � > 0 or|x−p |μ−p

→ 0 as p→+∞, up

to subsequences. We will exclude the latter case. We start with a preliminaryresult.

Lemma 5.24 For x ∈

|x−p | := {y ∈ R2 : y|x−p | ∈ } let us define the rescaled

function

w−p (x) := p

up(x−p )(up(|x−p |x)− up(x

−p ))

.

Then

w−p → γ in C2

loc(R2 \ {0}) as p→+∞, (5.78)

where γ ∈ C2(R2 \ {0}), γ ≤ 0, γ (x∞)= 0 for a point x∞ ∈ ∂B1(0) and it is asolution to

−�γ = �2eγ in R2 \ {0}.In particular γ ≡ 0 when �= 0.

Proof (5.77) implies that |x−p | → 0 as p → +∞, so it follows that the set

|x−p | →R2 as p→+∞.

By definition we have

w−p ≤ 0, wp

(x−p|x−p |

)= 0 and w−

p =−p on ∂

(

|x−p |

). (5.79)

Moreover, for x ∈

|x−p | we define ξp := |x−p |x and μξp as μ−2ξp

:= p|up(ξp)|p−1.

Thanks to (5.2) we then have

|−�w−p (x)| =

p|x−p |2|up(ξp)|p|up(x−p )|

= |up(ξp)||up(x−p )|

|x−p |2μ2

ξp

≤ c∞|x−p |2μ2

ξp

, (5.80)

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246 Francesca De Marchis, Isabella Ianni and Filomena Pacella

where c∞ := limp ‖up‖∞. Then, observing that|x−p |μξp

≤ C|x| by Proposition 5.22

applied to ξp, we have

|−�w−p (x)| ≤

c∞C2

|x|2 .

Specifically, for any R > 0

|−�w−p | ≤ c∞C2R2 in

|x−p |\B 1

R(0). (5.81)

So, similarly to the proof of Theorem 5.6(ii) (using now that w−p (

x−p|x−p | )= 0),

it follows that for any R > 1 (x−p|x−p | ∈ ∂B1(0)⊂ BR(0) \B 1

R(0) for R > 1), w−

p is

uniformly bounded in BR(0) \B 1R(0).

After passing to a subsequence, standard elliptic theory applied to thefollowing equation

−�w−p (x)=

|x−p |2(μ−p )2

(1+ w−

p (x)

p

)∣∣∣∣∣1+ w−p (x)

p

∣∣∣∣∣p−1

(5.82)

gives that w−p is bounded in C2,α

loc (R2 \ {0}) . Hence (5.78) and the properties of

γ follow.In particular when � = 0 it follows that γ is harmonic in R2 \ {0} and

γ (x∞) = 0 for some point x∞ ∈ ∂B1(0), therefore by the maximum principlewe obtain γ ≡ 0.

Proposition 5.25 There exists � > 0 such that

|x−p |μ−p

→ � as p→+∞.

Proof By Proposition 5.22 we know that|x−p |μ−p

→ � ∈ [0,+∞) as p→+∞. Let

us suppose by contradiction that �= 0. Then Lemma 5.24 implies that

w−p → 0 in C2

loc(R2 \ {0}) as p→+∞. (5.83)

By (5.2), applying the divergence theorem in B|x−p |(0) we get

p∫∂B|x−p |(0)

∇up(y) · y

|y| dσ(y)= p∫

B|x−p |(0)∩N−p

|up(x)|p dx

− p∫

B|x−p |(0)∩N+p

|up(x)|p dx. (5.84)

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 247

Scaling up with respect to |x−p | as in Lemma 5.24, by (5.83) we obtain∣∣∣∣∣∣p∫∂B|x−p |(0)

∇up(y) · y

|y| dσ(y)

∣∣∣∣∣∣=∣∣∣∣p∫

∂B1(0)|x−p |∇up(|x−p |x) ·

x

|x| dσ(x)

∣∣∣∣=∣∣∣∣∫

∂B1(0)up(x

−p )∇w−

p (x) ·x

|x| dσ(x)

∣∣∣∣ ≤ |up(x−p )|2π sup

|x|=1|∇w−

p (x)| = op(1).

(5.85)

Now we want to estimate the right-hand side in (5.84). We first observe thatscaling around |x−p | with respect to μ−p we get

p∫

B|x−p |(0)∩N−p

|up(x)|p dx= p∫

B1(0)∩N−

p

|x−p ||up(|x−p |y)|p|x−p |2 dy

≤ c∞∫

B1(0)∩N−

p

|x−p |

|up(|x−p |y)|p−1

|up(x−p )|p−1

|x−p |2(μ−p )2

dy = op(1), (5.86)

where in the last equality we have used that|up(|x−p |y)|p−1

|up(x−p )|p−1 ≤ 1, since |x−p |y ∈N−

p

and that by assumption|x−p |μ−p

→ 0 as p→+∞.

Next we claim that there exists p > 1 such that for any p≥ p

Bμ+p (x+p )⊂ B|x−p |(0). (5.87)

Indeed, Corollary 5.10 implies that

+∞= limp

d(x+p ,NLp)

μ+p≤ lim

p

|x+p − x−p |μ+p

≤ limp

|x+p |μ+p

+ limp

|x−p |μ+p

= limp

|x−p |μ+p

,

where the last equality follows from Lemma 5.23 (i.e.|x+p |μ+p

→ 0). Hence, for

any x ∈ B1(0) we have

|x+p +μ+p x||x−p |

≤ |x+p ||x−p |

+ μ+p|x−p |

≤ 2μ+p|x−p |

→ 0 as p→+∞,

and so (5.87) is proved.Hence, by (5.87) and scaling around x+p with respect to μ+p we obtain

p∫

B|x−p |(0)∩N+p

|up(x)|p dx≥ p∫

Bμ+p(x+p )

|up(x)|pdx= c∞∫

B1(0)eUdx+ op(1).

(5.88)Collecting (5.84), (5.85), (5.86) and (5.88) we clearly get a contradiction.

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248 Francesca De Marchis, Isabella Ianni and Filomena Pacella

Next we show that the nodal line shrinks to the origin faster than μ−p asp→+∞.

Proposition 5.26 We have

maxyp∈NLp

|yp|μ−p

→ 0 as p→+∞.

Proof By Proposition 5.25 it is enough to prove that

maxyp∈NLp

|yp||x−p |

→ 0 as p→+∞.

First we show that, for any yp ∈ NLp, the following alternatives hold:

either|yp||x−p |

→ 0 or|yp||x−p |

→+∞ as p→+∞. (5.89)

Indeed, assume by contradiction that |yp||x−p | →m∈ (0,+∞) as p→+∞. Then

w−p (

yp

|x−p | )=−p→−∞ as p→+∞. But we have proved in Lemma 5.24 that

w−p (

yp

|x−p | )→ γ (ym) ∈ R, where ym is such that |ym| = m > 0, and this gives a

contradiction.To conclude the proof we have then to exclude the second alternative in

(5.89). For yp ∈ NLp, let us assume by contradiction that |yp||x−p | → +∞ as p→

+∞ and let us observe that

∃ zp ∈ NLp such that|zp||x−p |

→ 0 as p→+∞. (5.90)

Indeed, in the previous section we have shown that O∈N+p , hence there exists

tp ∈ (0,1) such that zp := tpx−p ∈ NLp. Since |zp||x−p | < 1, by (5.89) we get (5.90).

Then for any M > 0 there exists αMp ∈ NLp such that

|αMp |

|x−p | →M as p→+∞and this is in contradiction with (5.89).

Finally, we can analyze the local behavior of up around the minimum pointx−p . Note that by Lemma 5.23 and Proposition 5.25 we can already claim thatthe rescaling v−p about x−p (see (5.64)) cannot converge to U in R2 \ {0}, whereU is the function in (5.12), indeed we have the following.

Proposition 5.27 Passing to a subsequence

v−p (x)−→ V(x− x∞) in C2loc(R

2 \ {x∞}), as p→+∞, (5.91)

where V is the radial singular function in (5.66) which satisfies the Liouville

equation (5.67) and x∞ := limpx−pμ−p

.

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Asymptotic Analysis for the Lane–Emden Problem in Dimension Two 249

Proof Let us consider the translations of v−p :

s−p (x) := v−p

(x− x−p

μ−p

)= p

up(x−p )(up(μ

−p x)− up(x

−p )), x ∈

μ−p,

which solve

−�s−p (x)=∣∣∣∣∣1+ s−p (x)

p

∣∣∣∣∣p−1(

1+ s−p (x)p

), s−p

(x−pμ−p

)= 0, s−p ≤ 0.

Observe that

μ−p→R2 as p→+∞.

We claim that for any fixed r > 0, |−�s−p | is bounded in

μ−p\Br(0).

Indeed, Proposition 5.26 implies that if x ∈ N+p

μ−p, then |x| ≤

maxzp∈NLp

|zp|μ−p

< r, for

p large, hence (

μ−p\Br(0)

)⊂ N−

p

μ−pfor p large

and so the claim follows by observing that for x ∈ N−p

μ−p, then |−�s−p (x)| ≤ 1.

Hence, by the arbitrariness of r > 0, s−p → V in C2loc(R

2 \ {0}) as p→+∞,where V is a solution of

−�V = eV in R2 \ {0}

which satisfies V ≤ 0 and V(x�) = 0, where x� := limpx−pμ−p

and |x�| = � by

Proposition 5.25. Moreover, by virtue of Theorem 5.6-(i) and by (5.65) it canbe seen that eV ∈ L1(R2).

Observe that if V was a classical solution of −�V = eV in the whole R2

then necessarily V(x)= U(x− x�). As a consequence v−p (x)= s−p (x+ x−pμ−p

)→V(x + x�) = U(x) in C2

loc(R2 \ {−x�}) as p → +∞. But then Lemma 5.23

would imply that |x�| = |x−p |μ−p

→ 0 as p→+∞, which is in contradiction with

Proposition 5.25. Thus, by [9, 10, 11] and the classification in [8] we have thatV solves, for some η > 0, the following entire equation:{ −�V = eV − 4πηδ0 in R2,∫

R2 eVdx= 8π(1+η),

where δ0 denotes the Dirac measure centered at the origin.By the classification given in [40], we have that either V is radial, or η ∈ N

and V is (η+ 1)-symmetric. Actually it turns out that the energy bound (5.65)

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250 Francesca De Marchis, Isabella Ianni and Filomena Pacella

forces V to be a radial solution, V(r), satisfying⎧⎨⎩−V ′′ − 1

r V ′ = eV in (0,+∞),V ≤ 0,V(�)= V ′(�)= 0.

.

The solution of this problem is

V(r)= log

(2α2βαrα−2

(βα+ rα)2

),

where α=√2�2+ 4 and β = �(α+2α−2

)1/α. The conclusion follows by observing

that v−p (x)= s−p(

x+ x−pμ−p

)and setting x∞ =−x�.

5.3.3 Conclusion of the Proof of Theorem 5.17

It follows by combining all the previous results. More precisely (i) and (ii)follow from Propositions 5.18 and 5.21. (iii) is due to Proposition 5.22. (iv) isin Proposition 5.26. Finally, (v) comes from Proposition 5.27.

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∗ P.le Aldo Moro 5, 00185 Roma, Italy† V.le Lincoln 5, 81100 Caserta, Italy‡ P.le Aldo Moro 5, 00185 Roma, Italy

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6

A Data Assimilation Algorithm: the Paradigm ofthe 3D Leray-α Model of Turbulence

Aseel Farhat∗, Evelyn Lunasin† and Edriss S. Titi‡

In this paper we survey the various implementations of a new data assimilation(downscaling) algorithm based on spatial coarse mesh measurements. As aparadigm, we demonstrate the application of this algorithm to the 3D Leray-αsubgrid scale turbulence model. Most importantly, we use this paradigm to showthat it is not always necessary to collect coarse mesh measurements of all the statevariables that are involved in the underlying evolutionary system, in order torecover the corresponding exact reference solution. Specifically, we show that inthe case of the 3D Leray-α model of turbulence, the solutions of the algorithm,constructed using only coarse mesh observations of any two components of thethree-dimensional velocity field, and without any information on the thirdcomponent, converge, at an exponential rate in time, to the corresponding exactreference solution of the 3D Leray-α model. This study serves as an addendum toour recent work on abridged continuous data assimilation for the 2D Navier–Stokesequations. Notably, similar results have also been recently established for the 3Dviscous Planetary Geostrophic circulation model in which we show that coarsemesh measurements of the temperature alone are sufficient for recovering, throughour data assimilation algorithm, the full solution; i.e. the three components ofvelocity vector field and the temperature. Consequently, this proves the Charneyconjecture for the 3D Planetary Geostrophic model; namely, that the history of thelarge spatial scales of temperature is sufficient for determining all the otherquantities (state variables) of the model.

This paper is dedicated to the memory of Professor Abbas Bahri

MSC Subject Classifications: 35Q30, 93C20, 37C50, 76B75, 34D06.Keywords: 3D Leray-α-model, subgrid scale turbulence models, continuousdata assimilation, downscaling, Charney conjecture, coarse measurements ofvelocity.

∗ Department of Mathematics, University of Virginia.† Department of Mathematics, United States Naval Academy.‡ Department of Mathematics, Texas A&M University and The Science Program, Texas A&M

University at Qatar.

253

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254 Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

6.1 Introduction

Data assimilation is a methodology to estimate weather or ocean variablesby combining (synchronizing) information from observational data with anumerical dynamical (forecast) model. In the context of control engineering,tracing back since the early 1970s, data assimilation was also applied tosimpler models in [59, 69, 77]. One of the classical methods of continuousdata assimilation, see e.g., [20, 37], is to insert observational measurementsdirectly into a model as the latter is being integrated in time. For example, onecan insert Fourier low mode observables into the evolution equation for thehigh modes, then the values for the low modes and high modes are combined toform a complete approximation of the state of the system. This resulting statevalue is then used as an initial condition to evolve the forecast model using highresolution simulation. For the 2D Navier–Stokes, this approach was consideredin [9, 10, 39, 41, 52, 58, 70, 72]. The problem when some state variableobservations are not available as an input to the numerical forecast model wasstudied in [13, 21, 37, 38, 42, 61] for simplified numerical forecast models.

Recent studies in [23, 24, 25, 26, 27] have established rigorous analyticalsupport pertaining to a continuous data assimilation algorithm similar to thatintroduced in [4], which is a feedback control algorithm applied to dataassimilation (see also [3, 64]), previously known as nudging or Newtonianrelaxation. In these cases, the authors have analyzed a data assimilationalgorithm for different systems when some of the state variable observationsare not supplied in the algorithm. In other words, a rigorous analytical supportto a data assimilation algorithm that can identify the full state of the systemknowing only coarse spatial mesh observational measurements not of full statevariables of the model, but only of the selected state variables in the system,were analytically justified. In this article we summarize our recent results toprovide support to the applicability of the scheme in several model equations.We then demonstrate the application of this algorithm to the Leray-α subgridscale turbulence model which serves as an addendum to our recent work onabridged continuous data assimilation for the 2D Navier–Stokes equations.

Starting from the work of [4], the recent works [23, 24, 25, 26, 27], men-tioned above provided some stepping stones to rigorous justification to some ofthe earlier conjectures of meteorologists in numerical weather prediction. Forexample, the systematic theoretical framework of the proposed global controlscheme allowed the authors in [4] and [23] to provide sufficient conditions onthe spatial resolution of the collected spatial coarse mesh data and the relax-ation parameter that guarantees that the approximating solution obtained fromthis algorithm converges to the corresponding unknown reference solution

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Abridged Data Assimilation for 3D α Models 255

over time (with the assumption that the observational data measurementsare free of noise). Without access to concrete theoretical analysis, earlierimplementation of the “nudging” algorithm left geophysicists searching forthe optimal or suitable nudging coefficient (or relaxation parameter) throughexpensive numerical experiments. Naturally, one wishes for the availabilityof a sharper estimate for the operational characteristic parameters than whatthe theoretical results give. However, these recent analytical results, althoughnot sharp, may allow one to track the correct parameter ranges without theexpensive numerical experiments. Computational studies implementing thesealgorithms under drastically more relaxed conditions were demonstrated in[2, 36] for example.

To understand the value of the development of these analytical resultsstemming from the series of studies, we should mention its valuable impactin meteorology. In weather prediction, we’ve mentioned earlier the need toanalyze the success of a data assimilation algorithm when some state variableobservations are not available. Charney’s question in [13, 37, 38] of whethertemperature observations are enough to determine all the dynamical state ofthe system is an important problem in meteorology and engineering. In [38],an analytical argument suggested that the Charney’s conjecture is correct,in particular, for the shallow water model. The authors of [38] derived adiagnostic system for the velocity field that gives the velocity in terms offirst- and second-time derivatives of the mass field (the geopotential of thefree surface of the fluid). A mathematical argument was then presented tojustify that the mass field and its first- and second-time derivatives determinethe velocity field fully. Similar diagnostic systems can be derived for othersimple primitive equation models. The work in [38] gave a precise theoreticalformulation of the Charney conjecture for certain simple atmospheric models.

Numerical tests in [38] suggested that in practice, it can be hard toimplement this method to solve for the velocity field using only measurementson the mass field. Further numerical testing in [37] affirmed that it is notcertain whether assimilation with temperature data alone will yield initialstates of arbitrary accuracy. The authors of [37] considered the primitiveequations (the main weather forecast model) and tested and compared differenttime-continuous data assimilation methods using temperature data alone.In their numerical experiments, they concluded that the accuracy of theassimilation is highly dependent on the assimilation method used and on theintegrity of the measured observational temperature data. A relevant recentnumerical study on a data assimilation algorithm for the 2D Benard system[2], inspired by the work in [23], showed that it is sufficient to use coarsevelocity measurements in the algorithm to recover the full true state of the

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256 Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

system. On the other hand, it was concluded in [2] that data assimilation usingcoarse temperature measurements only will not always recover the true stateof the system. It was observed that the convergence to the true state usingtemperature measurements only is actually sensitive to the amount of noise inthe measured data as well as to the spacing (the sparsity of the collected data)and the time-frequency of such measured temperature data. These results in[2] are consistent with the results of the earlier numerical experiments in [38]and [37].

These results may indicate that the answer to Charney’s question is negativefor the practical issues we have with our measuring equipments or our numer-ical solving techniques. More recently, in [27], we proposed an improvedcontinuous data assimilation algorithm for the 3D Planetary Geostrophicmodel that requires observations of the temperature only. We provided arigorous mathematical argument that temperature history of the atmospherewill determine other state variables for this specific planetary scale model,thus justifying theoretically Charney’s conjecture for this model. Numericalimplementation of our algorithm for the 3D Planetary Geostrophic model issubject to new work to compare with the numerical results obtained in [2],[37] and [38].

We will review relevant results starting from the algorithm introduced in[4]. The algorithm in [4] can be formally described as follows: suppose thatu(t) represents a solution of some dynamical system governed by an evolutionequation of the type

du

dt= F(u), (6.1)

where the initial data u(0)= uin is missing. Let Ih(u(t)) represent an interpolantoperator based on the observations of the system at a coarse spatial resolutionof size h, for t ∈ [0,T]. The algorithm proposed in [4] is to construct a solutionv(t) from the observations that satisfies the equations

dv

dt= F(v)−μ(Ih(v)− Ih(u)), (6.2a)

v(0)= vin, (6.2b)

where μ> 0 is a relaxation (nudging) parameter and vin is taken to be arbitraryinitial data. As mentioned in the previous literature, this particular algorithmwas designed to work for general dissipative dynamical systems of the form(6.1) that are known to have global, in time, solutions, a finite-dimensionalglobal attractor and a finite set of determining parameters (see, e.g., [18, 30,32, 33, 34, 35, 44, 47, 48] and references therein). Lower bounds on μ > 0

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Abridged Data Assimilation for 3D α Models 257

and upper bounds on h > 0 can be derived such that the approximate solutionv(t) converges in time to the reference solution u(t). These estimates are notsharp (see the numerical results in [2] and [36]) since their derivation usesthe existing estimates for the global solution in the global attractor of thesedissipative systems.

In the context of the incompressible 2D Navier–Stokes equations (NSE),the authors of [4] studied the conditions under which the approximate solu-tion v(t) obtained by the algorithm in (6.2) converges to the referencesolution u(t) over time (see also [36]). In [1], it was shown that approx-imate solutions constructed using observations on all three components ofthe unfiltered velocity field converge in time to the reference solution ofthe 3D Navier–Stokes-α model. Another application of this algorithm for thethree-dimensional Brinkman–Forchheimer–Extended Darcy model was intro-duced in [65]. The authors of [46] studied the convergence of the algorithmto the reference solution in the case of the two-dimensional subcritical surfacequasi-geostrophic (SQG) equation. The convergence of this synchronizationalgorithm for the 2D NSE, in higher-order (Gevery class) norm and in L∞

norm, was later studied in [8]. An extension of the approach in [4] to the casewhen the observational data contains stochastic noise was analyzed in [7]. Astudy of the the algorithm for the 2D NSE when the measurements are obtaineddiscretely in time and are contaminated by systematic error is presented in [31].More recently in [68], the authors obtain uniform in time estimates for the errorbetween the numerical approximation given by the Post-Processing Galerkinmethod of the downscaling algorithm and the reference solution, for the 2DNSE. Notably, this uniform in time error estimates provide strong evidence forthe practical reliable numerical implementation of this algorithm.

In [23], a continuous data assimilation scheme for the two-dimensionalincompressible Benard convection problem was introduced. The data assim-ilation algorithm in [23] constructed the approximate solutions for the velocityu and temperature fluctuations using only the observational data of the velocityfield and without any measurements for the temperature fluctuations. In [24],we introduced an abridged dynamic continuous data assimilation for the 2DNSE inspired by the recent algorithms introduced in [4, 23]. There we establishconvergence results for the improved algorithm where the observational dataneeding to be measured and inserted into the model equation are reduced orsubsampled. Our algorithm required observational measurements of only onecomponent of the velocity vector field. The underlying analysis was madefeasible by taking advantage of the divergence-free condition on the velocityfield. Our work in [24] was then applied and extended for the convergenceanalysis for a 2D Benard convection problem, where the approximate solutions

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258 Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

constructed using observations in only the horizontal component of thetwo-dimensional velocity field and without any measurements on the temper-ature converge in time to the reference solution of the 2D Benard system. Thiswas a progression from the recent result in [23] where convergence results wereestablished, given that observations are known on all of the components of thevelocity field and without any measurements of the temperature. In [25] wepropose that a data assimilation algorithm based on temperature measurementsalone can be designed for the Benard convection in a porous medium. In thiswork it was established that requiring a sufficient amount of coarse spatialobservational measurements of only the temperature measurements as inputis able to recover the full state of the system. Subsequently, in [27] weproposed an improved data assimilation algorithm for recovering the exactfull reference solution (velocity and temperature field) of the 3D PlanetaryGeostrophic model, at an exponential rate in time, by employing coarse spatialmesh observations of the temperature alone. In particular, we presented arigorous justification of an earlier conjecture of Charney which states that thetemperature history of the atmosphere, for certain simple atmospheric models,determines all the other state variables.

6.2 Application to Turbulence Models

All the analysis of the proposed data assimilation algorithm assumes the globalexistence of the underlying model and uses previously-known estimates. It isfor this reason that we are not able to prove similar results for the 3D NSE case,even though numerical testing may be applicable and feasible. Note, however,that we are able to formulate the analytical setting for a family of globallywell-posed subgrid scale turbulence models belonging to a family calledα-models of turbulence. These are simplified models through an averagingprocess that is designed to capture the large scale dynamics of the flow andat the same time provide a reliable closure model for the averaged equations.The first member of the family was introduced in the late 1990s and calledthe Navier–Stokes-α (NS-α) model (also known as the Lagrangian averagedNavier–Stokes-α (LANS-α) or viscous Camassa–Holm equations [14, 15, 16,28, 29]). It is written as follows:

∂tv+ u · ∇v+∇u · v+∇p= ν�v+ f , (6.3a)

∇ · u= 0, and v = u−α2�u. (6.3b)

Unlike other subgrid closure models which normally add some additionaldissipative process, this new modeling approach regularizes the NSE by

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Abridged Data Assimilation for 3D α Models 259

Table 6.1. Some special cases of the model (6.4) with α > 0, and withS = (I−α2�)−1 and Sθ2 = [I + (−α2�)θ2 ]−1.

Model NSE Leray-α ML-α SBM NSV NS-α NS-α-like

A −ν� −ν� −ν� −ν� −ν�S −ν� ν(−�)θ

M I S I S S S Sθ2

N I I S S S I Iχ 0 0 0 0 0 1 1

restructuring the distribution of the energy in the wave number k > 1/α ofthe inertial range [28]. In other words, NS-α smooths the nonlinearity ofthe NSE, instead of enhancing dissipation. Many other α-models, such asthe Leray-α [17], the Clark-α [11], the Navier–Stokes–Voigt (NSV) equation[49, 50, 73, 74], and the models we have introduced, namely, the modifiedLeray-α (ML-α) [45], the simplified Bardina model (SBM) [12, 55], and theNS-α-like models [71], were inspired by this regularization technique. Thesemodels can be represented by a generalized model of the form

∂tu+Au+ (Mu · ∇)(Nu)+χ∇(Mu)T · (Nu)+∇p= f (x), (6.4a)

∇ · u= 0, (6.4b)

u(0,x)= uin(x), (6.4c)

where A, M, and N are bounded linear operators having certain mappingproperties, χ is either 1 or 0, θ controls the strength of the dissipation operatorA, and the two parameters which control the degree of smoothing in theoperators M and N, respectively, are θ1 and θ2. Table 6.1 summarizes certainα-models of turbulence.

All of the models just mentioned have global regular solutions and possessfewer degrees of freedom than the NSE. Moreover, mathematical analysisalso proved that the solutions to these models converge to the solution ofNSE in the limit as the filter width parameter α tends to zero. In addition,several of the α-models of turbulence have been tested against averagedempirical data collected from turbulent channels and pipes, for a wide rangeof Reynolds numbers (up to 17× 106) [14, 15, 16]. The successful analytical,empirical and computational aspects (see for example [28, 40, 62, 63] andreferences therein) of the alpha turbulence models have attracted numerousapplications, see for example [5] for application to the quasi-geostrophicequations, [51] for application to the Birkhoff–Rott approximation dynamicsof vortex sheets of the 2D Euler equations, and [60, 66, 67] for applications

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260 Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

to incompressible magnetohydrodynamic equations. See also [53, 54] for theα-regularization of the inviscid 3D Boussinesq equation. A unified analysisof an additional family of α-type regularized models, also called a generalfamily of regularized Navier–Stokes and MHD models on n-dimensionalsmooth compact Riemannian manifolds with or without boundary, with n≥ 2,is studied in [43]. For approximate deconvolution models of turbulence see[56, 57]. For other closure models see [6] and references therein.

The proposed algorithms in [23, 24, 26, 27, 25] sparked an idea that perhapsfor this α-model one can construct approximate solutions, using only obser-vations in the horizontal components and without any measurements on thevertical component of the velocity field, which converge in time to the refer-ence solution. This is indeed the case, made possible by taking advantage of thedivergence-free condition for the velocity field. Similar results can be claimedfor certain other α-models. In this paper, we apply the data assimilationalgorithm for the case of the 3D Leray-α model which we recall below [17]:

∂tv− ν�v+ (u · ∇)v =−∇p+ f , (6.5a)

∇ · u=∇ · v = 0, (6.5b)

v = u−α2�u, (6.5c)

v(0,x,y,z)= vin(x,y,z), (6.5d)

where u,v and p are periodic, with basic periodic box = [0,L]3 = T3.(6.5e)

The nonlinearity is advected by the smoother velocity field, and noticethat, consistent with all the other alpha models, the above system is theNavier–Stokes system of equations when α = 0, i.e. u= v.

Recall that Ih(ϕ) represents an interpolant operator based on the observa-tional measurements of the scalar function ϕ at a coarse spatial resolutionof size h. Given the viscosity ν, the proposed algorithm for reconstructingu(t) and v(t) from only the horizontal observational measurements, which arerepresented by means of the interpolant operators Ih(v1(t)) and Ih(v2(t)) fort ∈ [0,T], is given by the system

∂tv∗1 − ν�v∗1 + (u∗ · ∇)v∗1 =−∂xp∗ +μ

(Ih(v1)− Ih(v

∗1))+ f1, (6.6a)

∂tv∗2 − ν�v∗2 + (u∗ · ∇)v∗2 =−∂yp∗ +μ

(Ih(v2)− Ih(v

∗2))+ f2, (6.6b)

∂tv∗3 − ν�v∗3 + (u∗ · ∇)v∗3 =−∂xp∗ + f3, (6.6c)

∇ · u∗ = ∇ · v∗ = 0, (6.6d)

v∗ = u∗ −α2�u∗, (6.6e)

v∗(0,x,y,z)= v∗in(x,y,z), (6.6f)

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Abridged Data Assimilation for 3D α Models 261

supplemented with periodic boundary conditions, where μ is again a positiveparameter which relaxes (nudges) the coarse spatial scales of v∗ toward thoseof the observed data.

Consequently, v∗(t,x,y,z) is the approximating velocity field, withv∗(0,x,y,z) = v∗in(x,y,z) taken to be arbitrary. We note that any dataassimilation algorithm using two out of three components of the velocity fieldalso works. Here, observational data of the horizontal components Ih(v1(t))and Ih(v2(t)) were chosen as an example.

We will consider interpolant observables given by linear interpolant oper-ators Ih : H1() → L2(), which approximate the identity and satisfy theapproximation property

‖ϕ− Ih(ϕ)‖L2() ≤ γ0h‖ϕ‖H1() , (6.7)

for every ϕ in the Sobolev space H1(). One example of an interpolantobservable of this type is the orthogonal projection onto the low Fourier modeswith wave numbers k such that |k| ≤ 1/h. A more physical example is thevolume elements that were studied in [47]. A second type of linear interpolantoperators Ih : H2()→ L2(), which satisfy the approximation property

‖ϕ− Ih(ϕ)‖L2() ≤ γ1h‖ϕ‖H1()+ γ2h2 ‖ϕ‖H2() (6.8)

for every ϕ in the Sobolev space H2(), can be considered with this algorithm.An example of this type of interpolant observables is given by the measure-ments at a discrete set of nodal points in (see Appendix A in [4]). Thetreatment for the second type of interpolant is slightly more technical (see,e.g. [23, 24]), and thus we won’t consider it here for the sake of keeping thisnote concise.

We prove an analytic upper bound on the spatial resolution h of theobservational measurements and an analytic lower bound on the relaxationparameter μ that is needed in order for the proposed algorithm (6.6) to recoverthe reference solution of the 3D Leray-α system (6.5) that corresponds to thecoarse measurements. These bounds depend on physical parameters of thesystem, the Grashof number as an example. We remark that extensions ofalgorithm (6.6), for the cases of measurements with stochastic noise and of dis-crete spatio-temporal measurements with systematic error, can be establishedby combining the ideas we present here with the techniques reported in [7] and[31], respectively.

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262 Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

6.3 Preliminaries

We define F to be the set of divergence-free L-periodic trigonometric polyno-mial vector fields from R3 →R3, with spatial average zero over . We denoteby L2(), Ws,p(), and Hs() ≡ Ws,2() the usual Sobolev spaces in threedimensions, and we denote by H and V the closure of F in L2() and H1(),respectively.

We denote the dual of V by V′

and the Helmholz–Leray projector fromL2() onto H by Pσ . The Stokes operator A : V → V

′can now be expressed as

Au=−Pσ�u,

for each u,v ∈ V . We observe that in the periodic boundary condition caseA = −�. The linear operator A is self-adjoint and positive definite withcompact inverse A−1 : H → H. Thus, there exists a complete orthonormal setof eigenfunctions wi in H such that Awi = λiwi, where 0 < λi ≤ λi+1 for i ∈N.The domain of A will be written as D(A)= {u ∈ V : Au ∈H}.

We define the inner products on H and V respectively by

(u,v)=3∑

i=1

∫T3

uivi dx and ((u,v))=3∑

i,j=1

∫T3

∂jui∂jvi dx,

and their associated norms (u,u)1/2 = ‖u‖L2(), ((u,u))1/2 = ∥∥A1/2u∥∥

L2().

Note that ((·, ·)) is a norm due to the Poincare inequality

‖φ‖2L2()

≤ λ−11 ‖∇φ‖2

L2(), for all φ ∈ V , (6.9)

where λ1 is the smallest eigenvalue of the operator A in three dimensions,subject to periodic boundary conditions.

We use the following inner products in H1() and H2(), respectively

((u,v))H1() = λ1[(u,v)+α2(A1/2u,A1/2v)

]and

((u,v))H2() = λ21

[(u,v)+ 2α2(A1/2u,A1/2v)+α4(Au,Av)

].

The above inner products were used in [17] so that the norms in H1()

and H2() are dimensionally homogeneous. Using these definitions, one canobserve that

λ1 ‖v‖L2() ≤ ‖u‖H2() ≤ 2λ1 ‖v‖L2() , (6.10)

where v = u−α2�u.

Remark 6.1 We will use these notations indiscriminately for both scalars andvectors, which should not be a source of confusion.

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Abridged Data Assimilation for 3D α Models 263

Let Y be a Banach space. We denote by Lp([0,T];Y) the space of (Bochner)measurable functions t �→w(t), where w(t)∈ Y , for a.e. t ∈ [0,T], such that theintegral

∫ T0 ‖w(t)‖p

Y dt is finite.Hereafter, c denotes a universal dimensionless positive constant. Our esti-

mates for the nonlinear terms will involve the Sobolev inequality in threedimensions:

‖u‖L∞() ≤ cλ−1/41 ‖u‖H2() . (6.11)

Furthermore, inequality (6.7) implies that

‖w− Ih(w)‖2L2()

≤ c20h2

∥∥A1/2w∥∥2

L2(), (6.12)

for every w ∈ V , where c0 = γ0.Here, G denotes the the Grashof number in three dimensions

G= ‖f‖L2()

ν2λ3/41

. (6.13)

We recall that the 3D Leray-α model (6.5), subject to periodic boundaryconditions, is well-posed and possesses a finite-dimensional compact globalattractor.

Theorem 6.2 (Existence and uniqueness) [17] If vin ∈ V and f ∈ H, then,for any T > 0, the 3D Leray-α model (6.5) has a unique global strong solutionv(t,x,y,z)= (v1(t,x,y,z),v2(t,x,y,z),v3(t,x,y,z)) that satisfies

v ∈ C([0,T];V)∩L2([0,T];D(A)), anddv

dt∈ L2([0,T];H).

Moreover, the system admits a finite-dimensional global attractor A that iscompact in H.

The following bounds on solutions v of (6.5) can be proved using theestimates obtained in [17].

Proposition 6.3 [17] Let τ > 0 be arbitrary, and let G be the Grashof numbergiven in (6.13). Suppose that v is a solution of (6.5), then there exists a timet0 > 0 such that for all t≥ t0 we have

‖v(t)‖2L2()

≤ 2ν2λ−1/21 G2 (6.14a)

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264 Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

and ∫ t+τ

t‖∇v(s)‖2

L2()ds≤ 2(1+ τνλ

1/21 )νG2. (6.14b)

We also recall the following bound on the solutions v in the global attractorof (6.5) that was proved in [22]. This estimate improves the estimate in [17] onthe enstrophy

∥∥A1/2v∥∥2

L2().

Proposition 6.4 [22] Suppose that v is a solution in the global attractor of(6.5), then ∥∥A1/2v(t)

∥∥2

L2()≤ c

ν2G4

α4λ3/21

, (6.15)

for large t > 0, for some dimensionless constant c > 0.

6.4 Analysis of the Data Assimilation Algorithm

We will prove that under certain conditions on μ, the approximate solution(v∗1 ,v∗2 ,v∗3) of the data assimilation system (6.6) converges to the solution(v1,v2,v3) of the 3D Leray-α (6.5), as t →∞, when the observable operatorssatisfy (6.7).

Theorem 6.5 Suppose Ih satisfy (6.7) and μ > 0 and h > 0 are chosen suchthat μc2

0h2 ≤ ν, where c0 is the constant in (6.7). Let v(t,x,y,z) be a strongsolution of the Leray-α model (6.5) and choose μ> 0 large enough such that

μ≥ 2ccνG4

α4λ1, (6.16)

and h > 0 small enough such that μc20h2 ≤ ν, where the constants c, c, and c0

appear in (6.30), (6.15) and (6.12), respectively.If the initial data v∗in ∈ V and f ∈ H, then the continuous data assimi-

lation system (6.6) possesses a unique global strong solution v∗(t,x,y,z) =(v∗1(t,x,y,z),v∗2(t,x,y,z), v∗3(t,x,y,z)) that satisfies

v∗ ∈ C([0,T];V)∩L2([0,T];D(A)), anddv∗

dt∈ L2([0,T];H).

Moreover, the solution v∗(t,x,y,z) depends continuously on the initial data v∗inand it satisfies ∥∥v(t)− v∗(t)

∥∥L2()

→ 0,

at exponential rate, as t →∞, where v = (v1,v2,v3) is the solution of the 3DLeray-α (6.5), with observed data on v1 and v2 only.

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Abridged Data Assimilation for 3D α Models 265

Proof Define p= p−p∗, u= u−u∗, and v= v−v∗, thus v= u−α2�u. Thenv1, v2 and v3 satisfy the equations

∂ v1

∂t−�v1+ u1∂xv1+ u2∂yv1+ u3∂zv1+ (u∗ · ∇)v1+ ∂xp=−μIh(v1),

(6.17a)

∂ v2

∂t−�v2+ u1∂xv2+ u2∂yv2+ u3∂zv2+ (u∗ · ∇)v2+ ∂yp=−μIh(v2),

(6.17b)

∂ v3

∂t−�v3+ u1∂xv3+ u2∂yv3+ u3∂zv3+ (u∗ · ∇)v3+ ∂zp= 0, (6.17c)

∂xv1+ ∂yv2+ ∂zv3 = ∂xu1+ ∂yu2+ ∂zu3 = 0. (6.17d)

Since we assume that v is a reference solution of system (6.5), then itis enough to show the existence and uniqueness of the difference v. In theproof below, we will derive formal a priori bounds on v, under appropriateconditions on μ and h. These a priori estimates, together with the globalexistence and uniqueness of the solution v, form the key elements for showingthe global existence of the solution v∗ of system (6.6). The convergence ofthe approximate solution v∗ to the exact reference solution v will also beestablished under the tighter condition on the nudging parameter μ as stated in(6.16). Uniqueness can then be obtained using similar energy estimates.

The estimates we provide in this proof are formal, but can be justified by theGalerkin approximation procedure and then passing to the limit while usingthe relevant compactness theorems. We will omit the rigorous details of thisstandard procedure (see, e.g., [19, 75, 76]) and provide only the formal a prioriestimates.

Taking the L2()-inner product of (6.17a), (6.17b) and (6.17c) with v1, v2

and v3, respectively, we obtain

1

2

d

dt‖v1‖2

L2()+ ν

∥∥A1/2v1

∥∥2

L2()≤ |J1|− (∂xp, v1)−μ(Ih(v1), v1),

1

2

d

dt‖v2‖2

L2()+ ν

∥∥A1/2v2

∥∥2

L2()≤ |J2|− (∂yp, v2)−μ(Ih(v2), v2),

1

2

d

dt‖v3‖2

L2()+ ν

∥∥A1/2v1

∥∥2

L2()≤ |J3|− (∂zp, v3),

where

J1 := J1a+ J1b+ J1c := (u1∂xv1, v1)+ (u2∂yv1, v1)+ (u3∂zv1, v1),

J2 := J2a+ J2b+ J2c := (u1∂xv2, v2)+ (u2∂yv2, v2)+ (u3∂zv2, v2),

J3 := J3a+ J3b+ J3c := (u1∂xv3, v3)+ (u2∂yv3, v3)+ (u3∂zv3, v3).

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266 Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

By the Holder inequality, inequality (6.10) and the Sobolev inequality(6.11), we can show that

|J1a| = |(u1∂xv1, v1)| ≤ ‖∂xv1‖L2() ‖u1‖L∞() ‖v1‖L2()

≤ cλ−1/41 ‖∂xv1‖L2() ‖u1‖H2() ‖v1‖L2()

≤ cλ3/41 ‖∂xv1‖L2() ‖v1‖2

L2()

≤ cλ1/41 ‖∂xv1‖L2() ‖v1‖L2()

∥∥A1/2v1

∥∥L2()

≤ ν

8

∥∥A1/2v1

∥∥2

L2()+ cλ1/2

1

ν‖∂xv1‖2

L2()‖v1‖2

L2().

(6.18)

Using similar analysis to that used above, we obtain the following estimates:

|J1b| =∣∣(u2∂yv1, v1)

∣∣≤ ν

8

∥∥A1/2v2

∥∥2

L2()+ cλ1/2

1

ν

∥∥∂yv1

∥∥2L2()

‖v1‖2L2()

,

(6.19)

|J1c| = |(u3∂zv1, v1)| ≤ ν

20

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν‖∂zv1‖2

L2()‖v1‖2

L2(),

(6.20)

|J2a| = |(u1∂xv2, v2)| ≤ ν

8

∥∥A1/2v1

∥∥2

L2()+ cλ1/2

1

ν‖∂xv2‖2

L2()‖v2‖2

L2(),

(6.21)

|J2b| =∣∣(u2∂yv2, v2)

∣∣≤ ν

8

∥∥A1/2v2

∥∥2

L2()+ cλ1/2

1

ν

∥∥∂yv2

∥∥2L2()

‖v2‖2L2()

,

(6.22)

|J2c| = |(u3∂zv2, v2)| ≤ ν

20

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν‖∂zv2‖2

L2()‖v2‖2

L2(),

(6.23)

|J3a| = |(u1∂xv3, v3)| ≤ ν

20

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν‖∂xv3‖2

L2()‖v1‖2

L2(),

(6.24)

|J3b| =∣∣(u2∂yv3, v3)

∣∣≤ ν

20

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν

∥∥∂yv3

∥∥2L2()

‖v2‖2L2()

.

(6.25)

Next, using integration by parts and the divergence-free condition (6.17d),we obtain

J3c = (u3∂zv3, v3)=−(v3,∂z(u3v3))

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Abridged Data Assimilation for 3D α Models 267

=−(v3,∂zu3v3)− (v3, u3∂zv3)

= (v3,(∂xu1+ ∂yu2)v3)+ (v3, u3(∂xv1+ ∂yv2))

=: J3d + J3e.

Integration by parts once again implies

J3d = (v3,(∂xu1+ ∂yu2)v3)

=−(v3, u1∂xv3)− (v3, u2∂yv3)− (∂xv3, u1v3)− (∂yv3, u2v3)

=: J3d1+ J3d2+ J3d3+ J3d4

and

J3e = (v3, u3(∂xv1+ ∂yv2))

=−(v3,∂xu3v1)− (v3,∂yu3v2)− (∂xv3, u3v1)− (∂yv3, u3v2)

=: J3e1+ J3e2+ J3e3+ J3e4.

Using the Holder inequality, inequality (6.10) and the Sobolev inequality(6.11), we have

|J3d1| = |(v3, u1∂xv3)| ≤ ‖v3‖L2() ‖u1‖L∞() ‖∂xv3‖L2()

≤ cλ−1/41 ‖v3‖L2() ‖u1‖H2() ‖∂xv3‖L2()

≤ cλ3/41 ‖v3‖L2() ‖v1‖L2() ‖∂xv3‖L2()

≤ cλ1/41

∥∥A1/2v3

∥∥L2()

‖v1‖L2() ‖∂xv3‖L2()

≤ ν

20‖∂xv3‖2

L2()+ cλ1/2

1

ν

∥∥A1/2v3

∥∥2

L2()‖v1‖2

L2()

and similarly,

|J3d2| = |(v3, u2∂yv3)| ≤ ν

20

∥∥∂yv3

∥∥2L2()

+ cλ1/21

ν

∥∥A1/2v3

∥∥2

L2()‖v2‖2

L2().

By a similar argument as in (6.18), we can show that

|J3d3| = |(∂xv3, u1v3)| ≤ ν

20

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν‖∂xv3‖2

L2()‖v1‖2

L2()

and

|J3d4| =∣∣(∂yv3, u3v2)

∣∣≤ ν

20

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν

∥∥∂yv3

∥∥2L2()

‖v2‖2L2()

.

Thus,

|J3d| ≤ ν

5

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν

∥∥A1/2v3

∥∥2

L2()

(‖v1‖2

L2()+‖v2‖2

L2()

).

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268 Aseel Farhat, Evelyn Lunasin and Edriss S. Titi

We apply similar calculations to J3e and obtain

|J3e| ≤ ν

5

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν

∥∥A1/2v3

∥∥2

L2()

(‖v1‖2

L2()+‖v2‖2

L2()

).

This yields

|J3c| = |(u3∂zv3, v3)|

≤ 2ν

5

∥∥A1/2v3

∥∥2

L2()+ cλ1/2

1

ν

∥∥A1/2v3

∥∥2

L2()

(‖v1‖2

L2()+‖v2‖2

L2()

).

(6.26)

Young’s inequality and the assumption μc20h2 ≤ ν imply that

−μ(Ih(vi), vi)=−μ(Ih(vi)− vi, vi)−μ‖vi‖2L2()

≤μc0h‖vi‖L2()

∥∥A1/2vi

∥∥L2()

−μ‖vi‖2L2()

≤ ν

2

∥∥A1/2vi

∥∥2

L2()− μ

2‖vi‖2

L2(), i= 1,2. (6.27)

Also we note that

(∂xp, v1)+ (∂yp, v2)+ (∂zp, v3)= 0, (6.28)

due to integration by parts, the boundary conditions, and the divergence-freecondition (6.17d).

Combining all the bounds (6.18)–(6.28) and denoting ‖vH‖2L2()

:=‖v1‖2

L2()+‖v2‖2

L2(), we obtain

d

dt‖v‖2

L2()+ ν

2

∥∥A1/2v∥∥2

L2()≤(

cλ1/21

ν

∥∥A1/2v∥∥2

L2()−μ

)‖vH‖2

L2(),

or, using the Poincare inequality (6.9), we have

d

dt‖v‖2

L2()+ νλ1

2‖v‖2

L2()+β(t)‖vH‖2

L2()≤ 0, (6.29)

where

β(t) :=μ− cλ1/21

ν

∥∥A1/2v∥∥2

L2(). (6.30)

Since v(t) is a solution in the global attractor of (6.5), then∥∥A1/2v

∥∥2L2()

satisfies the bound (6.15) for t > t0, for some large enough t0 > 0. Now, theassumption (6.16) yields

d

dt‖v‖2

L2()+min

{νλ1

2,μ

2

}‖v‖2

L2()≤ 0,

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Abridged Data Assimilation for 3D α Models 269

for t > t0. By Gronwall’s lemma, we obtain∥∥v(t)− v∗(t)∥∥2

L2()= ‖v(t)‖L2()→ 0,

at an exponential rate, as t→∞.

Acknowledgements

We would like to thank Professor M. Ghil for the stimulating exchangeconcerning the Charney conjecture and for pointing out to us some of therelevant references. E.S.T. is thankful for the kind hospitality of ICERM,Brown University, where part of this work was completed. The work of A.F. issupported in part by the NSF grant DMS-1418911. The work of E.L. is sup-ported in part by the ONR grant N0001416WX01475, N0001416WX00796and HPC grant W81EF61205768. The work of E.S.T. is supported in partby the ONR grant N00014-15-1-2333 and the NSF grants DMS-1109640 andDMS-1109645.

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[57] W. Layton and L. Rebholz, Approximate Deconvolution Models of Turbulence:Analysis, Phenomenology and Numerical Analysis, Lecture Notes in Mathemat-ics, Springer, 2010.

[58] K. J. H Law, A. Shukla and A. M. Stuart, Analysis of the 3DVAR filter forthe partially observed Lorenz’63 model, Disc and Cont. Dyn. Sys. 34, (2014),1061–1078.

[59] D. Leunberger, An introduction to observers, IEEE. T. A. Control 16, (1971),596–602.

[60] J. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-alphamodels, J. Math. Phys. 48(6), (2007).

[61] A. Lorenc, W. Adams and J. Eyre, The treatment of humidity in ECMWF’sdata assimilation scheme, Atmospheric Water Vapor, Academic Press New York,(1980), 497–512.

[62] E. Lunasin, S. Kurien, M. Taylor and E. S. Titi, A study of the Navier–Stokes-αmodel for two-dimensional turbulence, Journal of Turbulence 8, (2007), 751–778.

[63] E. Lunasin, S. Kurien and E. S. Titi, Spectral scaling of the Leray-α modelfor two-dimensional turbulence, Journal of Physics A: Math. Theor. 41, (2008),344014.

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Abridged Data Assimilation for 3D α Models 273

[64] E. Lunasin and E. S. Titi, Finite determining parameters feedback control fordistributed nonlinear dissipative systems – a computational study, EvolutionEquations and Control Theory 6(4), (2017), 535–557.

[65] P. Markowich, E. S. Titi, and S. Trabelsi, Continuous data assimilation for thethree-dimensional Brinkman–Forchheimer–Extended Darcy model, Nonlinearity29(4), (2016), 1292–1328.

[66] P. D. Mininni, D. C. Montgomery and A. Pouquet, Numerical solutions of thethree-dimensional magnetohydrodynamic alpha-model, Phys. Rev. E 71, (2005),046304.

[67] P. D. Mininni, D. C. Montgomery and A. Pouquet, A numerical study of the alphamodel for two dimensional magnetohydrodynamic turbulent flows, Phys. Fluids17, (2005), 035112.

[68] C. Mondaini and E. S. Titi, Uniform in time error estimates for the postprocessingGalerkin method applied to a data assimilation algorithm, SIAM Journal onNumerical Analysis 56(1), (2018), 78–110.

[69] H. Nijmeijer, A dynamic control view of synchronization, Physica D 154, (2001),219–228.

[70] E. Olson and E. S. Titi, Determining modes for continuous data assimilation in2D turbulence, Journal of Statistical Physics 113(5–6), (2003), 799–840.

[71] E. Olson and E. S. Titi, Viscosity versus vorticity stretching: global well-posednessfor a family of Navier–Stokes-alpha-like models, Nonlinear Anal. 66(11), (2007)2427–2458.

[72] E. Olson and E. S. Titi, Determining modes and Grashoff number in 2D turbu-lence, Theoretical and Computation Fluid Dynamics 22(5), (2008), 327–339.

[73] A. P. Oskolkov, The uniqueness and solvability in the large of the boundary valueproblems for the equations of motion of aqueous solutions of polymers, Zap.Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 38, (1973), 98–136.

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∗ Aseel Farhat, Department of Mathematics, University of Virginia, Charlottesville, VA 22904,USA. [email protected].

† Evelyn Lunasin, Department of Mathematics, United States Naval Academy, Annapolis, MD,21402 USA. [email protected].

‡ Edriss S. Titi, Department of Mathematics, Texas A&M University, 3368 TAMU, CollegeStation, TX 77843-3368, USA. Also, The Science Program, Texas A&M University at Qatar,Doha, Qatar. [email protected]

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7

Critical Points at Infinity Methodsin CR Geometry

Najoua Gamara∗

Sub-Riemannian spaces are spaces whose metric structure may be viewed as aconstrained geometry, where motion is only possible along a given set ofdirections, changing from point to point. The simplest example of such spaces isgiven by the so-called Heisenberg group. The characteristic constrained motion ofsub-Riemannian spaces has numerous applications ranging from robotic control inengineering to neurobiology where it arises naturally in functional magneticresonance imaging (FMRI). It also arises naturally in other branches of puremathematics as Cauchy–Riemann geometry, complex hyperbolic spaces, and jetspaces. In this paper, we review the use of the relationship between Heisenberggeometry and Cauchy–Riemann (CR) geometry. More precisely, we focus on theresolution of the Yamabe Conjecture which was definitely solved by techniquesrelated to the theory of critical points at infinity. These techniques were firstintroduced by A. Bahri and H. Brezis for the Yamabe conjecture in the Riemanniansettings. We also review the problem of the prescription of the scalar curvatureusing the same techniques which were studied first by A. Bahri and J. M. Coron aswell as the multiplicity of solutions. Finally, we announce in this direction newexistence results for Cauchy–Riemann spheres.

Key words: Yamabe Problem, CR manifold, Webster scalar curvature, Criticalpoints at infinity, Euler–Poincare Characteristic, Flatness condition.MSC[2000]: 58E05, 57R70, 53C21, 53C15.

7.1 Introduction

In 1995, Professor A. Bahri proposed to R. Yacoub and N. Gamara tosolve the remaining cases left open by D. Jerison and J.M. Lee of theCauchy–Riemann–Yamabe Conjecture [36, 37, 38]. In 1987, D. Jerison andJ.M. Lee formulated in [38] the CR Yamabe conjecture and developed

∗College of Science, Taibah University, Saudi Arabia and Faculty of Science, University Tunis ElManar, Tunisia.

274

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Critical Points at Infinity Methods in CR Geometry 275

the analogy between it and the Yamabe problem in conformal Riemanniangeometry, which had already been solved by T. Aubin [1] and R. Schoen [44].Besides the proof of T. Aubin and R. Schoen, another proof by A. Bahri [4],A. Bahri and H. Brezis [5] was available using methods related to the theoryof critical points at infinity. Based on his experience and his numerous worksin that direction [2, 3, 4, 5, 6, 7], A. Bahri was convinced that these topologicalmethods are well adapted to solve this conjecture in the Cauchy–Riemannsettings. This theory involves variational methods, algebraic topology andMorse theory. In 2001, R. Yacoub and N. Gamara [33] solved the sphericalcase of the CR Yamabe Conjecture and N. Gamara finalized the resolution ofthe CR Yamabe conjecture [24].

This paper has three purposes, first, we give a review of the Yamabeproblem [40, 49] on compact Cauchy–Riemannian manifolds and present itsextension to the problem of the prescription of the scalar curvature on compactCauchy–Riemannian manifolds without boundary. Multiplicity results forconformal contact forms admitting a given prescribed scalar curvature are alsoreviewed. Second, we announce recent improvements of the scalar curvatureprescription problem in the case of CR spheres.

To fix the notation in our context, let M be a real orientable 2n +1-dimensional manifold. A CR structure on M is given by a distinguishedn-dimensional complex subbundle of the complexified tangent bundle of M,T1,0 ⊂ CTM, called the holomorphic tangent bundle satisfying T1,0 ∩T1,0 = 0,where overbars denote complex conjugation and [T1,0,T1,0] ⊂ T1,0; this axiomis often referred to as the Frobenius integrability property of the CR structure.Standard examples of CR manifolds are those of real hypersurface type, in thiscase M is a hypersurface of Cn+1 and the CR structure defined on M is the oneinduced by the CR structure of the ambient space

T1,0(M)x =CTMx ∩T1,0(Cn+1)x, x ∈M.

T1,0(Cn+1) denotes the holomorphic tangent bundle over Cn+1: the span of

{ ∂

∂zj , 1≤ j≤ n+ 1}, where (z1, . . . .zn+1) are the cartesian complex coordinates

on Cn+1.The Levy distribution of the CR manifold M is H = Re(T1,0 + T1,0). Since

M is orientable H is oriented by its complex structure( J : H →H, J(v+ v)=i(v− v)) and H⊥ has a global non-vanishing section θ . The Levy form of θ isthe non-degenerate hermitian form defined by

Lθ (X, Y)=−2idθ(X∧Y)X , Y ∈ T1,0.

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276 N. Gamara

If Lθ is positive definite, M is said to be strictly pseudoconvex and θ definesa contact structure on M; θ ∧ dθn is a volume form on M.

A pseudohermitian structure on M is a CR structure together with a contactform θ . There is a unique globally defined tangent vector field T transverse tothe Levy distribution, determined by

θ(T)= 1, T�dθ = 0

(where � means interior product, so this equation means dθ(T ,X) = 0 for allvector fields X on M). T is referred to as the Reeb vector field of (M,θ).For a strictly pseudoconvex CR manifold the Reeb vector field extends theLevy form Lθ to a Riemannian metric gθ on M : the Webster metric on (M,θ),given by

gθ (X,Y)= Lθ (X,Y), gθ (X,T)= 0, gθ (T ,T)= 1, X,Y ∈H.

The horizontal gradient is given by ∇Hu=∏H∇u, where

∏H∇ : T(M)→

H is the projection associated with the natural sum decomposition T(M)=H⊕RT and the gradient of u, ∇u, is given by gθ (∇u,Y)= Y(u) for any vector fieldY on M. If div denote the divergence operator associated with the volume formθ ∧ dθn, we define a natural self adjoint, positive, second-order differentialoperator

*bu=−div(∇Hu), u ∈ C2(M),

called the sub-Laplacian of (M,θ). If {Wα}, α = 1, . . . ,n is any local frame forT1,0, the admissible coframe dual to Wα is the collection of (1,0) forms

{θα}, θα(Wβ)= δαβ ,θα(T)= θβ(Wα)= 0,

where α = α + n , and Wα = Wα . {T ,Wα ,Wβ} is a frame for CTM and its

admissible dual frame is {θ ,θα ,θβ}. dθ = ihαβθα ∧ θβ , Lθ (XαWα ,YβWβ) =

hαβXαYβ .If we denote again {Wα}, α = 1, . . . ,2n a local Lθ− orthonormal frame in

H, then the sub-Laplacian operator is locally given by

*bu=−2n∑α=1

(Wα(Wαu)−∇WαWα(u)).

More precisely, in a local coordinate (U,wi) system on M, for α = 1, . . . ,2n,we have

Wα = aiα

∂wi, ai

α ∈ C∞(U,R), 1≤ i≤ 2n+ 1

and

*bu=−2n+1∑ij=1

∂wi

(αij ∂u

∂wj

)+

2n+1∑j=1

αj ∂u

∂wj,

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Critical Points at Infinity Methods in CR Geometry 277

with αij =∑2nα=1 ai

αajα and αj = ∂αij

∂wi + αik�jik, where �

jik are the Christoffel

symbols associated with the CR connection. Now, if we denote by W∗α the

formal adjoint of the vector field Wα given by

W∗αu=− ∂

∂wi(ai

αu)− aiα�

jiku, u ∈ C1

0(U,R)

and use the Hormander operator associated with the system of vector fieldsW = {Wα , α = 1, . . . ,2n} given by

HWu=2n∑α=1

WαW∗αu,

it is straightforward that locally, *b =HW , see [22].As an example, we consider the Heisenberg group Hn, it is the Lie group

whose underlying manifold is Cn×R with the following group law:

(z, t)∗ (z′, t′)= (z+ z′, t+ t′ + 2Im(zz′)) ∀ z,z′ ∈ Cn and t, t′ ∈ R,

where zz′ =∑1≤j≤n zjz′j. We define a norm in Hn by

|(z, t)|Hn = (|z|4+ t2)14 .

The complex vectors fields on Hn,

Zj = ∂

∂zj+ izj ∂

∂t,

∂zj= 1

2

(∂

∂xj− ∂

∂yj

), zj = xj+ iyj, 1≤ j≤ n,

are left invariant with respect to the group law and homogeneous with degree−1 with respect to the dilations

(z, t)→ (λz,λ2t),(λ ∈R).

The space (T1,0)(z,t) is spanned by Zj,(z,t), 1≤ j≤ n and gives a left invariantCR structure on Hn. The form

θ0 = dt+ 2∑

1≤j≤n

(xjdyj− yjdxj)

annihilates T1,0; we take it to be the contact form of the CR structure on Hn.The sub-laplacian operator on Hn associated with the contact form θ0 is

�b = Lθ0 =−1

2

j=n∑j=1

(ZjZj+ ZjZj).

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278 N. Gamara

7.2 The Yamabe Problem and Related Topics

The Yamabe problem goes back to Yamabe himself [49] who claimed in 1960to have a solution, but in 1968, N. Trudinger [46] discover an error in his proofand corrected Yamabe’s proof. T. Aubin [1] improved Trudinger’s result, usingvariational methods and Weyl’s tensor characteristics. In 1984, R. Schoen [44]solved the remaining cases using variational methods and the positive masstheorem. We have also to point out the work of J.M. Lee and T.H. Parker in[40], which is a detailed discussion on the Yamabe problem unifying the workof T. Aubin [1] with that of R. Schoen [44]. Besides the proof by Aubin andSchoen for the Riemannian Yamabe conjecture, another proof by A. Bahri [4],A. Bahri and H. Brezis [5] was available by techniques related to the theory ofcritical points at infinity.

7.2.1 The Riemannian Yamabe Problem: Overview

Given a compact Riemannian manifold (M,g) without boundary, of dimensionn ≥ 3, the Yamabe conjecture states that there is a metric g conformal to g

which has a constant scalar curvature Rg = λ. We write g= u4

n−2 g, u > 0. Weobtain the following transformation law for the scalar curvature of the metricsg and g:

Rg = u−n+2n−2 (cn�u+Ru).

Hence the Yamabe problem is equivalent to solving

cn�u+Ru= λun+2n−2 , u > 0.

Let p= 2nn−2 and L= cn�+R be the conformal Laplacian of (M,g).

The last equation can be rewritten as

Lu= λup−1, u > 0.

The Yamabe problem has a variational formulation with Euler functional

J(u)=∫

M uLudvg(∫M updvg

) 2p

.

Let u be a positive function in C∞(M) and a critical point of J, then λ= J(u).The infimum of the functional J,

λ(M)= inf{J(u)/u ∈ C∞(M), u > 0

},

is a conformal invariant called the Yamabe invariant of (M,g).

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Critical Points at Infinity Methods in CR Geometry 279

We have the following results.

Theorem 7.1 (Yamabe, Trudinger, Aubin) For any compact Riemannianmanifold (M,g) without boundary, we have

λ(M)≤ λ(Sn)= n(n− 1)ω2nn .

Theorem 7.2 (Yamabe, Trudinger, Aubin) The Yamabe problem can be solvedon any compact manifold M if λ(M) < λ(Sn).

Hence, Theorem 7.2 reduces the resolution of the Yamabe problem tothe estimate of the Yamabe invariant λ(M). In this way, T. Aubin provedthe conjecture in the cases where (M,g) is not a conformally flat compactRiemannian manifold of dimension n ≥ 6. All the remaining cases of theYamabe problem were solved by R. Schoen using the positive mass theorem.

7.2.2 CR Yamabe Problem

Let (M,θ) be a real compact orientable and integrable pseudohermitianmanifold of dimension 2n+ 1. We denote by L = Lθ = (2+ 2

n )�b + Rθ theconformal CR Laplacian on M, where �b is the sub-Laplacian operator and Rθ

the Webster scalar curvature associated with θ .The CR Yamabe conjecture states that there is a contact form θ on M, CR

conformal to θ , which has a constant Webster scalar curvature Rθ . We have

then to find θ in the form θ = u2n θ , where u is a positive function defined

on M. This problem is equivalent to solving the following partial differentialequation:

(P) :

{Lu = u1+ 2

n on M,u > 0.

This equation is a particular case of more general equations of the type:

Lu+Qu= u1+ 2n , u > 0 on M, Q ∈ L∞(M).

In [38], D. Jerison and J.M. Lee have extensively studied the CR Yamabeproblem and showed that there is a deep analogy between the CR Yamabeproblem and the Riemannian one. Their results can be formally compared tothe partial completion of the proof of the Riemannian Yamabe conjecture by

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280 N. Gamara

T. Aubin. In 1986, D. Jerison and J.M. Lee proved some properties of the CRYamabe invariant:

λ(M)= infu∈S2

1(M)

{Aθ (u)/Bθ (u)= 1} ,

where S21(M) is a Folland–Stein space, Aθ (u) =

∫M Lu u θ ∧ dθn; Bθ (u) =∫

M | u |2+ 2n θ ∧ dθn, and gave a necessary condition on it to have the existence

of solutions for the CR Yamabe problem.

Theorem 7.3 1. λ(M) depends on the CR structure on M, not on the choiceof θ .

2. λ(M) ≤ λ(S2n+1), where S2n+1 ⊂ Cn+1 is the unit sphere with its standardCR structure.

3. If λ(M) < λ(S2n+1), then equation (P) has a solution.

In 1987, D. Jerison and J.M. Lee proved the following result.

Theorem 7.4 Let M be a compact strictly pseudoconvex 2n+ 1-dimensionalCR manifold, n≥ 2, not locally CR equivalent to S2n+1, then λ(M) < λ(S2n+1).Hence, the CR Yamabe problem can be solved on M.

The remaining cases left open by D. Jerison and J.M. Lee should by analogybe solved by using some CR positive mass theorem. Unfortunately, such aCR version of the positive mass theorem did not exist at that time. Besidesthe proof of T. Aubin and R. Schoen of the Riemannian Yamabe conjecture,another proof by A. Bahri [4] and A. Bahri and H. Brezis [5] of the sameconjecture was available by techniques related to the theory of critical pointsat infinity. This proof is completely different in spirit as well in techniques anddetails from the proof of T. Aubin and R. Schoen. It does not require the useof any theory of minimal surfaces, nor the use of a CR positive mass theorem.It turns out that this proof can be carried to the CR framework.

The cases left open by D. Jerison and J.M. Lee have been the subject of twopapers.

1. In 2001, R. Yacoub and N. Gamara in [33] solved the CR Yamabe problemfor spherical CR manifolds.

2. In the same year, N. Gamara in [24] completed the resolution of the CRYamabe conjecture for all dimensions by solving the three-dimensionalsubcase of the non-conformally flat case.

The proofs of the results of [33] and [24] are based on a contradictionargument.

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Critical Points at Infinity Methods in CR Geometry 281

7.2.3 Critical Points at Infinity Method

We consider the subspace of S21(M), defined by

H ={

u ∈ S21(M)/

∫M|du|2θ θ ∧ dθn <∞,

∫M|u|2+ 2

n θ ∧ dθn <∞}

.

Let � = {u ∈H,s.t.‖u‖H = 1} , ‖u‖H =(∫

M((2 + 2n ) |du|2θ + Rθu2)θ ∧

dθn) 1

2, and let

�+ = {u ∈�,/u > 0} .For u ∈H, we define the CR Yamabe functional:

J(u)=∫

M((2+ 2n ) |du|2θ +Rθu2)θ ∧ dθn

(∫

M |u|2+2n θ ∧ dθn)

nn+1

.

If u is a critical point of J on �+, then J(u)n2 u is a solution of the Yamabe

equation (P).Let us recall that the standard solutions of the CR Yamabe equation on the

Heisenberg group Hn are obtained by left translations and dilations

(z, t)→ (λz,λ2t),(λ ∈R)

of the functions u(z, t)= K |w+ i|−n, w= t+ i |z|2 (z, t) ∈Hn, K ∈C.Since the injection S2

1(Hn)→ L2+ 2

n (Hn) is continuous but not compact, thefunctional J does not satisfy the Palais–Smale condition denoted by (PS). Moreprecisely, one can see that the standard solutions on Hn after superpositionare the good candidate sequences which violate (PS). Therefore, the classicalvariational theory based on compactness arguments does not apply in this case.

7.2.4 The Case of a Spherical CR Manifold

General SettingsLet (M,θ) be a compact spherical CR manifold, we show the existence of

a conformal factor u2na depending differentiably on a ∈ M, such that if θ is

replaced by u2na θ in a ball B(a,ρ), then (M, u

2na θ) is locally (Hn,θ0). We may

use in B(a,ρ) the usual multiplication of Hn and the standard solutions of theCR Yamabe problem, which we denote by δ(b,λ), where λ ∈ R. The functionδ(b,λ) satisfies {

Lθ0δ(b,λ)= δ(b,λ)1+ 2n on B(a,ρ),

b ∈ B(a,ρ).

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282 N. Gamara

We then define on M a family of “almost solutions”, which we denote byδ(a,λ). These functions are the solutions of

Lδ(a,λ)= δ′(a,λ)1+ 2n ,

where {δ′(a,λ)= ωauaδ(a,λ) on B(a,ρ),

δ′(a,λ)= 0 on Bc(a,ρ).

Here ωa is a cut-off function used to localize our function near the base point aas λ goes to infinity, we show that these “almost solutions” closely approximateat infinity the Yamabe solutions of the Heisenberg group∣∣∣δ(a,λ)− δ′(a,λ)

∣∣∣=O

(1

λn

),∣∣∣δ(a,λ)− δ′(a,λ)

∣∣∣C2=O

(1

λn

), when λ→∞.

Neighborhoods of Critical Points at InfinityFollowing A. Bahri (see [2, 4]), we set the following definitions and notations.

Definition 7.5 A critical point at infinity of J on �+ is a limit of a flow lineu(s) of the equation: ⎧⎨⎩

∂u

∂s=−∂J(u),

u(0)=u0.

We define neighborhoods of critical points at infinity of J as follows:

V(p,ε)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

u ∈�+ s.t. thereexistpconcentrationpoints a1, . . . ,ap inM and

p concentrationsλ1, . . . ,λp s.t.∥∥∥∥u− 1

p12 S

n2

∑i=pi=1 δ(ai,λi)

∥∥∥∥H

< ε

with λi >1ε, and for i = j

εij = (λiλj+ λj

λi+λiλjd2(ai,aj))

−n ≥ 1ε,

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭where d(x,y), if x and y are in a small ball of M of radius ρ is∥∥exp−1

x (y)∥∥

Hn expx is the CR exponential map for the point x and d(x,y)is equal to ρ

2 otherwise. (S is the Sobolev constant for the inclusion

S21(H

n)→ L2+ 2n (Hn)).

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Critical Points at Infinity Methods in CR Geometry 283

Because of the estimates of the “almost solutions” given above, we canreplace, in the analysis of the (PS) condition, the functions δ′ or δ by thefunctions δ. Hence, we will be able to characterize the sequences of functionswhich violate the Palais–Smale condition.

In fact, we prove that, if (uk) is a sequence of H satisfying ∂J(uk)→ 0 andJ(uk) is bounded, then (uk) has a weak limit u in H. Hence, if u is non-zero,we prove that u is a critical point of J. Since we will prove the CR Yamabeproblem using a contradiction argument, we suppose that equation (P) has nosolutions. Then we have the following characterization of the sequences failingthe (PS) condition.

Proposition 7.6 Let {uk} be a sequence such that ∂J(uk)→ 0 and J(uk) isbounded. Then there exist an integer p ∈N+, a sequence εk → 0 (εk > 0) andan extracted subsequence of (uk) such that uk

‖uk‖ ∈ V(p,εk).

This proposition was first introduced in the Riemannian settings in [5, 2];the proof follows from iterated blow-up around the concentration points. Forthe CR settings, a complete proof is given in [33].

The CR Yamabe Problem: Ideas about the ProofConsidering for p ∈N the formal barycentric sets:

B0(M)= ∅,

Bp(M)={

p∑1

αiδxi ,p∑

i=1

αi = 1,αi > 0, xi ∈M

},

where δxi is the Dirac mass at the point xi, and the following level sets of thefunctional J:

Wp ={

u ∈�+ / J(u) < (p+ 1)1n S

}.

We define a map fp(λ) from Bp(M) to �+ by

fp(λ)

(i=p∑i=1

αiδxi

)=

∑i=pi=1αiδ(xi,λi)

‖∑i=pi=1 αiδ(xi,λi)‖

.

The following theorem is proved in [33].

Theorem 7.7 1. For any integer p ≥ 1, there exists a real λp > 0, such thatfp(λ) sends Bp(M) in Wp, for any λ > λp.

2. There exists an integer p0 ≥ 1, such that for any integer p ≥ p0 and forany λ > λp0 , the map of pairs fp(λ) : (Bp(M),Bp−1(M))→ (Wp,Wp−1) ishomologically trivial, i.e.

fp∗(λ)= 0,

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284 N. Gamara

where

fp∗(λ) : H∗(Bp(M),Bp−1(M))→H∗(Wp,Wp−1)

and H∗(•) is the homology group with Z /2Z coefficients of •.

On the other hand, arguing by contradiction, we will assume that the weaklimit u of any (PS) sequences (uk) of H satisfying ∂J(uk) → 0 with J(uk)

bounded is zero; otherwise, our problem would be solved, since we wouldhave found a solution. Then, assuming that (uk) is non-negative, we provethat we can extract from (uk) a subsequence denoted again by (uk), such that

uk‖uk‖H

∈ V(p,εk) with εk > 0 and limk→∞ εk = 0.In this case, we proved that the pair (Wp,Wp−1) retracts by deformation on

the pair (Wp−1 ∪Ap,Wp−1), where Ap ⊂ V(p,ε). More precisely, we prove thatthe elements of Ap are of the form

∑i=pi=1αiδxi,λi + v, with v small in the norm

‖‖H . Therefore, the model . . .⊂ Bp−1(M)⊂ Bp(M)⊂ . . . can be compared viafp to . . .⊂Wp−1 ⊂Wp, and we proved that fp∗(λ) = 0 , for every p ∈N∗, whichis a contradiction with the result of Theorem 7.7 and therefore achieves theproof of the CR Yamabe problem in this case.

7.2.5 The CR 3-dimensional Non-spherical Case

In the paper [24], it is shown how the techniques of critical points at infinitycan settle the case of a strictly pseudoconvex CR manifold (M,θ) of dimension2n+1, without assuming that M is locally conformally flat. In fact, in [24] wefocus on the case n= 1, but the techniques apply to higher CR dimensions withno more assumptions. We just follow the sketch of the proof given for the casen= 1, introducing where required some modifications due to the dimension ofthe CR manifold.

Here, we will give some ideas about the proof for the CR Yamabe problemin the case of a generic 3-dimensional non-spherical CR manifold.

The proof of the result in this case is similar to the one given for the CRspherical case. It is obtained by using a contradiction argument.

We use the same techniques given by A. Bahri and H. Brezis in [5]. How-ever, in this case the study of “the almost solutions” δa is not straightforwardas in the CR spherical case, where we have locally a relation between theconformal Laplacians of M and Hn. Here, we have to use the Green’s functionassociated with L to derive a good asymptotic expansion of the Yamabefuntional J near the sets of its critical points at infinity. Finally, to computethe numerator and denominator of J, we used the approach of D. Jerison andJ.M. Lee who refined in [36] the notion of normal coordinates by constructingnew intrinsic CR normal coordinates for an abstract CR manifold. These

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Critical Points at Infinity Methods in CR Geometry 285

coordinates are called pseudohermitian normal coordinates. The notions andresults introduced and proved by D. Jerison and J.M. Lee are parallel, withdrastically different techniques to the ones introduced by J.M. Lee and T.Parker [40] for the Riemannian Yamabe problem. In pseudohermitian normalcoordinates, D. Jerison and J.M. Lee gave the Taylor series of θ and {θα}to high order at a point q ∈ M, in terms of the pseudohermitian curvatureand torsion. Since the problem is CR invariant they had to choose θ soas to simplify the curvature and the torsion at a base point q as much aspossible. Using the results of [36], we proved the following estimates inpseudohermitian normal coordinates near a base point q ∈M:

R=O(2), W = Z+O(3), W = Z+O(3), L=−2(ZZ+ZZ)+O(2),

Gq(z, t)= C(ρ−2(z, t))+A+O(ρ(z, t)), Gq > 0,

where O(m) is a homogenous polynomial in ρ of degree a least m. Using theseestimates and the topological method based on the theory of critical points atinfinity explained earlier, we derive the result in this case.

7.2.6 Kazdan–Warner-like Problems

For the Yamabe problem, we have to look for a new contact form ofthe Cauchy–Riemannian manifold, conformal to the basic one which hasa constant scalar curvature. In this section, we prescribe on a compactCauchy–Riemannian manifold without boundary positive functions with dif-ferent behavior and we explore the relationship between the critical points ofsuch functions and the existence of solutions for the problem.

For the Riemannian settings, the problem of prescribing the scalar curvatureis known as the Kazdan–Warner problem, and it has been studied by variousauthors for dimensions 2, 3 and 4 as well as in high dimensions. There aremany papers devoted to this problem as well as to the multiplicity of solutionsfor the related differential equation, we can mention [12, 13, 14, 15, 16, 34, 35,39, 41]. For the CR settings, see [42] and [23]. Here, we will merely refer to themost directly related literature, using the method based on the theory of criticalpoints at infinity. For the Riemannian settings, we refer to [3, 6, 8, 10, 11]and more recently [17]. Concerning the Cauchy–Riemann setting the pioneerarticle is due to N. Gamara [25] and for the recent ones, see [20, 21, 26, 27,30, 31, 32].

In this section, we will review the first work in this direction in CRmanifolds: the case of a compact spherical CR manifold (M,θ) of dimension3, without boundary, as displayed in [25]. Let K : M → R∗+ be a positive C2

function; our objective is to find suitable conditions on K for which there exists

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286 N. Gamara

a contact form θ conformal to θ such that K is its Webster scalar curvature:K = Rθ , θ = u2θ , where u : M → R is positive. The problem of the scalarcurvature is equivalent to the resolution of the partial differential equation

(PK) :

{Lu = Ku3 on M,

u > 0,

where L is the conformal Laplacian of the manifold M.Problem (PK) has a variational structure, with associated Euler functional:

J(u),u∈ S21(M). A solution u of (PK) is a critical point of J. As for the Yamabe

problem, the functional J fails to satisfy the Palais–Smale condition, that is,there exist non-compact sequences along which the functional is bounded andits gradient goes to zero. The failure of the (PS) condition has been analyzedfor the Riemannian case throughout the works of [2, 3, 6, 8, 10, 39, 41, 43, 45].For the CR case, a complete description of sequences failing to satisfy (PS) isgiven in [33].

Since this problem has been formulated, obstructions have to be pointedout. The main difficulty encountered when one tries to solve equations of thetype (PK) consists of the failure of the Palais–Smale condition, which leads tothe failure of classical existence mechanisms. We will use a gradient flow toovercome the non-compactness. Thinking of the sequences failing to satisfythe Palais–Smale condition as “critical points”, our objective was to try to findsuitable parameters, in order to complete a Morse lemma at infinity analogousto the one given for the Riemannian case. The Morse lemma is crucial to provethe existence of solution for equation (PK); more precisely, the method weused to prove the existence of solutions for problem (PK) is based on thework of A. Bahri [3, 6, 7]. This method involves a Morse lemma at infinity,which establishes, near the set of critical points at infinity of the functionalJ, a change of variables in the space (ai,αi,λi,v), 1 ≤ i ≤ p to (ai, αi, λi,V),(αi = αi), where V is a variable completely independent of ai and λi suchthat J(

∑αiδai,λi) behaves like J(

∑αiδai ,λi

)+‖V‖2. The Morse lemma relieson the construction of a suitable pseudogradient for the associated variationalproblem, which is based on the expansion of J and its gradient ∂J near infinity.We define also a pseudogradient for the V variable with the aim of making thisvariable disappear by setting ∂V

∂s = −νV , where ν is taken to be a very largeconstant. Then at s = 1, V(s) = exp(−νs)V(0) will be as small as we wish.This shows that, in order to define our deformation, we can work as if V waszero. The deformation will be extended immediately with the same propertiesto a neighborhood of zero in the V variable.

We prove that the Palais–Smale condition is satisfied along the decreasingflow lines of this pseudogradient, as long as these flow lines do not enter the

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Critical Points at Infinity Methods in CR Geometry 287

neighborhood of a finite number of critical points of K. This method allowsus to study the critical points at infinity of the variational problem, by com-puting their total index and comparing this total index to the Euler–Poincarecharacteristic of the space of variations. This procedure was extensively usedin earlier Riemannian work and has displayed the role of the Green’s functionin solving equations of the type (PK).

It is important to recall that for the case we review, we have a balance phe-nomenon between the self interactions and interactions between the functionsfailing to satisfy the Palais–Smale condition.

To state our results, we set up the following conditions and notation.Let G(a,) be a Green’s function for L at a∈M and Aa the value of the regular

part of G evaluated at a.We assume that K has only non-degenerate critical points ξ1,ξ2, . . . ,ξr such

that

−�θK(ξi)

3K(ξi)− 2Aξi = 0, i= 1, . . . ,r.

Assume that ξi, i = 1, . . . ,r1 are all the critical points of K with −�θK(ξi)3K(ξi)

−2Aξi > 0. Let τl = (i1, . . . , il) denote any l-tuple of (1, . . . ,r1), 1 ≤ l ≤ r1. Wedefine the following matrix M(τl)= (Mst) with

Mss =−�θK(ξs)

3K2(ξs)− 2

Aξs

K(ξs),

Mst =−2G(ξs,ξt)√K(ξs)K(ξt)

, for 1≤ s = t≤ l.

Assume that for any τl, 1≤ l≤ r1, M(τl) is non-degenerate. If we denote bykij the index of the critical point ξij with respect to K, i(τl)= 4l− 1−∑l

j=1 kij

is the index of the critical point at infinity τl. We obtain the following result.

Theorem 7.8

Ifr1∑

l=1

∑τl, M(τl)>0

(−1)i(τl) = 1,

then (PK) has a solution.

This result means that if the total contribution of the critical points at infinityto the topology of the level sets of the associated functional J is not trivial, thenwe have a solution for (PK).

To close this section, let us recall some results concerning multiplicityresults for problem (PK). The first paper in this direction is due to H. Chtioui,M. Ould Ahmedou and R. Yacoub. In [20] the authors generalized the resultof N. Gamara [25]; they addressed the case where the total sum given above

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288 N. Gamara

is equal to 1 but a partial sum is not equal to 1, and proved existence andmultiplicity results for (PK). More precisely, we have the following result.

Theorem 7.9 Let ρ(τl) denotes the least eigenvalue of the matrix M(τl) if thereexists a positive integer k ∈N such that:

1.∑

τl ,ρ(τl) > 0i(τl)≤ k− 1

(−1)i(τl) = 1,

2. ∀τl such that ρ(τl) > 0, i(τl) = k.

Then, there exists a solution ω to the problem (PK) such that

m(ω)≤ k,

where m(ω) denotes the Morse index of ω, defined as the dimension of thespace of negativity of the linearized operator L(δ) := Lθ (δ)− 3ω2δ.

Moreover, for generic K, if ∑τl ,ρ(τl) > 0

(−1)i(τl) = 1

and if we denote by S the set of all the solutions of (PK), a lower bound of Sis given by

#S ≥∣∣∣∣∣∣1−

∑τl ,ρ(τl) > 0

(−1)i(τl)

∣∣∣∣∣∣ .Recall that many mathematicians have worked on these problems in the

Riemannian settings as well in the CR settings, we can mention for example[6, 8, 9, 10, 11, 18, 19] and [21, 26, 27, 29, 47, 48, 30, 31, 32].

7.3 Curvature “Flatness Condition” on CR Spheres

In this section, we announce new existence results on Cauchy–Riemannspheres concerning the prescription of the scalar curvature [28]. Our aim is toprescribe on S2n+1, the unit sphere of Cn+1 endowed with its standard contactform θ1 a C2 positive function K satisfying the so called β-flatness condition.As we have seen in Section 7.2, this problem is equivalent to solving thefollowing partial differential equation:

(Pβ) :

{Lθ1 u= K u1+ 2

n on S2n+1,u > 0,

(7.1)

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Critical Points at Infinity Methods in CR Geometry 289

where Lθ1 is the conformal Laplacian of S2n+1, Lθ1 = (2+ 2n )�θ1 +Rθ1 , where

�θ1 =�S2n+1 and Rθ1 = n(n+1)2 are respectively the sub-Laplacian operator and

the Webster scalar curvature of (S2n+1,θ1).We will focus here on the case n= 1: the 3-dimensional CR sphere. To state

our results, we set up the following conditions and notation.Let G(a,) be a Green’s function for L at a ∈ S3.We denote by

K={(ξi)(1≤i≤r), such that∇K(ξi)= 0

}the set of all critical points of K. We say that K satisfies the β-flatnesscondition if for all ξi ∈K, there exist

2≤ β = β(ξi) < 4 and b1 = b1(ξi), b2 = b2(ξi), b0 = b0(ξi) ∈R∗

such that in some pseudohermitian normal coordinates system centered at ξi,we have

K(x)= K(ξi)+ b1|x1|β + b2|x2|β + b0|t| β2 +R(x), (7.2)

where2∑

k=1

bk+ κb0 = 0,2∑

k=1

bk+ κ′b0 = 0 with

κ =

∫H1|t| β2 1−||z|2− it|2∣∣∣1+|z|2− it

∣∣∣6 θ0∧ dθ0

∫H1|x1|β 1−||z|2− it|2∣∣∣1+|z|2− it

∣∣∣6 θ0∧ dθ0

, κ′ =

∫H1

|t| β2∣∣∣1+|z|2− it∣∣∣4 θ0∧ dθ0

∫H1

|x1|β∣∣∣1+|z|2− it∣∣∣4 θ0∧ dθ0

.

The function[β]∑p=0

∣∣∇pR(x)∣∣ ‖x‖−β−r

H1 = o(1) as x approaches ξi, ∇r denotes all

possible partial derivatives of order r and [β] is the integer part of β.Let

K1 :={ξi ∈K such that β = β(ξi)= 2 and b1+ b2+ κ

′b0 < 0

},

K2 :={ξi ∈K such that β = β(ξi) > 2 and b1+ b2+ κ

′b0 < 0

}.

The index of the function K at ξi ∈ K, denoted by m(ξi), is the number ofstrictly negative coefficients bk(ξi):

m(ξi)= #{

bk(ξi);bk(ξi) < 0}

.

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290 N. Gamara

For each p-tuple (ξi1 , . . . ,ξip)∈ (K1)p ( ξil = ξij if l = j), we associate the matrix

M(ξi1 , . . . ,ξip)= (Mst)1≤s,t≤p:

Mss=−∑2

k=1 bk+ κ ′b0

3K2(ξs),

Mst=−2G(ξs,ξt)

[K(ξs)K(ξt)]1/2, for s = t.

(7.3)

We say that K satisfies condition (C) if

for each p-tuple (ξi1 , . . . ,ξip) ∈ (K1)p the corresponding matrix (Mst)

is non degenerate. (7.4)

We define the sets

• K+1 :=⋃

p

{(ξi1 , . . . ,ξip) ∈ (K1)

p, &(ξi1 , . . . ,ξip) > 0}

and

• l+ :=max{p ∈N s.t ∃ (ξi1 , . . . ,ξip) ∈K+

1

}.

For (ξi1 , . . . ,ξip) ∈K+1 , let i(ξi1 , . . . ,ξip) := 4p− 1−∑p

j=1 m(ξij).The main result of this paper is the following.

Theorem 7.10 Let K be a C2 positive function on S3 satisfying the β- flatnesscondition and condition (C), if

∑ξ∈K2

(−1)3−m(ξ)+l+∑

p=1

∑(ξi1 ,.,ξip )∈K+1

(−1)i(ξi1 ,.,ξip ) = 1.

Then, there exists at least one solution of (Pβ).

In the present case, we have the presence of multiple blow-up points. Infact, looking at the possible formations of blow-up points, it comes out that theinteraction of two different bubbles given by < δai,λi ,δaj,λj >L, i = j dominatesthe self interaction < δai,λi ,δai,λi >L, in the case where 2 < β < 4, while in thecase where β = 2, we have a balance phenomenon, that is any interaction oftwo bubbles is of the same order with respect to the self interaction.

Problem (Pβ) has a nice variational structure, with associated Euler func-tional

J(u)=∫S2n+1 Lu u θ ∧ dθ

(∫S2n+1 K u2+ 2

n θ ∧ dθ)12

, u ∈ S21(S

2n+1).

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Critical Points at Infinity Methods in CR Geometry 291

As in the case reviewed in Section 7.2, the functional J fails to satisfy thePalais–Smale condition on the set �+ = {

u∈ S21(S

2n+1)/‖u‖= 1 u≥ 0}. Using

the CR equivalence F induced by the Cayley transform (see definition below)between S2n+1 minus a point and the Heisenberg group Hn, equation (Pβ) isequivalent up to an influent constant to

(PHn) :

{(2+ 2

n )�Hn u= K u1+ 2n on Hn ,

u > 0,

where �Hn is the sub-Laplacian of Hn and K = K ◦F−1.Next, we will introduce the Cayley transform.Let Bn+1 = {

z∈Cn+1 / |z|< 1}

be the unit ball in Cn+1 and Dn+1 ={(z,w)∈

Cn×C / Im(w) > |z|2} be the Siegel domain, where ∂Dn+1 ={(z,w) ∈ Cn×

C / Im(w)= |z|2}.

Definition 7.11 [22] The Cayley transform is the correspondence between theunit ball Bn+1 in Cn+1 and the Siegel domain Dn+1, given by

C(ζ )=( ζ

1+ ζn+1, i

1− ζn+1

1+ ζn+1

); ζ = (ζ

′,ζn+1) , 1+ ζn+1 = 0.

The Cayley transform gives a biholomorphism of Bn+1 onto the Siegeldomain Dn+1. Moreover, when restricted to the sphere minus a point, C givesa CR diffeomorphism:

C : S2n+1\(0, . . . ,0,−1)−→ ∂Dn+1.

Let us recall the CR diffeomorphism

f : Hn −→ ∂Dn+1,(z, t) �−→ f (z, t)= (z, t+ i|z|2) ,

with the obvious inverse f−1(z,w)= (z,Re(w)), z ∈ Cn, w ∈ C. We obtain theCR equivalence with this mapping:

F : S2n+1\(0, . . . ,0,−1) −→ Hn,ζ = (ζ1, . . . ,ζn+1) �−→ (z, t)= (

ζ11+ζn+1

, . . . , ζn1+ζn+1

, i 2Imζn+1|1+ζn+1|2

)with inverse

F−1 : Hn −→ S2n+1\(0, . . . ,0,−1)

(z, t) �−→ ζ = ( 2z11+|z|2−it

, . . . , 2zn1+|z|2−it

, i 1−|z|2+it1+|z|2−it

),

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292 N. Gamara

and choose the standard contact form of S2n+1 as

θ1 = in+1∑j=1

(ζjdζ j− ζ jdζj

).

Then, we have F∗(4(c−10 δ(0,1))

2n θ0)= θ1.

Let us differentiate and take into account that δ(0,1)(F(ζ )) = c0|1+ ζn+1|2;we obtain

dθ1 =(

dζn+1

1+ ζn+1+ dζn+1

1+ ζn+1

)∧ θ1+|1+ ζn+1|2F∗(dθ0)

and

θ1∧ dθn1 = |1+ ζn+1|2(n+1)F∗(θ0∧ dθn

0 ).

We introduce the following function for each (ζ0,λ) on S2n+1×]0,+∞[:ω(ζ0,λ)(ζ )= |1+ ζn+1|−nδ(F(ζ0),λ) ◦F(ζ ).

We have Lθ1ω(ζ0,λ) =ω1+ 2

n(ζ0,λ), i.e. ω(ζ0,λ) is a solution of the Yamabe problem on

S2n+1.We also have∫

S2n+1Lθ1ω(ζ0,λ) ω(ζ0,λ) θ1∧ dθn

1 =∫Hn

Lθ0δ(g0,λ) δ(g0,λ) θ0∧ dθn0 ,

and ∫S2n+1

|ω(ζ0,λ)|2+ 2n θ1∧ dθn

1 =∫Hn|δ(g0,λ)|2+ 2

n θ0∧ dθn0 ,

where g0 = F(ζ0), and g= F(ζ ).As a consequence, the variational formulation for (Pβ) is equivalent to the

variational formulation for (PHn). So, in this case, we don’t need to construct“almost solutions” to solve (Pβ), we go through the Heisenberg group usingdirectly the solutions of the Yamabe problem in Hn.

Acknowledgements

The author would like to express her deep gratitude to Professor Abbas Bahrifor his valuable research supervision during the planning and development ofthe resolution of the CR Yamabe conjecture. His willingness to give his time sogenerously has been very much appreciated. She also would like to thank theLaboratory of Non-Linear Analysis of the Mathematics Department, RutgersUniversity, New Jersey, USA, where most of this work was accomplished.

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[20] H. Chtioui, M. Ould Ahmedou and R. Yacoub: Existence and multiplicity resultsfor the prescribed Webster scalar curvature on 3 CR manifolds, J. Geo. Anal.,23(2), (2013), 878–894.

[21] H. Chtioui, M. Ould Ahmedou and R. Yacoub: Topological methods for theprescribed Webster scalar curvature problem on CR manifolds, Dif. Geo. andAppl. 28 (2010), 264–281.

[22] S. Dragomir and G. Tomassini: Differential Geometry and Analysis on CRManifolds, Progress in Mathematics, Volume 246.

[23] V. Felli and F. Uguzzoni: Some existence results for the Webster scalar curvatureproblem in presence of symmetry, Ann. Math. 183 (2004), 469–493.

[24] N. Gamara: The CR Yamabe conjecture in the case n = 1, J. Eur. Math. Soc. 3,(2001), 105–137.

[25] N. Gamara: The prescribed scalar curvature on a 3-dimensional CR manifold,Advanced Nonlinear Studies 2 (2002), 193–235.

[26] N. Gamara, H. Chtioui and K. Elmehdi: The Webster scalar curvature problem onthe three dimentional CR manifolds, Bull. Sci. Math. 131, (2007), 361–374.

[27] N. Gamara and S. El Jazi: The Webster scalar curvature revisited – the case ofthe three dimensional CR sphere, Calculus of Variations and Partial Differentialequations Vol. 42, no. 1–2 (2011), 107–136.

[28] N. Gamara and B. Hafassa: The β-Flatness Condition in CR Spheres, Adv.Nonlinear Stud. v.17, issue 1, (2017), 193–213.

[29] N. Gamara, H. Guemri and A. Amri: Optimal control in prescribing Websterscalar curvature on 3-dimensional pseudoHermitian manifold, Nonlin. Anal.TMA, 127 (2015), 235–262.

[30] N. Gamara and M. Riahi: Multiplicity results for the prescribed Webster scalarcurvature on the three CR sphere under “flatness condition”, Bull. Sci. Math.,136, (2012), 72–95.

[31] N. Gamara and M. Riahi: The impact of the flatness condition on the prescribedWebster scalar curvature, Arab. J. Math., King Fahd University, (2013) 2,381–392.

[32] N. Gamara and M. Riahi: The interplay between the CR flatness condition andexistence results for the prescribed Webster scalar curvature, Adv. Pure. App.Math., APAM, v. 3, issue 3, (2012), 281–291.

[33] N. Gamara and R. Yacoub: CR Yamabe conjecture: the conformally flat case, Pac.J. Math., vol. 201, no. 1 (2001), 121–175.

[34] Z. Han: Prescribing Gaussian curvature on S2, Duke. Math. J. 61 (1990), 679–703.[35] E. Hebey: Changements de metriques conformes sur la sphere, le probleme de

Nirenberg, Bull. Sci. Math. 114 (1990), 215–242.[36] D. Jerison and J.M. Lee: Extremals for the Sobolev inequality on the Heisenberg

group and the CR Yamabe problem, J. Amer. Math. Soc., 1, (1988), 1–13.[37] D. Jerison and J.M. Lee: Intrinsic CR normal coordinates and the CR Yamabe

problem, J. Diff. Geo., 29 (1989), 303–343.[38] D. Jerison and J.M. Lee: The Yamabe problem on CR manifolds, J. Diff. Geom.,

25 (1987), 167–197. MR 88i:58162, Zbl 661 32026.[39] J. Kazdan and F. Warner: Existence and conformal deformation of metrics with

prescribed Gaussian and scalar curvature, Ann. Math. (2) 101 (1975), 317–331.

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Critical Points at Infinity Methods in CR Geometry 295

[40] J.M. Lee and T.H. Parker: The Yamabe problem, Bull. Amer. Math. Soc. (NS) 17(1987),no.1, 37–91.

[41] Y.Y. Li: Prescribing scalar curvature on Sn and related problems, Part I, J. Diff.Equa. 120 (1995), 319–410.

[42] A. Malchiodi and F. Uguzzoni: A perturbation result for the Webster scalarcurvature problem on the CR sphere, J. Math. Pures. Appl., 81 (2002), 983–997.

[43] J. Sacks and K. Uhlenbeck: The existence of minimal immersions of 2-spheres,Ann. Math. 113 (1981), 1–24.

[44] R. Schoen: Conformal deformation of a metric to constant scalar curvature,J. Diff. Geo. 20 (1984), 479–495.

[45] M. Struwe: A global compactness result for elliptic boundary value problemsinvolving limiting nonlinearities, Math. Z. 187 (1984), 511–517.

[46] N.S. Trudinger: Remarks concerning the conformal deformation of Riemannianstructures on comapact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968),265–274.

[47] R. Yacoub: On the Scalar Curvature Equations in high dimension, Adv. Non-lin.Studies 2(4) (2002), 373–393.

[48] R. Yacoub: Existence results for the Prescribed Webster Scalar Curvature onhigher dimensional CR manifolds, Adv. Non-lin. Studies 13 (3) (2013), 625–661.

[49] H. Yamabe: On a deformation of Riemannian structures on compact manifolds,Osaka Math. J. 12 (1960), 21–37.

[email protected], [email protected]

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8

Some Simple Problems for theNext Generations

Alain Haraux∗

A list of open problems on global behavior in time of some evolution systems,mainly governed by partial differential equations, is given together with somebackground information explaining the context in which these problems appeared.The common characteristic of these problems is that they appeared a long time agoin the personal research of the author and received almost no answer till now, withthe exception of very partial results which are listed to help the readers’understanding of the difficulties involved.

AMS classification numbers: 35B15, 35B40, 35L10, 37L05, 37L15Keywords: Evolution equations, bounded solutions, compactness, oscillationtheory, almost periodicity, weak convergence, rate of decay.

8.1 Introduction

Will future generations go on studying mathematical problems? This in itselfis an open question, but the growing importance of computer applicationsin everyday life together with the fundamental intricacies of computer sci-ence, abstract mathematical logic and the developments of new mathematicalmethods makes the positive answer rather probable.

This text does not comply with the usual standards of mathematical papersfor two reasons: it is a survey paper in which no new result will be presentedand the results which we recall to motivate the open questions will be givenwithout proof.

It is not so easy to introduce an open question in a few lines. Giving thestatement of the question is not enough, we must also justify why we considerthe question important and explain why it could not be solved until now. Bothpoints are delicate because the importance of a problem is always questionableand the difficulty somehow disappears when the problem is solved.

∗Sorbonne Universites and Laboratoire Jacques-Louis Lions, Paris

296

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Some Simple Problems for the Next Generations 297

The questions presented here concern the theory of differential equationsand mostly the case of PDEs. They were encountered by the author during hisresearch and some of them are already 40 years old. They might be consideredpurely academic by some of our colleagues more concerned with real worldapplications, but they are selected, among a much wider range of openquestions, since their solution probably requires completely new approachesand will likely open the door towards a new mathematical landscape.

8.2 Compactness and Almost Periodicity

Throughout this section, the terms “maximal monotone operator” and “almostperiodic function” will be used without having been defined. Although bothterms are by now rather well known, the definitions and main properties ofthese objects will be found respectively in the reference texts [8, 2, 6, 29].To help the understanding of readers who are not experts in the field, we justpoint out that the concept of maximal monotone operator is a generalizationin the Hilbert space framework of the idea of a nondecreasing continuousfunction, sufficiently large to encompass all positive or skew-adjoint (possiblyunbounded) linear operators. On the other hand, the concept of an almostperiodic function is a topological extension for the finite sum of periodicfunctions with arbitrary periods. An almost periodic function does not enjoyany specific local property and can be “recognized” or identified only whenexamined on an infinite interval, a half-line or the whole real line.

One of my first fields of investigation was, in connection with the abstractoscillation theory, the relationship between (pre-)compactness and asymptoticalmost periodicity for the trajectories of an almost periodic contractive process.The case of autonomous processes (contraction semigroups on a metric space)had been studied earlier in the Hilbert space framework by Dafermos andSlemrod [15], the underlying idea being that on the omega-limit set of aprecompact trajectory, the semi-group becomes an isometry group. Then thesituation resembles the simpler case of the isometry group generated on aHilbert space H by the equation

u′ +Au(t)= 0

with

A∗ = −A,

for which almost periodicity of precompact trajectories was known alreadyfrom L. Amerio quite a while ago (the case of vibrating membranes and

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298 Alain Haraux

vibrating plates with fixed bounded edge are special cases of this generalresult).

Remark 8.1 The problem solved by Dafermos and Slemrod [15] gives ageneral answer to asymptotic behavior of mechanical systems subject to onlyinterior elastic forces and (exterior or interior) damping forces. When anexterior force is added, the problem under study becomes the adaptation of theresponse to the exterior source and can become very complicated; resonancephenomena may occur if the damping is incomplete or too weak.

As expected, the case of a nonautonomous process, associated with atime-dependent evolution equation of the form

u′ +A(t)u(t) 0 0

is not so good in general. In [17] (see also [18] for a related new almostperiodicity criterion) I established an almost periodicity result for precompacttrajectories of a periodic contraction process on a complete metric space, andin the same paper I exhibited a simple almost periodic (linear) isometry processon R2, generated by an equation of the form

u′ + c(t)Ju(t)= 0

with J a π2 -rotation around 0, for which no trajectory except 0 is almost

periodic.Actually, while writing my thesis dissertation, I was specifically interested

in the so-called “quasi-autonomous” problem, and I met the following generalquestion.

Problem 8.2 (1977) Let A be a maximal monotone operator on a real Hilbertspace H, let f : R−→H be almost periodic and let u be a solution of

u′ +Au(t) 0 f (t)

on [0,+∞) with a precompact range. Can we conclude that u is asymptoti-cally almost periodic?

After studying a lot of particular cases in which the answer is positive (A=Llinear, A a subdifferential ∂� and some operators of the form L+∂�), I provedin [19] that the answer is positive if H = RN with N ≤ 2. But the answer isunknown for general maximal monotone operators even if H =R3.

Remark 8.3 In [28] it is stated that the answer is positive for all N, butthere is a mistake in the proof, relying on a geometrical property which isnot valid in higher dimensions, more specifically in 3D the intersection of the

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Some Simple Problems for the Next Generations 299

(relative) interiors of two arbitrarily close isometric proper triangles can beempty. Therefore the argument from [19] cannot be used in the same way forN ≥ 3.

Remark 8.4 The problem is also open even when A∈C1(H,H), in which casethe monotonicity just means

∀u ∈H, ∀v ∈H, (A′(u),v)≥ 0.

Remark 8.5 The answer is positive if f is periodic, as a particular case of themain result of [17].

Since an almost periodic function has precompact range, studying the exis-tence of almost periodic solutions requires some criteria for precompactness ofbounded orbits. In the case of an evolution PDE, precompactness is classicallyderived from higher regularity theory. For parabolic equations the smoothingeffect provides some higher-order regularity for t > 0 for bounded semi-orbitsdefined on R+. In the hyperbolic case, although there is no smoothing effectin finite time, precompactness of orbits was derived by Amerio and Prouse[1] from higher regularity of the source and strong coercivity of the dampingoperator g in the case of the semilinear hyperbolic problem

utt −�u+ g(ut)= f (t,x) in R+ ×, u= 0 on R+ × ∂,

where is a bounded domain of RN . But this method does not apply evenin the simple case g(v) = cv3 for c > 0,N ≤ 3, a case where boundedness ofall trajectories is known. The following question makes sense even when thesource term is periodic in t and g is globally Lipschitz continuous.

Problem 8.6 (1978) Let be a bounded domain of RN and g a nonincreasingLipschitz function. We consider the semilinear hyperbolic problem

utt −�u+ g(ut)= f (t,x) in R+ ×, u= 0 on R+ × ∂.

We assume that f : R−→ L2() is continuous and periodic in t. Assuming

u ∈ Cb(R+,H1

0())∩C1b(R

+,L2()),

can we conclude that⋃t≥0

{(u(t, .),ut(t, .))} is precompact in H10()×L2()?

Remark 8.7 The answer is positive in the following extreme cases.

(1) If g = 0 (by Browder–Petryshyn’s theorem, there is a periodic solution,hence compact, and all the others are precompact by addition).

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300 Alain Haraux

(2) If g−1 is uniformly continuous, see [22], the result does not requireLipschitz continuity of g and applies for instance to g(v) = cv3 for c >

0,N ≤ 3.

It would be tempting to “interpolate”, but even the case g(v)= v+ and N= 1already seems to be nontrivial.

Remark 8.8 The same question is of course also relevant when f is almostperiodic, and the result of [22] is true in this more general context. Moreover,precompactness of bounded trajectories when g = 0 is also true when f isalmost periodic. This is related to a fundamental result of Amerio statingthat if the primitive of an almost periodic function: R −→ H is bounded, itis also almost periodic. More precisely, if H is a Hilbert space and A is a(possibly unbounded) skew-adjoint linear operator with compact resolvent, letus consider a bounded solution (on R with values in H) of the equation

U′ +AU = F,

where F : R −→ H is almost periodic. Then exp(tA)U := V is a boundedsolution of

V ′ = exp(tA)F

and, since exp(tA)ψ is almost periodic as well as exp(−tA)ψ for any ψ ∈H, bya density argument on generalized trigonometric polynomials, it is immediateto check that a function W : R −→ H is almost periodic if and only ifexp(tA)W : R −→ H is almost periodic. Then Amerio’s Theorem appliedto V gives the result, and this property applies in particular to the waveequation written as a system in the usual energy space. Then, starting from asolution bounded on R+, a classical translation-(weak)compactness argumentof Amerio gives a solution bounded on R of the same equation. We skip thedetails since this remark is mainly intended for experts in the field.

Remark 8.9 Historically, the work of Amerio and Prouse [1] was motivatedby the study of vibrations of a membrane under the action of elastic forces, anexterior oscillating force and a damping term of the form

g(ut)= c1ut+ c2|ut|ut.

For small velocities the viscous damping g1(ut) = c1ut is prevalent whilefor large velocities the quadratic hydrodynamical response g2(ut) = c2|ut|ut

prevails. When studying the autonomous damped oscillations, generally onlyg1(ut) = c1ut is significant and the hydrodynamical term is important for theso-called transient oscillations leading, after a time depending on the initial

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Some Simple Problems for the Next Generations 301

state, to a situation where the total energy is small enough for g1 to becomeprevalent.

8.3 Oscillation Theory

Apart from the almost periodicity of solutions, which provides a startingpoint to describe precisely the global time behavior of vibrating strings andmembranes with fixed edge, it is natural to try a description of sign changesof the solutions on some subset of the domain. Let us first consider the basicequation

u′′ +Au(t)= 0, (8.1)

where V is a real Hilbert space, A ∈ L(V ,V ′) is a symmetric, positive, coerciveoperator and there is a second real Hilbert space H for which V ↪→H =H′ ↪→V ′ where the imbedding on the left is compact. In this case it is well knownthat all solutions u∈C(R,V)∩C1(R,H) of (8.1) are almost periodic : R−→Vwith mean-value 0. Then for any form ζ ∈ V ′, the function g(t) := 〈ζ ,u(t)〉 is areal-valued continuous almost periodic function with mean-value 0. It is theneasy to show that either g≡ 0, or there exists M > 0 such that on each intervalJ with |J| ≥M, g takes both positive and negative values. We shall say that anumber M > 0 is a strong oscillation length for a numerical function g∈L1

loc(R)

if the following alternative holds: either g(t)= 0 almost everywhere, or for anyinterval J with |J| ≥M, we have

meas{t ∈ J, f (t) > 0}> 0 and meas{t ∈ J, f (t) < 0}> 0.

As a consequence of the previous argument, under the above conditions onH,V and A, for any solution u∈C(R,V)∩C1(R,H) of (8.1) and for any ζ ∈V ′,the function g(t) := 〈ζ ,u(t)〉 has some finite strong oscillation length M =M(u,ζ ).

In the papers [9, 10, 21, 26] the main objective was to obtain a strongoscillation length independent of the solution and the observation in variouscases, including nonlinear perturbations of equation (8.1). A basic example isthe vibrating string equation

utt − uxx+ g(t,u)= 0 in R× (0, l), u= 0 on R×{0, l}, (8.2)

where l > 0 and g(t, .) is an odd nondecreasing function of u for all t. Herethe function spaces are H = L2(0, l) and V = H1

0(0, l). Since any functionof V is continuous, a natural form ζ ∈ V ′ is the Dirac mass δx0 for somex0 ∈ (0, l). It turns out that 2l is a strong oscillation length independent of thesolution and the observation point x0, exactly as in the special case g= 0, the

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302 Alain Haraux

ordinary vibrating string. Since in this case all solutions are 2l-periodic withmean-value 0 functions with values in V , it is clear that 2l is a strong oscillationlength independent of the solution and the observation point x0. The slightlymore complicated g(t,u) = au with a > 0 is immediately more difficult sincethe general solution is no longer time-periodic, it is only almost periodic int. The time-periodicity is too unstable and for an almost periodic function,the determination of strong oscillation lengths is not easy in general, as wasexemplified in [26]. The oscillation result of [9, 10] is consequently not soimmediate even in the linear case. In the nonlinear case, it becomes even moreinteresting because the solutions are no longer known to be almost periodic.

In dimensions N ≥ 2, even the linear case becomes difficult. It has beenestablished in [26] that even for analytic solutions of the usual wave equationin a rectangle, there is no uniform pointwise oscillation length common to allsolutions at some points of the domain. One would imagine that it becomestrue if the point is replaced by an open subset of the domain, but apparentlynobody knows the answer to the following exceedingly simpler question.

Problem 8.10 (1985) Let = (0,2l)× (0,2l) ⊂ R2. We consider the linearwave equation

utt−�u= 0 in R×, u= 0 on R× ∂.

Given T > 0, can we find a solution u for which

∀(t,x) ∈ [0,T]× (0, l)× (0, l), u(t,x) > 0?

Or does this become impossible for T large enough?

Another simple-looking intriguing question concerns the pointwise oscil-lation of solutions to semilinear beam equations, since the solutions of thecorresponding linear problem oscillate at least as fast as those of the stringequation.

Problem 8.11 (1985) We consider the semilinear beam equation

utt+ uxxxx+ g(u)= 0 in R× (0,1), u= uxx = 0 on R×{0,1}with g odd and nonincreasing with respect to u. Is it possible for a solutionu(t, .) to remain positive at some point x0 on an arbitrarily long (possiblyunbounded) time interval?

Finally, let us mention a question on spatial oscillation of solutions toparabolic problems. Since the heat equation has a very strong smoothing effecton the data, and all solutions are analytic inside the domain for t > 0, it seemsnatural to think that they do no accumulate oscillations and, for instance in 1D,

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Some Simple Problems for the Next Generations 303

the zeroes of u(t, .) will be isolated for t > 0. A very general result of this type,valid for semilinear problems as well, has been proved by Angenent [3]. Butas soon as N ≥ 2, even the linear case is not quite understood. The answer tothe following question seems to be unknown.

Problem 8.12 (1997) Let ⊂RN be a bounded open domain and f ∈ C1(R).We consider the heat equation

ut−�u+ f (u)= 0 in R×, u= 0 on R× ∂.

For t > 0 , we consider

E = {x ∈, u(t,x) = 0}.Is it true that E has a finite number of connected components?

Remark 8.13 The solutions u of the elliptic problem

−�u+ f (u)= 0 in , u= 0 on ∂

are such that {x ∈ , u(x) = 0} has a finite number of connected componentsfor a large class of functions f , see e.g. [14]. Hence stationary solutions cannotprovide a counterexample.

Remark 8.14 Actually Problem 8.12 is open even for f = 0 and N = 2, themost difficult aspect being the behavior of the solution near the boundary.

8.4 A Semilinear String Equation

There are in the literature a lot of results on global behavior of solutions toHamiltonian equations in finite and infinite dimensions. Apart from Poincare’srecurrence theorem and the classical results of Liouville on quasi-periodiciltyfor most solutions of completely integrable finite-dimensional Hamiltonians,none of the recent results is easy and there is essentially nothing on PDEsexcept in 1D. Even the case of semilinear string equations is not at all wellunderstood. While looking for almost-periodic solutions (trying to generalizethe Rabinowitz theorem on nontrivial periodic solution) I realized that evenprecompactness of general solutions is unknown for the simplest semilinearstring equation in the usual energy space.

Problem 8.15 (1976) For the simple equation

utt − uxx+ u3 = 0 in R× (0,1), u= 0 on R×{0,1}the following simple-looking questions seem to be still open.

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304 Alain Haraux

Question 1. Are there solutions which converge weakly to 0 as time goes toinfinity?

Question 2. If (u(0, .),ut(0, .)) ∈ H2((0,1)) ∩H10((0,1))×H1

0((0,1)) := V ,does (u(t, .),ut(t, .)) remain bounded in V for all times?

Remark 8.16 To understand the difficulty of the problem, let us just mentionthat the equation

iut+|u|2u= 0 in R× (0,1), u= 0 on R×{0,1}has many solutions tending weakly to 0 and, although the calculations are lessobvious, the same thing probably happens to

utt + u3 = 0 in R× (0,1), u= 0 on R×{0,1}.Hence the problem appears as a competition between the “good” behaviorof the linear string equation and the bad behavior of the distributed ODEassociated with the cubic term.

Remark 8.17 If the answer to question 2 is negative, it means that, followingthe terminology of Bourgain [7], the cubic wave equation on an interval is aweakly turbulent system. Besides, weak convergence to 0 might correspondto an accumulation of steep spatial oscillations of weak amplitude, notcontradictory with the energy conservation of solutions.

Remark 8.18 In [11, 12, 13], the authors investigated the problem

utt − uxx+ u∫ l

0u2(t,x)dx= 0 in R× (0, l), u= 0 on R×{0, l}, (8.3)

which can be viewed as a simplified model to aid understanding of the aboveequation. In this case, there is no solution tending weakly to 0, and the answerto question 2 is positive. Interestingly enough, in this case the distributed ODEtakes the form utt + c2(t)u = 0, so that the solution has the form a(x)u1(t)+b(x)u2(t) and remains in a two-dimensional vector space! This precludes bothweak convergence to 0 and weak turbulence.

8.5 Rate of Decay for Damped Wave Equations

Let us consider the semilinear hyperbolic problem

utt −�u+ g(ut)= 0 in R+ ×, u= 0 on R+ × ∂,

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Some Simple Problems for the Next Generations 305

where is a bounded domain of RN and g is a nondecreasing function withg(0)= 0. Under some natural growth conditions on g, the initial value problemis well-posed and can be put in the framework of evolution equations generatedby a maximal monotone operator in the energy space

H10()×L2().

An immediate observation is the formal identity

d

dt[∫

(u2t +|∇u|2)dx] = −2

g(ut)utdx≤ 0,

showing that the energy of the solution is nonincreasing. When g(s) = cswith c > 0, one can prove the exponential decay of the energy by a simplecalculation involving a modified energy function

Eε(t)=∫

(u2t +|∇u|2)dx+ ε

uutdx.

The exponential decay is of course optimal since

d

dt[∫

(u2t +|∇u|2)dx] = −2

cu2t dx≥−2c

(u2t +|∇u|2)dx.

A similar calculation can be performed if 0 < c ≤ g′(s) ≤ C, and the resultis even still valid for g(s)= cs+ a|s|αs under a restriction on α > 0 dependingon the dimension.

More difficult, and somehow more interesting, is the case

g(s)= a|s|αs, a > 0, α > 0,

in which under a restriction relating α and N, various authors (see e.g. [30],[27] and the references therein) obtained the energy estimate∫

(u2t +|∇u|2)dx≤ C(1+ t)−

2α .

But now the energy identity only gives

d

dt[∫

(u2t +|∇u|2)dx] = −2

a|ut|α+2dx,

while to prove the optimality of the decay we would need something like

d

dt[∫

(u2t +|∇u|2)dx] ≥ −C(

u2t dx)1+ α

2 .

Unfortunately the norm of ut in Lα+2 cannot be controlled in terms of the L2

norm, even if strong restrictions on ut are known. If ut is known to be bounded

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306 Alain Haraux

in a strong norm, let us say an Lp norm with p large, we can derive a lowerestimate of the type

[∫

(u2t +|∇u|2)dx] ≥ δ(1+ t)−β

for some β > 2α

. But even p=∞ does not allow us to reach the right exponent.In 1994, using special Lyapunov functions only valid for N = 1, the author

(see [20]) showed that for all sufficiently regular nontrivial initial data, we havethe estimate ∫

(u2t +|∇u|2)dx≥ C(1+ t)−

3α .

In general, for N > 2 , some estimate of the form∫

(u2t +|∇u|2)dx≥ C(1+ t)−K

will be obtained if the initial data belong to D(−�)×H10() and α < 4

N−2 . Butwe shall have in all cases K > 4

αand K tends to infinity when α approaches the

value 4N−2 .

Remark 8.19 It is perfectly clear that none of the above partial results issatisfactory, since for analogous systems in finite dimensions, of the type

u′′ +Au+ g(u′)

with A symmetric, coercive, (g(v),v)≥ c|v|α+2 and |g(v)| ≤C|v|α+1, the exactasymptotic of any nontrivial solution is

|u′|2+|u|2 ∼ (1+ t)−2α .

Moreover, an optimality result of the decay estimate has been obtained in 1Dby J. Vancostenoble and P. Martinez [33] in the case of a boundary damping forwhich the same upper estimate holds. The difference is that inside the domain,an explicit formula gives a lot of information on the solution.

Problem 8.20 For the equation

utt−�u+ g(ut)= 0 in R+ ×, u= 0 on R+ × ∂

withg(s)= a|s|αs, a > 0,α > 0,

Question 1. can we find a solution u for which

|∫

(u2t +|∇u|2)dx∼ (1+ t)−

2α ?

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Some Simple Problems for the Next Generations 307

Question 2. can we find a solution u for which the above property is notsatisfied?

Remark 8.21 Both questions seem to be still open for any domain and anyα > 0.

Remark 8.22 One might ask why we are interested in arbitrary values of α >

0, since the basic mechanical model corresponds to the damping induced byair and given by g(ut) = c1ut + c2|ut|ut, in which case we may simplify byconsidering only one of the two terms. However, for “average” values of thevelocity a more accurate approximation would be c|ut|αut for some α ∈ (0,1).

8.6 The Resonance Problem for Damped WaveEquations with Source Term

To close this short list, we consider the semilinear hyperbolic problem withsource term

utt −�u+ g(ut)= f (t,x) in R+ ×, u= 0 on R+ × ∂,

where is a bounded domain of RN . We assume that the exterior force f (t,x)is bounded with values in L2(), In this case, all solutions U = (u,ut) arelocally bounded on (0,T) with values in the energy space H1

0()×L2(). Thequestion is what happens as t tends to infinity.

When g(s) behaves like a super-linear power |s|αs for large values of thevelocity, it follows from a method introduced by G. Prouse [32] and extendedsuccessively by many authors, among which are M. Biroli [4], [5] and theauthor of this survey in [25], that the energy of any weak solution remainsbounded for t large, under the restriction α(N − 2) ≤ 4. Then many attemptswere tried to avoid this growth assumption. Many partial results were obtainedunder additional conditions (f bounded in stronger norms, f anti-periodic,higher growths for N ≤ 2, see e.g. [23], [24], [16]). But the following basicquestion remains open.

Problem 8.23 Assume N ≥ 3,

g(s)= a|s|αs, a > 0, α >4

N− 2.

Is it still true that the energy of all solutions remains bounded for any exteriorforce f (t,x) bounded with values in L2()?

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308 Alain Haraux

Remark 8.24 One might ask why we are interested in arbitrary values ofα > 0. The same observation as in Remark 8.22 applies here. In addition, theexistence of 3D loudspeakers (which did not exist at the time of Amerio andProuse’s seminal paper) gives us another motivation to extend the study tohigher dimensions and large values of α, even if the problem with α > 2 looksfor the moment purely “academic”. We consider it important to understandwhat happens and a counterexample proving that overdamping may occur herewould change substantially our understanding of mechanical vibrations andenergy transfer phenomena.

Remark 8.25 The positive boundedness results require a weaker boundednesscondition on f ; it is sufficient that it belongs to a Stepanov space Sp(R,L2()

with p > 1. The first results in this direction were actually published by G.Prodi in 1956 (see [31]), so that the problem is about 60 years old.

References

[1] L. Amerio and G. Prouse, Uniqueness and almost-periodicity theorems for a nonlinear wave equation, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8,46 (1969) 1–8.

[2] L. Amerio and G. Prouse, Almost-periodic functions and functional equations,Van Nostrand Reinhold Co., New York–Toronto, Ont.–Melbourne 1971 viii+184pp.

[3] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew.Math. 390 (1988), 79–96.

[4] M. Biroli, Bounded or almost periodic solution of the non linear vibratingmembrane equation, Ricerche Mat. 22 (1973), 190–202.

[5] M. Biroli and A. Haraux, Asymptotic behavior for an almost periodic, stronglydissipative wave equation, J. Differential Equations 38 (1980), no. 3, 422–440.

[6] S. Bochner and J. Von Neumann, On compact solutions of operational-differentialequations, Ann. of Math. 2, 36 (1935), 255–291.

[7] Jean Bourgain, On the growth in time of higher Sobolev norms of smooth solutionsof Hamiltonian, Internat. Math. Res. Notices 6 (1996), 277–304.

[8] H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dansles espaces de Hilbert (in French), North-Holland Mathematics Studies, No. 5.,Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1973.vi+183 pp.

[9] T. Cazenave and A. Haraux, Proprietes oscillatoires des solutions de certainesequations des ondes semi-lineaires, C. R. Acad. Sci. Paris Ser. I Math. 298 (1984),no. 18, 449–452.

[10] T. Cazenave and A. Haraux, Oscillatory phenomena associated to semilinearwave equations in one spatial dimension, Trans. Amer. Math. Soc. 300 (1987),no. 1, 207–233.

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Some Simple Problems for the Next Generations 309

[11] T. Cazenave, A. Haraux and F.B. Weissler, Une equation des ondes completementintegrable avec non-linearite homogene de degre trois, C. R. Acad. Sci. Paris Ser.I Math. 313 (1991), no. 5, 237–241.

[12] T. Cazenave, A. Haraux and F.B. Weissler, A class of nonlinear, completelyintegrable abstract wave equations, J. Dynam. Differential Equations 5 (1993),no. 1, 129–154.

[13] T. Cazenave, A. Haraux and F.B. Weissler, Detailed asymptotics for a con-vex Hamiltonian system with two degrees of freedom., J. Dynam. DifferentialEquations 5 (1993), no. 1, 155–187.

[14] M. Comte, A. Haraux and P. Mironescu, Multiplicity and stability topics insemilinear parabolic equations, Differential Integral Equations 13 (2000), no.7–9, 801–811.

[15] C.M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contractionsemigroups, J. Functional Analysis 13 (1973), 97–106.

[16] T. Gallouet, Sur les injections entre espaces de Sobolev et espaces d’Orliczet application au comportement a l’infini pour des equations des ondessemi-lineaires (in French), Portugal. Math. 42 (1983/84), no. 1, 97–112 (1985).

[17] A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almostperiodic processes, J. Diff. Eq. 49(1983), no. 3, 473–483.

[18] A. Haraux, A simple almost-periodicity criterion and applications, J. Diff. Eq. 66(1987), no. 1, 51–61.

[19] A. Haraux, Asymptotic behavior for two-dimensional, quasi-autonomous, almostperiodic evolution equations, J. Diff. Eq. 66 (1987), no. 1, 62–70.

[20] A. Haraux, Lp estimates of solutions to some non-linear wave equations in onespace dimension, Int. J. Math. Modelling and Numerical Optimization 1 (2009),Nos 1–2, p. 146–154.

[21] A. Haraux, On the strong oscillatory behavior of all solutions to some secondorder evolution equations, Port. Math. 72 (2015), no. 2, 193–206.

[22] A. Haraux, Almost-periodic forcing for a wave equation with a nonlinear, localdamping term, Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), no. 3–4, 195–212.

[23] A. Haraux, Nonresonance for a strongly dissipative wave equation in higherdimensions, Manuscripta Math. 53 (1985), no. 1–2, 145–166.

[24] A. Haraux, Anti-periodic solutions of some nonlinear evolution equations,Manuscripta Math. 63 (1989), no. 4, 479–505.

[25] A. Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep. 3(1987), no. 1, i–xxiv and 1–281.

[26] A. Haraux and V. Komornik, Oscillations of anharmonic Fourier series and thewave equation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 57–77.

[27] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyper-bolic problems, Arch. Rational Mech. Anal. 100 (1988), no. 2, 191–206.

[28] Z. Hu and A.B. Mingarelli, Almost periodicity of solutions for almost periodicevolution equations, Differential Integral Equations 18 (2005), no. 4, 469–480.

[29] C. F. Muckenhoupt, Almost periodic functions and vibrating systems, Journal ofMathematical Physics 8 (1928–1929), 163–199.

[30] M. Nakao, Asymptotic stability of the bounded or almost periodic solution of thewave equation with nonlinear dissipative term, J. Math. Anal. Appl. 58 (1977),no. 2, 336–343.

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310 Alain Haraux

[31] G. Prodi, Soluzioni periodiche di equazioni a derivate parziali di tipo iperboliconon lineari (in Italian), Ann. Mat. Pura Appl. (4) 42 (1956), 25–49.

[32] G. Prouse, Soluzioni limitate dell’equazione delle onde non omogenea contermine dissipativo quadratico (in Italian), Ricerche Mat. 14 (1965), 41–48.

[33] J. Vancostenoble and P. Martinez, Optimality of energy estimates for the waveequation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim. 39(2000), no. 3, 776–797 (electronic).

∗UPMC Universite Paris 06, CNRS, UMR 7598. 4, place Jussieu 75005, Paris, [email protected]

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9

Clustering Phenomena for Linear Perturbationof the Yamabe EquationAngela Pistoia∗ and Giusi Vaira†

This paper is warmly dedicated to Professor Abbas Bahrion the occasion of his 60th birthday

Let (M,g) be a non-locally conformally flat compact Riemannian manifold withdimension N ≥ 7. We are interested in finding positive solutions to the linearperturbation of the Yamabe problem

−Lgu+ εu= uN+2N−2 in (M,g),

where the first eigenvalue of the conformal Laplacian −Lg is positive and ε is asmall positive parameter. We prove that for any point ξ0 ∈M which isnon-degenerate and non-vanishing minimum point of the Weyl’s tensor and for anyinteger k there exists a family of solutions developing k peaks collapsing at ξ0 as εgoes to zero. In particular, ξ0 is a non-isolated blow-up point.

Keywords: Yamabe problem, linear perturbation, blow-up points

AMS subject classification: 35J35, 35J60

9.1 Introduction

Let (M,g) be a smooth, compact Riemannian manifold of dimension N ≥ 3.The Yamabe problem consists of finding metrics of constant scalar curvaturein the conformal class of g. It is equivalent to finding a positive solution to theproblem

Lgu+ κuN+2N−2 = 0 in M, (9.1)

∗ Angela Pistoia, Dipartimento di Scienze di Base e Applicate per l’Ingegneria, SapienzaUniversita di Roma

† Giusi Vaira, Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Universitadi Roma

311

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312 Angela Pistoia and Giusi Vaira

for some constant κ . Here Lgu :=�gu− N−24(N−1)Rgu is the conformal Laplacian,

�g is the Laplace–Beltrami operator and Rg is the scalar curvature of themanifold .

In particular, if u solves (9.1), then the scalar curvature of the metric

g= u4

N−2 g is nothing but 4(N−1)N−2 κ . The Yamabe problem was completely solved

by Yamabe [25], Aubin [1], Trudinger [24] and Schoen [19] (see also theproof given by Bahri [2]). The solution is unique in the case of negative scalarcurvature and it is unique (up to a constant factor) in the case of zero scalarcurvature. The uniqueness is not true any more in the case of positive scalarcurvature. Indeed, Schoen [20] and Pollack in [15] exhibit examples where alarge number of high energy solutions with high Morse index exist. Thus it isnatural to ask if the set of solutions is compact or not, as was raised by Schoenin [21]. It is also useful to point out that in t