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이학박사학위논문

On the small data scatteringproblem for the fractional

Schrodinger equation with cubicHartree-type nonlinearity

(삼차하트리형분수차슈뢰딩어방정식의작은초깃값산란)

2018년 8월

서울대학교대학원

수리과학부

양창훈



(삼차하트리형분수차슈뢰딩어방정식의작은초깃값산란)

지도교수이상혁

이논문을이학박사학위논문으로제출함

2018년 4월

서울대학교대학원

수리과학부

양창훈양창훈의이학박사학위논문을인준함

2018년 6월

위 원 장 김 판 기 (인)

부위원장 이 상 혁 (인)

위 원 조 용 근 (인)

위 원 홍 영 훈 (인)

위 원 Sebastian Herr (인)



A dissertationsubmitted in partial fulfillment

of the requirements for the degree ofDoctor of Philosophy

to the faculty of the Graduate School ofSeoul National University

by

Changhun Yang

Dissertation Director : Professor Sanghyuk Lee

Department of Mathematical SciencesSeoul National University

August 2018

c© 2018 Changhun Yang

All rights reserved.

AbstractOn the small data scatteringproblem for the fractional


Changhun Yang

Department of Mathematical SciencesThe Graduate School

Seoul National University

We consider three topics of the initial value problem for the fractional Schrodingerequation with cubic Hartree type nonlinearity.

The first and the second ones are on the low regularity problem for well-posedness and scattering. We consider the mass-critical and mass-supercriticalequation and prove the small data scattering in the scaling invariant spaces withradial or angular regularity assumption. The results are shown to be optimal byproviding the ill-posedness result in opposite side. The main ingredients of proofare Strichartz estimates, Bilinear estimates and theory of U p,V p spaces.

The third one is on the modified scattering when the potential is coulomb typewhich is referred to as ”scattering critical” problem. We describe the behavior ofsmall solutions at infinity by deriving a suitable nonlinear asymptotic correctionto scattering. We prove the refined time decay estimates and weighted estimates.Using these we perform an asymptotic analysis in fourier side by exploiting thenull structure in the equation.

Key words: Fractional Schrodinger equation, Hartree-type nonlinearity, Low reg-ularity well-posedness, Scattering, Linear and Bilinear estimatesStudent Number: 2012-20249

i

Publications and Priori presentation

This thesis consists of joint works which have been previously published at:

1. Joint work with Sebastian Herr, Critical well-posedness and scattering re-sults for fractional Hartree-type equations, to appear in Differential and In-tegral Equations.

2. Joint work with Yonggeun Cho and Thoru Ozawa, Small data scatteringof Hartree type fractional Schrodinger equations in a scaling critical space,submitted.

3. Joint work with Yonggeun Cho and Gyeongha Hwang, On the modifiedscattering of 3-d Hartree type fractional Schrodinger equations with Coulombpotential, to appear in Advances in Differential Equations.

The material of this thesis has been previously presented at:

1. 5th East asian conference in Harmonic analysis and application. June, 11th,2017. Hangzhou, China

2. Topics in Harmonic Analysis - Intensive Lectures Series. November, 7th,2017. Seoul National University, Seoul, Korea

Contents

Abstract i

Publication and Priori presentation ii

1 Introduction 11.1 The fractional Schrodinger equations . . . . . . . . . . . . . . . . 11.2 Main theorem 1: Radial symmetry . . . . . . . . . . . . . . . . . 51.3 Main theorem 2: Angular regularity . . . . . . . . . . . . . . . . 61.4 Main theorem 3: Modified scattering . . . . . . . . . . . . . . . . 7

2 Preliminaries 112.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 U p, V p spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Auxiliary estimates . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Scattering for radial case 183.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Bilinear estimates for radial functions . . . . . . . . . . . . . . . 203.3 Proofs of the main results . . . . . . . . . . . . . . . . . . . . . . 28

4 Scattering with angular derivative 324.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . 34

iii

CONTENTS

5 Modified scattering 455.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Global well-posedness . . . . . . . . . . . . . . . . . . . . . . . 485.3 Time decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 Weighted energy estimates . . . . . . . . . . . . . . . . . . . . . 595.5 Modified scattering . . . . . . . . . . . . . . . . . . . . . . . . . 74

Appendix A Ill-posedness 92

Abstract (in Korean) i

iv

Chapter 1

Introduction

1.1 The fractional Schrodinger equations

Let d ∈ N, 1 ≤ α ≤ 2, 0 < γ < d and κ ∈ R/{0}. We consider the following initialvalue problem for a fractional Schrodinger equation with a cubic Hartree-typenonlinearity: −i∂tu + (−∆)

α2 u = κ(| · |−γ ∗ |u|2)u

u(0, ·) = ϕ(1.1.1)

Here, the unknown is a function u : (−T,T ) × Rd → C, the initial datum isϕ : Rd → C and (−∆)

α2 is defined as the spatial Fourier multiplier with symbol | · |α

on Rd, and ∗ denotes spatial convolution. We will consider initial data ϕ ∈ H s(Rd)and solutions will be continuous curves in H s(Rd).

The linear fractional Schrodinger equations have been derived to describe nat-ural phenomena in the context of fractional quantum mechanics [26, 27], systemof long-range lattice interaction [24], water waves [29], turbulence [2] and so on.Heuristically, Hartree nonlinearity can be interpreted as an interaction betweenparticles or waves with potential [13].

We may rescale solutions according to

u(t, x)→ uλ(t, x) := λd−γ+α

2 u(λαt, λx), (1.1.2)

1

CHAPTER 1. INTRODUCTION

for fixed λ > 0. We verify that the scaling invariant space is Hγ−α

2 (Rd). We de-note the critical index by sc =

γ−α

2 . Then the equation is referred to being Hγ−α

2 -subcritical if we find a solution u in H s for s > γ−α

2 , Hγ−α

2 (Rd)-critical for s < γ−α

2

and Hγ−α

2 (Rd)- supercritical for s < γ−α

2 .There are two significant conserved quantities which are particularly useful

for investigating the global dispersive behavior of large solutions. For sufficientlysmooth and decaying solutions u of (1.1.1) the mass and energy defined by

M(u(t)) := ‖u(t)‖2L2(Rd)

E(u(t)) :=12〈(−∆)

α2 u(t), u(t)〉 +

κ

4〈(| · |−γ ∗ |u(t)|2)u(t), u(t)〉

are conserved as time varies. Here, 〈·, ·〉 is the complex inner product in L2(Rd).The equation is called mass-critical if mass is invariant under the scaling (1.1.2),that is γ = α, called mass-subcritical if γ < α, and called mass-supercritical ifγ > α in (1.1.1). The energy space for (1.1.1) is H

α2 (Rd). Similarly we use the

terminology the energy-critical if energy space is scaling invariant space, that isγ = 2α, and subcritical, supercritical if γ < 2α and γ > 2α respectively.

We address the question of well-posedness, scattering and ill-posedness of(1.1.1). Especially we want to prove the optimal well-posedness results with lowregularity condition. That is, we want to find s0 ∈ R such that if ϕ ∈ H s(Rd) fors > s0 the well-posedness result is established, on the other hand if s < s0 theill-posedness result holds true. We expect that the scaling invariant spaces serveas the minimal regularity spaces. Indeed, there are many other dispersive PDEshaving this property.

By Duhamel’s formula, (1.1.1) is written as an integral equation

u = Sα(t)ϕ − i∫ t

0Sα(t − t′)(F(u(t′))) dt′. (1.1.3)

Here we define the linear propagator Sα(t) given by the solution to the linearproblem i∂tu = (−∆)

α2 u with initial datum u(0) = ϕ. It is formally given by

Sα(t)ϕ := e−it|∇|αϕ = F −1(e−it|ξ|αF ϕ) = (2π)−d∫Rd

ei(x·ξ−t|ξ|α)ϕ(ξ) dξ, (1.1.4)

2


where ϕ = F (ϕ) denotes the Fourier transform of ϕ and F −1 the inverse Fouriertransform such that

F ( f )(ξ) =

∫Rd

e−ix·ξ f (x) dx, F −1(g)(x) = (2π)−d∫Rd

eix·ξg(ξ) dξ.

The formal definition of scattering is follow.

Definition 1.1.1. We say that a solution u to (1.1.1) scatters (to u±) in a Hilbertspace H if there exist ϕ± ∈ H (with u±(t) = Sα(t)ϕ±) such that limt→±∞ ‖u(t) −u±‖H = 0.

1.1.1 Previous results

For the case α = 1 we introduce a related equation which is called a “semi-relativistic equation”

− i∂tu +√

m − ∆u = κ(| · |−γ ∗ |u|2)u, m > 0. (1.1.5)

The Cauchy problem with Hartree-type nonlinearities has been studied intensivelyand is well-arranged in [4]. We list some of them. Concerning the range of 1 <

α < 2 and 0 < γ < d, Cho, Hajaiej, Hwang, and Ozawa [4] showed local well-posedness for initial data in s > γ

2 −min(γ, 2)α4 . For d ≥ 2 and 0 < γ < 2 this resultis sharp since on the other side s < γ(2−α)

4 we provide the counter example whichimplies illposedness (See Appendix for a precise statement). This result impliesthat we cannot obtain the positive result in scaling mass-invariant spaces given ageneral data. In this paper we will consider giving further assumption to initialdata and prove the well-posedness in scaling invariance space.

Further results on scattering

In this paragraph, we focus on the case where a dimension d is greater than orequal to 3 if there is no explicit comments.

(1) Denote V = κ|x|−γ for 0 < γ ≤ 1 (or 1 < γ < d), then V is referred to be oflong-range (or short-range, respectively) interaction. If V has a long range,

3


Cho, Hwang and Ozawa [5] proved a negative result that many smooth so-lutions may not scatter even in L2. The short-range scattering in H s can beshown simply by Strichartz estimates when 2 < γ < d and s > γ−α

2 since thedispersion of solution is fast enough [4]. This is also the case for Hartree(α = 2) and semi-relativistic equation (1.1.5). See [15, 20, 19, 8].

For (1.1.5) with γ = 1, which is called Boson star equation, a modifiedscattering result has been established by Pusateri [28]. In Chapter 4 we willconsider the fractional equation with the same Hartree potential γ = 1.

(2) In case 1 < γ ≤ 2, the dispersion of solution to (1.1.1) is not enough forStrichartz estimate on the whole time interval and it is still open whether thescattering result could be established or not. In view of the scattering theoryof Hartree and semi-relativistic equations, the scattering is expected to beshown in this range via radial symmetry assumption or weighted energyestimates. The result with radial assumption is one of our main topic whichis well-arranged in Chapter 2 and the result with angular regularity is inChapter 3.

(3) The other way is to use a weighted energy estimate for the norm ‖Ju‖Hs asin Hartree Schrodinger equations [20] and semi-relativistic equations [19].Let us denote Sα(t)(xSα(t)u) = Ju. Then we compute

J = Sα(t)xSα(t) = x + iαt|∇|α−2∇.

In the weighted space regime one can take advantage of the smoothing effectof Hartree potential in the high frequency analysis. If the initial data is ina weighted space, then the solution could be dispersive enough to scatter.Recently, in [3, 9] the small data scattering was shown in weighted spacewhen 1 < γ ≤ 2 and α0 := max

(6−4γ2−γ , 1

)< α < 2. The authors used

a commutator estimate based on Balakrishnan’s formula to get around thedifficulty caused by the non-locality, the lack of dispersiveness of Sα andthe lack of smoothness of |∇|α. The cost for the commutator estimate is torestrict the range α and d ≥ 3 and to require high regularity for using thesmoothness of potential.

4


(4) The Yukawa potential defined by V = κ e−µ|x||x| for µ > 0 is also important

potential and extensively investigated. For α = 1, (1.1.5) the authors [22]proved small data scattering when s > 1

2 . In [5] they consider generalizedpotentials including the Yukawa potentials and proved scattering for smalldata when s > 2−α

2 .

In this paper, we concentrate on the 2 or 3 dimensional case from now. But ourresults might be extended to all dimensions greater than 3.

1.2 Main theorem 1: Radial symmetry

In the first part of this paper, we restrict our consideration into the mass criticaland supercritical range, that is, α ≤ γ < 3. And the initial data is assumed to haveradial symmetry. First we introduce previous results for the radial case. If α =

γ = 1, in which case (1.1.1) arises as a model system for the dynamics of bosonstars, Herr and Lenzmann [21] proved local well-posedness in the full subcriticalrange s > 0 using Xs,b-spaces and ill-posedness in supercritical range s < 0.Concerning the scattering problem Herr and Tesfahun [22] proved scattering ofsolutions for small radial initial data with s > 0 in the case of Yukawa potentials.And if 6

5 < α < 2, the local well-posedness and small data scattering in the scalinginvariant space H

γ−α2 (R3) are established by using radial Strichartz estimates [4].

Our first main result fills the gap of the range 1 < α ≤ 65 .

Let us state our first main result on the global well-posedness and scatteringfor radial initial data which is small in the critical space of sc =

γ−α

2 .

Theorem 1.2.1. Let 1 < α ≤ 2. There exists δ > 0, such that for all ϕ ∈ H scrad(R3)

satisfying ‖ϕ‖Hsc ≤ δ, there exist a global solution u ∈ Cb(R,H scrad(R3)) of (1.1.1).

u is unique in a certain subspace and the flow map ϕ 7→ u is smooth.Moreover, the solution scatters as t → ±∞, i.e. there exist ϕ± ∈ H sc

rad(R3), suchthat

‖u(t) − Sα(t)ϕ±‖Hsc (R3) → 0 (t → ±∞).

5


To get the result in the full range, we apply a contraction argument in a func-tion space constructed based on the space V2 of bounded quadratic variation andextend bilinear estimates for free solutions to this space. In part, the strategy ofproof is similar to [22], but here we work in the critical regime.

To show the contraction we construct the resolution spaces employing theLittlewood-Paley decomposition. When we estimate the Hartree nonlinear term,main difficulty arises from high-high frequency interactions making low frequencyoutputs because of the singularity of potential near zero. Here the radial assump-tion enables us to overcome the difficulty by obtaining more improved bilinearestimates for all 1 < α ≤ 2.

In the super-critical range, i.e. s < 0, we provide a counterexample whichimplies in particular that the method based on the contraction mapping principlefails. Our next result shows that Theorem 3.1.2 is optimal.

Theorem 1.2.2. Let s < sc and T > 0. The flow map is discontinuous in H sc at theorigin, i.e. there exists a sequence (ϕn) of initial data in H sc

rad(R3) with solutions(un), such that ‖ϕn‖Hsc (R3) → 0 while sup0≤t≤T ‖un(t)‖Hsc (R3) 6→ 0.

The proof is based on an adaptation of the counterexample for the case α = 1from [21] and the abstract ill-posedness result from [1].

1.3 Main theorem 2: Angular regularity

A goal of the second part in this paper is to ease the radial symmetry assump-tion. We continue to consider the mass-critical and mass-supercritical equation(α ≤ γ). But the dimension we treat here is 2 or 3. We will consider much largerdata spaces containing all radial functions.

To state the main theorem let us introduce angularly regular Sobolev spaceH s,1. It is the set of all H s functions whose angular derivative is also in H s. Thenorm is defined by ‖ f ‖Hs,1 := ‖ f ‖Hs + ‖∇S f ‖Hs . Here ∇S is the gradient on the unitsphere and it can be represented as x × ∇.

6


Theorem 1.3.1. Let d = 2, 3, 1 < α < 2, α ≤ γ < 2 and γ > 2d2d−1 . Then there

exists δ > 0 such that for any ϕ ∈ H sc,1 with ‖ϕ‖Hsc ,1 ≤ δ, (1.1.1) has a uniquesolution u ∈ (C ∩ L∞)(R; H sc,1) which scatters in H sc,1.

The homogeneous Sobolev space H sc,1 can be replaced by inhomogeneousspace. The angular regularity is not optimal and taken by the technical reason ofLeibniz rule on the unit sphere. We believe the angular regularity threshold belower. But for the present, it is not obvious whether the angular regularity can beremoved or not.

Recently, the authors of [22], [5] treated Hartree problem by using U p-V p

space argument based on the localized bilinear estimates and endpoint Strichartzestimates. Our strategy for proof is similar, but we employed the spherical Strichartzestimates. In [16] they extended the range of endpoint estimates and also obtainedless regularity loss compared to existing Strichartz estimates by weakening the an-gular integrability which we call spherical Strichartz estimates. Thanks to this andsome radial lemmas we can circumvent the singularity in the low frequency partfrom high-high interaction, the range of γ, however, is restricted to 2d

2d−1 < γ < 2.

1.4 Main theorem 3: Modified scattering

In this part we focus on the case γ = 1. As explained in “Previous results”above, it is known that for the long range (0 < γ ≤ 1) no scattering occurs even inL2 and for the short-range (1 < γ < d = 3) scattering problems were investigatedwith initial data in H s or some weighted spaces. We consider the critical case(γ = 1) and prove a modified scattering in L∞ on the frequency to the Cauchyproblem with small initial data. Heuristically speaking, the time decay of L2-normof nonlinear term, which is computed on a linear solution, is t−γ. The decay ofnonlinear term is non-integrable in time if γ = 1, which we refer as the “scattering-critical”. For this purpose we investigate the global behavior of xeit|∇|αu, x2eit|∇|αuand 〈ξ〉5eit|∇|αu. Due to the non-smoothness of |∇|α near zero frequency the rangeof α is restricted to ( 17

10 , 2). Our approach is inspired by the work [28] of Pusaterideveloped to study semi-relativistic equations.

7


The equation (1.1.1) has the scaling invariance in H1−α

2 and thus it is referredto mass(energy)-subcritical. The subcritical nature readily leads us to the globalwell-posedness. This can be done by a simple energy estimate in HN . In ad-dition, a weighted energy estimate enables us to show the global evolution ofxeit|∇|αu, x2eit|∇|αu, eit|∇|αu such that

xeit|∇|αu ∈ C(R; H3), x2eit|∇|αu ∈ C(R; H2), 〈ξ〉5eit|∇|αu ∈ C(R; Cb(R3)),

provided the same regularity conditions are imposed to the initial data.We focus on an asymptotic behavior of solution to (1.1.1) as time goes to in-

finity. It is said that the solution u scatters to a linear asymptotic state if the effectof the nonlinear term becomes negligible as time goes to infinity. But our equationmay not scatter even though the initial data is arbitrarily small [5]. Instead, we canobserve the phenomenon of “modified scattering” for small solutions by identify-ing a proper nonlinear logarithmic correction. This nonlinear modified scatteringalso happens similarly for standard Hartree equation(α = 2 case) [19] and Bosonstar equation [28]. More precisely, our last topic can be stated as follows: For suf-ficiently small initial data u0 which is defined in a weighted Sobolev space, thereexist a global solution to (1.1.1) which decays in L∞ but behaves in nonlinearfashion over time.

Theorem 1.4.1 (Modified scattering). Let 1710 < α < 2 and N = 1500. Suppose ϕ

satisfies that‖ϕ‖HN + ‖xϕ‖H3 + ‖x2ϕ‖H2 + ‖〈ξ〉5ϕ‖L∞ ≤ ε0.

Then there exists ε0 such that for all ε0 ≤ ε0, the Cauchy problem (1.1.1) has aunique global solution u(t, x) such that

supt>0〈t〉

32 ‖u(t)‖L∞ . ε0.

Moreover, u satisfies the asymptotic behavior as follows: Let

Bα(t, ξ) := −κ

α(2π)3

∫ t

0

∫R3

∣∣∣∣∣ ξ

|ξ|2−α−

σ

|σ|2−α

∣∣∣∣∣−1

|u(ξ)|2dσρ(s−θξ)1〈s〉

ds, (1.4.1)

8


where ρ is a smooth compactly supported function and θ = 3α−540(α+1) . Then there

exist asymptotic state v+, such that for all t > 0

‖〈ξ〉5[e−iBα(t,ξ)v(t, ξ) − v+(ξ)]‖L∞ξ . 〈t〉−δ (1.4.2)

for some 0 < δ < min(2−α3α ,

1100 ). Similar result holds for t < 0.

We can describe the solution of (1.1.1) in the frequency space. To do so let usdefine

v(t, x) := eit|∇|αu(x). (1.4.3)

By Duhamel’s formula (1.1.3) is written as

v = ϕ(ξ) +

∫ t

0I(s, ξ) ds,

whereI(s, ξ) = ic0

∫R3

eisφα(ξ,η)|η|−2 |u|2(s, η)v(s, ξ − η) dηds. (1.4.4)

Here c0 = −2(2π)−2κ (for this we used |x|−1 = 4π|η|−2) and the phase function φαis defined by

φα(ξ, η) = |ξ|α − |ξ − η|α.

The formula of I can be rewritten as

I(s, ξ) := ic1

"R3×R3

eisφ(ξ,η,σ)|η|−2v(s, ξ − η)v(s, η + σ)v(s, σ)dηdσ, (1.4.5)

where c1 = −2(2π)−5κ and

φ(ξ, η, σ) = |ξ|α − |ξ − η|α − |η + σ|α + |σ|α.

These formulae play a crucial role in the proof of weighted energy estimates.When we consider the weighted estimates for the term xv and x2v, we should con-trol the multiplier s∇ξφ(ξ, η, σ) = s∇ξ(|ξ|α − |ξ − η|α). Note that ∇ξφ(ξ, 0, σ) = 0,which is referred to null structure in (1.1.1). This structure enables us to recoverthe loss of the factor s, i.e., as far as estimates are concerned, we can thinks∇ξφ(ξ, η, σ)|η|−2 behaves similarly as the original potential |η|−2.

9


This method which is based on dealing with space-time resonances have beensystematically studied in [14], where general approach of using Fourier analysismethods to investigate the long-time behavior of dispersive PDEs is arranged.Especially for the modified scattering with this technique, see also [23].

Let us briefly give some intuition for the formula (1.4.1). For simplicity weassume that |ξ| ∼ 1 and |η| . 2n with n < 0. Applying the Taylor expansion to thephase function φ of (1.4.5), we approximate I by

ic1

"R3×R3

eiαs(η·z)|η|−2v(s, ξ + η)v(s, ξ + η + σ)v(s, ξ + σ)dηdσ

= ic1v(s, ξ)∫R3F −1(|η|−2)(sz)|v(s, σ)|2dσ + [err],

where z =ξ

|ξ|2−α− σ|σ|2−α

and c1F−1(|η|−2) = −κ(2π)−3|x|−1. This formula yields an

insight for (1.4.1) and (1.4.2). In Chapter 4 below [err] will turn out to be O(s−1−)as s → ∞. The contribution of remaining region |η| & 2n is shown to decay fasterthan s−1 by making integration by parts twice.

We could not obtain a modified scattering in the whole range 1 < α < 2 forthe present. This is due to the lack of smoothness of |∇|α near zero frequency.This is the main difference of fractional equations from the usual Schrodinger orsemi-relativistic ones. The drawback can be overcome by the refined time-decayestimate. But it is inevitable to control at least the L2 norm of x2v for the requestedtime decay. Among the terms from taking ∇2

ξ to v and I in (1.4.4), the following isthe most worst case when we consider small α:

c0

∫ t

0s∫R3

[∇ξ ⊗ ∇ξφα]eisφα(ξ,η)|η|−2 |u|2(η)v(ξ − η)dηds.

In this expression, twice differentiation of the phase function φα rises to singularitynear 0 of order α − 2, which makes a problem in bounding the low frequency partwhen α becomes closer to 1. In the derivation of the asymptotic correction term asabove on the region |η| . 2n, we require the integral of |η|−2+α over this region tobe O(s−2−). This condition also restricts the range of α. One of our next subjectswill be to remove the gap on α.

10

Chapter 2

Preliminaries

2.1 Notations

• As usual, different positive constants depending only on d, α are denoted by thesame letter C, if not specified. A . B and A & B mean that A ≤ CB and A ≥ C−1B,respectively for some C > 0. A ∼ B means that A . B and A & B.• Littlewood-Paley operators: Let ρ ∈ C∞c (−2, 2) be even and satisfy ρ(s) = 1 for|s| ≤ 1. For β(ξ) := ρ(|ξ|)−ρ(2|ξ|) define βk(ξ) = β(2−kξ) for k ∈ Z. Then,

∑k∈Z βk =

1 on Rn \ {0} at it is locally finite. We define the (spatial) Fourier localizationoperator Pk f = F −1(βkF f ). Further, we define β≤k =

∑j∈Z: j≤k β j and P≤k f =

F −1(β≤kF f ), P>k f = f − P≤k f . Let βk = βk−1 + βk + βk+1 and Pk f = F −1(βkF f ).Then PkPk = PkPk = Pk. For a concise expression we abuse a notation; Fork1, k2 ∈ Z, 2k1 . 2k2 , 2k1 & 2k2 and 2k1 ∼ 2k2 are denoted by k1 � k2, k1 � k2 andk1 ∼ k2 respectively.• Fractional derivatives: |∇|s = (−∆)

s2 = F −1|ξ|sF , (1−∆)

s2 = F −1〈ξ〉sF for s > 0.

• Function spaces: H sr = |∇|−sLr, H s = H s

2, H sr = (1 − ∆)−s/2Lr, H s = H s

2, Lr =

Lrx(R

d) for s ∈ R and 1 ≤ r ≤ ∞.• Mixed-normed spaces: For a Banach space X, u ∈ Lq

I X iff u(t) ∈ X for a.e.t ∈ I and ‖u‖Lq

I X := ‖‖u(t)‖X‖LqI< ∞. Especially, we denote Lq

I Lrx = Lq

t (I; Lrx(R

d)),Lq

I,x = LqI Lq

x and Lqt Lr

x = LqRLr

x.• For any b ∈ R we use 〈b〉 = (1 + b2)

12 and also 〈x〉 = (1 + |x|2)

12 for any x ∈ Rd.

11

CHAPTER 2. PRELIMINARIES

• Let A = (Ai) and B = (B j) be any vectors in Rd. Then A ⊗ B denotes theusual tensor product such that (A ⊗ B)i j = AiB j. The same notation is used for thederivatives, i.e. ∇ ⊗ ∇ = (∂i∂ j)i, j=1,··· ,d. We also use ∇ ⊗ x, x ⊗ ∇.• For any positive integer `, and for any vector or derivative A, A` denotes the`-times product A ⊗ · · · ⊗ A.• Let T = (T j1,··· , jk),S = (S i1,··· ,il) be k-times and l-times product of tensors andderivatives. Then we define their dot product by

T; S :=∑

1≤ j1,··· , jk≤d1≤i1,··· ,il≤d

T j1,··· , jkS i1,··· ,il .

• For tensor-valued function F = (Fi1,···il) we use the norm

‖F‖X :=∑

1≤i1,··· ,il≤d

‖Fi1,··· ,il‖X.

2.2 U p, V p spaces

For the general theory of U p and V p, see e.g. [17, 18, 25]. We introduce a definitionand list some useful properties of U p and V p spaces.

Let 1 ≤ p < ∞. We call a finite set {t0, . . . , tK} a partition if −∞ < t0 < t1 <

. . . < tK ≤ ∞, and denote the set of all partitions by T .

Definition 2.2.1. Let 1 ≤ p < ∞. A U p-atom is defined by a step function a :R→ L2

x of the form

a(t) =

K∑k=1

χ[tk−1,tk)(t)φk−1,

where χ is the characteristic function,

{tk} ∈ T , {φk}K−1k=0 ⊂ L2

x withK−1∑k=0

‖φk‖pL2

x= 1.

The atomic space U p(R; L2x) is defined as the set of functions u : R → L2

x of theform

U p(R; L2x) =

u =

∞∑j=1

λ j a j

∣∣∣∣ a j are U p − atoms and {λ j} ∈ `1

,12


with the norm

‖u‖U p := inf

∞∑j=1

|λ j|

∣∣∣∣ u =∑

λ ja j

.Definition 2.2.2. Let 1 ≤ p < ∞. We define V p(R; L2

x) as the normed space of allfunctions v : R→ L2

x such that limt→±∞ v(t) exist and for which the norm

‖v‖V p := sup{tk}∈Z

K∑k=1

‖v(tk) − v(tk−1)‖pL2

x

1p

is finite. V p−(R; L2

x) denotes the normed space of all function v ∈ V p(R; L2x) with

v(−∞) = 0. V p−,rc(R; L2

x) is the closed subspace of all right continuous V p−(R; L2

x)functions.

Lemma 2.2.3. (1) U p(R; L2x) and V p(R; L2

x) are Banach spaces.

(2) For 1 ≤ p < q < ∞ the embedding U p(R; L2x) ↪→ Uq(R; L2

x) ↪→ L∞t L2x is

continuous.

(3) Every u ∈ U p(R; L2x) is right-continuous and limt→−∞ u(t) = 0.

(4) For 1 ≤ p < ∞ the embedding U p(R; L2x) ↪→ V p

−,rc(R; L2x) is continuous.

(5) For 1 < p < ∞ (U p(R; L2x))∗ = V p′(R; L2

x), where 1p + 1

p′ = 1.

Let B(u, v) denote the duality form between U p(R; L2x) and V p′(R; L2

x).

Lemma 2.2.4. Let 1 < p < ∞. Let u ∈ V1−(R; L2

x) be absolutely continuous oncompact intervals and v ∈ V p′(R; L2

x). Then

B(u, v) = −

∫ ∞

−∞

⟨u′(t), v(t)

⟩dt.

Lemma 2.2.5. Let 2 < p < ∞. Let v ∈ V2(R; L2x). Then there exists κ = κ(p) > 0

such that for all K ≥ 1, there exist w ∈ U2(R; L2x) and z ∈ U p(R; L2

x) with v = w+ zand

κ

K‖w‖U2 + eK‖z‖U p . ‖v‖V2 .

13


Lemma 2.2.6. Let 1 ≤ p < q and X be a Banach space and T : Uq → X be abounded, linear operator with ‖Tu‖X ≤ Cq‖u‖Uq for all u ∈ Uq. In addition assumethat there exists a constant Cp ∈ (0,Cq] such that the estimate ‖Tu‖X ≤ Cp‖u‖U p

hold true for all u ∈ U p. Then

‖Tu‖X ≤4Cp

cp,q

(ln

Cq

Cp+ 2cp,q + 1

)‖u‖V p , u ∈ V p

−,rc

where cp,q = (1 − pq ) ln 2.

Now we introduce adapted space U pα and V p

α to Sα. Recall that Sα(t) f =

e−it|∇|α f .

Definition 2.2.7. We define U pα(R; L2

x) as the spaces of all functions u : R → L2x

such that Sα(−t)u ∈ U p(R; L2x) with norm ‖u‖U p

α:= ‖Sα(−t)u‖U p . Likewise, we

define V pα (R; L2

x) and its norm ‖v‖V pα

:= ‖Sα(−t)v‖V p .

Lemma 2.2.3 is extended to the spaces U pα(R; L2

x) and V pα (R; L2

x).

Lemma 2.2.8. For any v ∈ L∞t L2x we have ‖v‖L∞t L2

x. ‖v‖V2

α.

Lemma 2.2.9 (Transfer principle). Let T : L2x×· · ·×L2

x → L1loc(R

d;C) be a n-linearoperator satisfying that

‖T (Sα(t)ϕ1, · · · ,Sα(t)ϕ1)‖Lqt X .

n∏i=1

‖ϕi‖L2(Rd).

for some 1 ≤ q ≤ ∞ and a Banach space X ⊂ L1loc. Then

‖T (u1, · · · , un)‖Lqt X .

n∏i=1

‖ui‖Uqα.

2.3 Auxiliary estimates

Lemma 2.3.1. Assume that m ∈ L1(Rd × Rd), d ≥ 1 satisfies∥∥∥∥ ∫Rd×Rd

m(η, σ)eixηeiyσdηdσ∥∥∥∥

L1x,y

≤ A(m). (2.3.1)

14


Then for any (p, q, r) with 1 ≤ p, q, r ≤ ∞ and 1p + 1

q + 1r = 1,∣∣∣∣∣∫

Rd×Rdm(η, σ)v(η)g(σ)h(±η ± σ)dηdσ

∣∣∣∣∣ . A(m)‖v‖Lp‖g‖Lq‖h‖Lr .

Moreover, for all p, q with 1p + 1

q = 12 , one has∥∥∥∥ ∫

Rdm(ξ, η)v(ξ ± η)g(η)dη

∥∥∥∥L2ξ

. A(m)‖v‖Lp‖g‖Lq .

For this see Appendix B.2 in [28]. As a corollary we have

Lemma 2.3.2. Suppose P,Q ∈ L1(Rd×Rd) satisfies (2.3.1) with A(P), A(Q). Then,∥∥∥∥"Rd×Rd

P(η, σ) ⊗Q(η, σ)eixηeiyσdσdη∥∥∥∥

L1x,y

. A(P)A(Q).

Lemma 2.3.3. Let 1 < α < 2 and d ≥ 2. Then for any ξ, σ ∈ Rd we have∣∣∣ ξ

|ξ|2−α−

σ

|σ|2−α

∣∣∣ & min(|σ|α−1,

|ξ − σ|

|σ|2−α

).

Proof of Lemma 2.3.3. Set

z :=ξ

|ξ|2−α−

σ

|σ|2−α

and let ξτ = τξ+(1−τ)σ for τ ∈ [0, 1]. If ξτ = 0 for some τ = τ0, then since ξ , σ,ξ , 0 and σ , 0, τ0 = |σ|/|ξ − σ| ∈ (0, 1) and ξ = −[(1 − τ0)/τ0]σ. Substitutingthis into z, we have

z = −α

(1 +

(1 − τ0

τ0

)α−1)

σ

|σ|2−α,

which means |z| & |σ|α−1. Note that 1 > τ0 = |σ|/|ξ − σ|.From now we assume that ξτ , 0 for all τ ∈ [0, 1]. Then

z =

∫ 1

0∂τ

[ξτ|ξτ|2−α

]dτ =

∫ 1

0|ξτ|−(2−α)Mξτ(ξ − σ) dτ, (2.3.2)

where Mξτ is the d × d symmetric matrix

Mξτ = I − (2 − α)ξτ ⊗ ξτ|ξτ|2

.

15


By Cauchy-Schwarz inequality we have that for any x ∈ Rd

x ·Mξτ x = |x|2 − (2 − α)(x · ξτ)2

|ξτ|2≥ (α − 1)|x|2.

Thus the eigenvalues of Mξτ is at least (α − 1), which implies that

|Mξτ x| ≥ (α − 1)|x|.

Using this fact, we get that

|z| ≥∫ 1

0|ξτ|−(2−α) dτ|ξ − σ| ≥

∫ 1

0|1 + τ

|ξ − σ|

|σ||−(2−α) dτ

|ξ − σ|

|σ|2−α.

If |ξ−σ||σ|≤ 1, then we have

|z| &|ξ − σ|

|σ|2−α.

If |ξ−σ||σ|

> 1, then

|z| &∫ |σ|

|ξ−σ|

0

(1 + τ

|ξ − σ|

|σ|

)−(2−α)dτ|ξ − σ|

|σ|2−α& |σ|α−1.

Therefore we estimate

|z| & (α − 1) min(|σ|α−1, |ξ − σ|/|σ|2−α).

�

Lemma 2.3.4. Let φ, φ be as in (5.5.8), (5.5.16), respectively. Then we have

|φ(ξ, η, σ) − φ(ξ, η, σ)| . |η|α.

Proof of Lemma 2.3.4.

|φ(ξ, η, σ) − φ(ξ, η, σ)|

.∣∣∣|ξ + η|α − |ξ|α − α

(ξ · η)|ξ|2−α

∣∣∣ +∣∣∣|ξ + σ + η|α − |ξ + σ|α − α

((ξ + σ) · η)|ξ + σ|2−α

∣∣∣.16


We only consider the first term. If |ξ| . |η|, then a direct calculation gives us∣∣∣|ξ + η|α − |ξ|α − α(ξ · η)|ξ|2−α

∣∣∣ . |η|α.If |ξ| � |η|, then MVP gives us

|ξ + η|α − |ξ|α =

∫ 1

0α

ξσ|ξσ|2−α

· η dσ,

where ξσ = ξ + ση. By another MVP as in (2.3.2) we get

|ξ + η|α − |ξ|α − α(ξ · η)|ξ|2−α

= α

"[0,1]2|ξστ|

α−2[Mξστ(στη)] · η dσdτ,

where ξστ = ξ + στη. Since |ξ| � |η|, |ξστ| ∼ |ξ| and hence |ξστ|α−2 . |η|α−2. On theother hand, |[Mξστη] · η| . |η|2. Therefore

||ξ + η|α − |ξ|α| . |η|α.

�

17

Chapter 3

Well-posedness and scattering forradial data

3.1 Main results

In this chapter, scattering for the mass-critical fractional Schrodinger equationwith a cubic Hartree-type nonlinearity for initial data in small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data isestablished. For this, we prove a bilinear estimate for free solutions and extend itto perturbations of bounded quadratic variation. This result is shown to be sharpby providing an example which implies the existence of unbounded third orderdirectional derivatives of the flow map in the super-critical range.

Our aim is to prove the existence and scattering of solutions to the IVP (1.1.1)in the critical space L2(R3), i.e. we will focus on mass-critical case and n = 3. Ourresults can be extended to mass-supercritical case and higher dimension greaterthan 3 using the similar method used in this and next chapter. But we do not pursuehere.

We will consider the subspace of radial functions. For s ∈ R define

H srad(R3) := {ϕ ∈ H s(R3) : ∃ϕ0 : [0,∞)→ R s.th. ϕ(x) = ϕ0(|x|) a.e.},

with norm ‖ · ‖Hs .

18

CHAPTER 3. SCATTERING FOR RADIAL CASE

Let us restate our first main theorem for mass-critical and 3-dimension case.

Theorem 3.1.1. Let 1 < α ≤ 2. There exists δ > 0, such that for all ϕ ∈ L2rad(R3)

satisfying ‖ϕ‖L2 ≤ δ, there exist a global solution u ∈ Cb(R, L2rad(R3)) of (1.1.1). u

is unique in a certain subspace and the flow map ϕ→ u is smooth.Moreover, the solution scatters as t → ±∞, i.e. there exist ϕ± ∈ L2

rad(R3), suchthat

‖u(t) − Sα(t)ϕ±‖L2(R3) → 0 (t → ±∞).

Our second result states that for large radial initial data in critical space, wehave local well-posedness. To state our results let us define

Br,Λ :={ϕ ∈ L2

rad(R3) : ‖ϕ‖L2 ≤ r, ‖P>Λϕ‖L2 ≤ ηr−1}, (3.1.1)

for r,Λ ≥ 1 and some parameter 0 < η � 1, which will be fixed in Subsection3.3.2 independently of r,Λ. Notice that for each ϕ ∈ L2

rad(R3) and η > 0 thereexist Λ > 0 such that ‖P>Λϕ‖L2 ≤ ηr−1. Therefore, for any ϕ ∈ L2

rad(R3) and anyr ≥ ‖ϕ‖L2 there exists Λ ≥ 1, such that ϕ ∈ Br,Λ. For large radial initial data in thecritical space, we have local well-posedness.

Theorem 3.1.2. Let 1 < α ≤ 2. For all r,Λ ≥ 1 and all ϕ ∈ Br,Λ ⊂ L2rad(R3), there

exists T = T (r,Λ) and a solution u ∈ C([0,T ], L2rad(R3)) of (1.1.1). u is unique in

a certain subspace and the flow map ϕ→ u is smooth.

Our next result shows that Theorem 3.1.2 is optimal.

Theorem 3.1.3. Let s < 0 and T > 0. The flow map is discontinuous in H s at theorigin, i.e. there exists a sequence (ϕn) of initial in L2

rad(R3) with solutions (un),such that ‖ϕn‖Hs(R3) → 0 while sup0≤t≤T ‖un(t)‖Hs(R3) 6→ 0.

The proof is based on an adaptation of the counterexample for the case α = 1from [21] and the abstract ill-posedness result from [1].

19


3.2 Bilinear estimates for radial functions

3.2.1 Free solutions

Since the characteristic hypersurface in R1+d defined by the phase function |ξ|α hasd nonvanishing principal curvatures in the case α > 1, there are similar Strichartzestimates as for the Schrodinger equation, up to a loss of derivatives dictated byscaling. For the following Lemma, its proof and more information we refer to[10].

Lemma 3.2.1. Let 1 < α ≤ 2, q > 2, r ≥ 2, 2q + d

r = d2 , θ = d

2 (2 − α)( 12 −

1r ). Then,

‖Sαϕ‖Lqt (R,Lr

x(Rd)) . ‖ϕ‖Hθ(Rd).

Next, we adapt [22, Lemma 3.2] to the case of generalized dispersion.

Lemma 3.2.2. Consider the integral

I(φ, ψ)(τ, ξ) =

∫φ(|η|)ψ(|ξ − η|)δ(τ − |η|α + |ξ − η|α)dη

for smooth φ and ψ supported in [−r, r] and [−R,R], for some r,R > 0. Then for0 ≤ τ ≤ αmax{r,R}α−1|ξ|,

I(φ, ψ)(τ, ξ) =2πα|ξ|

∫ ∞

a(τ,|ξ|)φ(ρ)ψ(ω(τ, ρ))ω(τ, ρ)2−αρdρ (3.2.1)

where

a(τ, |ξ|) =|ξ|2 + τ2/α

2|ξ|and ω(τ, ρ) = (ρα − τ)1/α. (3.2.2)

Furthermore,I(φ, ψ)(τ, ξ) = 0, if τ > αmax{r,R}α−1|ξ|. (3.2.3)

Proof. As in [22, pp. 8–9], the proof is an straight-forward modification of theargument for α = 1 from [12, Lemma 4.4]. First, we check that the delta function

20


and support condition on φ, ψ restrict the range of τ. By the mean-value theoremwe have

|τ| =∣∣∣|η|α − |ξ − η|α∣∣∣ ≤ αmax{|η|, |ξ − η|}α−1|ξ|,

implying the second claim (3.2.3).Now, let 0 ≤ τ ≤ αmax{r,R}α−1|ξ|. Using δ−calculus, we can write

δ(τ − |η|α + |ξ − η|α) =∣∣∣τ − |η|α − |ξ − η|α∣∣∣δ(|ξ − η|2α − (τ − |η|α)2)

= 2(|η|α − τ

)δ((|ξ|2 − 2ξ · η + |η|2)α − (τ − |η|α)2), (3.2.4)

having used 0 ≤ |ξ − η|α = |η|α − τ within the support of δ. Introduce polarcoordinates for η = ρθ, where ρ = |η| and θ =

η

|η|∈ S 2. Then dη = ρ2dS θdρ. We

set further b = θ · ξ|ξ|

, then

dS θ = dS θ′da, for θ′ ∈ S 1 and dη = ρ2dρdS θ′db.

Because of |ξ − η|α = |η|α − τ = |ρ|α − τ, we have |ξ − η| = ω(τ, ρ). Now, with

gτ,ξρ (b) = (|ξ|2 − 2|ξ|ρb + ρ2)α − (|η|α − τ)2,

the integrand is independent of θ′ and we obtain

I(φ, ψ)(τ, ξ) = 4π∫ ∞

τ1α

∫ 1

−1φ(ρ)ψ(ω(τ, ρ))(ρα − τ)ρ2δ

(gτ,ξρ (b)

)dbdρ.

We use the delta function to set the value of b to

bτ,ξρ :=|ξ|2 + ρ2 − (ρα − τ)2/α

2|ξ|ρ, (3.2.5)

with the condition b ≤ 1 that forces

|ξ|2 + ρ2 ≤ 2|ξ|ρ + (ρα − τ)2/α

and since (ρα−τ)2/α ≤ ρ2−τ2/α, the domain of the ρ-integration is further restrictedto {

ρ ≥|ξ|2 + τ2/α

2|ξ|

}. (3.2.6)

21


We compute the integral over b as∫ 1

−1δ(gτ,ξρ (b)

)db =

∣∣∣ ddb

gτ,ξρ (bτ,ξρ )∣∣∣−1

= (2α|ξ|ρ(ρα − τ)2(α−1)α )−1, (3.2.7)

where bτ,ξρ is the value in (3.2.5). With (3.2.6) and (3.2.7), we finally obtain

I(φ, ψ)(τ, ξ) = 4π∫ ∞

a(τ,|ξ|)φ(ρ)ψ(ω(τ, ρ))

ρ(ρα − τ)

2α|ξ|(ρα − τ)2(α−1)α

dρ,

which reduces to the desired form. �

Proposition 3.2.3. Let 1 ≤ α ≤ 2. Consider u+(t) = Sα(t) f and v−(t) = Sα(−t)g,where f and g are radial. Then for any k, k1, k2 ∈ Z with k1 ≥ k2, we have

‖Pk(Pk1u+Pk2v

−)‖L2t,x(R1+3) . 2k2k2( 1−α

2 )‖Pk1 f ‖L2x(R3)‖Pk2g‖L2

x(R3) (3.2.8)

Proof. The Fourier transform of a radial function is a radial function, and we mayassume f , g ≥ 0. We denote fβk1 , gβk2 by ψk1 , φk2 , respectively, and compute thespace-time Fourier transform

Ft,x{Pk(Pk1u+Pk2v

−)}(τ, ξ)

=

∫R

e−itτβk(|ξ|)∫R3βk1(|ξ − η|)e

−it|ξ−η|α f (ξ − η)βk2(|η|)eit|η|α g(η)dηdt

= βk(|ξ|)∫R3ψk1(|ξ − η|)φk2(|η|)δ(τ − |η|

α + |ξ − η|α)dη.

For 0 ≤ τ ≤ Cα2k1(α−1)2k, Lemma 3.2.2 implies

Ft,x{Pk(Pk1u+Pk2v

−)}(τ, ξ) ≈ χµ(ξ)1|ξ|

∫ ∞

a(τ,|ξ|)φk2(ρ)ψk1(ω(τ, ρ))ω(τ, ρ)2−αρdρ,

while for τ > Cα2k1(α−1)2k there is no contribution. If −Cα2k1(α−1)2k ≤ τ ≤ 0, wesimilarly obtain

Ft,x{Pk(Pk1u+Pk2v

−)}(τ, ξ) ≈ βk(ξ)1|ξ|

∫ ∞

a(−τ,|ξ|)φk2(ω(−τ, ρ))ψk1(ρ)ω(−τ, ρ)2−αρdρ,

while for τ < −Cα2k1(α−1)2k there is no contribution. Hence,

‖Pk(Pk1u+Pk2v

−)‖2L2t,x(R1+3) := I1 + I2,

22


where, by Plancherel’s theorem,

I1 ≈

∫R3

∫ Cα2k1(α−1)2k

0

β2k(|ξ|)|ξ|2

∣∣∣∣∣ ∫ ∞

a(τ,|ξ|)[ρφk2(ρ)][ω(τ, ρ)2−αψk1(ω(τ, ρ))]dρ

∣∣∣∣∣2dτdξ,

I2 ≈

∫R3

∫ Cα2k1(α−1)2k

0

β2k(|ξ|)|ξ|2

∣∣∣∣∣ ∫ ∞

a(τ,|ξ|)[ρψk1(ρ)][ω(τ, ρ)2−αφk2(ω(τ, ρ))]dρ

∣∣∣∣∣2dτdξ.

In polar coordinates ξ → (r, θ) ∈ [0,∞) × S 2, the first term is

I1 ≈

∫ ∞

0χ2µ(r)

∫ Cα2k1(α−1)2k

0

∣∣∣∣∣ ∫ ∞

a(τ,r)[ρφk2(ρ)][ω(τ, ρ)2−αψk1(ω(τ, ρ))]dρ

∣∣∣∣∣2dτdr.

The Cauchy-Schwarz inequality implies∣∣∣∣∣ ∫ ∞

a(τ,|ξ|)[ρφk2(ρ)][ω(τ, ρ)2−αψk1(ω(τ, ρ))]dρ

∣∣∣∣∣2.

( ∫R

|φk2(ρ)ρ|2dρ)( ∫

R

|ω(τ, ρ)2−αψk1(ω(τ, ρ))βk2(ρ)|2dρ)

. ‖gk2‖2L2(R3)

( ∫R

|σ2−αψk1(σ)|22(k1−k2)(α−1)dσ).

Here, we used the change of variables σ = ω(τ, ρ), hence σα = ρα − τ and dρ =

(σρ)α−1dσ, and in the domain of integration we have |σ

α−1

ρα−1 | . 2(k1−k2)(α−1). We obtain

I1 . 2k‖Pk2g‖2L2

x(R3)2k1(α−1)2k( 1

2k12k2

)α−1‖Pk1 f ‖2L2

x(R3)

. 22k2k2(1−α)‖Pk2g‖2L2

x(R3)‖Pk1 f ‖2L2x(R3).

Concerning I2, we obtain

I2 . 2k‖Pk1 f ‖2L2x(R3)

∫ Cα2k1(α−1)2k

0

∫R

|ω(τ, ρ)2−αφk2(ω(τ, ρ))|2dρdτ

along the same lines. Again,

I2 . 22k2k2(1−α)‖Pk2g‖2L2

x(R3)‖Pk1 f ‖2L2x(R3),

by the same change of variables as above. �

23


Corollary 3.2.4. Let 1 ≤ α ≤ 2. For all k � k1 ∼ k2 and spatially radial functionsu j = Sα(·)ϕ j, we have

‖P≤k(Pk1u1Pk2u2)‖L2(R1+3) . 23−α

2 k2α−1

2 (k−k1)‖Pk1ϕ1‖L2(R3)‖Pk2ϕ1‖L2(R3). (3.2.9)

Proof. This follows by dyadic summation over k′ ≤ k from Proposition 3.2.3. �

3.2.2 Transference

By the atomic structure of U2α, estimates in L2 for free solutions transfer to U2

α-functions, hence to V p

α for p < 2. However, transference to V2α does not follow

from the general theory of these spaces. Nevertheless, we prove below that incase of the bilinear estimate of the previous section it does hold true. This mightalso have applications in the case α = 1, and the proof applies to certain othermultilinear estimates.

Proposition 3.2.5. Let 1 ≤ α ≤ 2. For all k � k1 ∼ k2 and spatially radialfunctions u1, u2 ∈ V2

α, we have

‖P≤k(Pk1u1Pk2u2)‖L2(R1+3) . 23−α

2 k2α−1

2 (k−k1)‖Pk1u1‖V2α‖Pk2u2‖V2

α. (3.2.10)

Proof. 1. Step: Let P =∑

k∈F Pk, with a finite set F of integers k of size k1 ∼ k2,such that PPk j = Pk j for j = 1, 2. We claim that

‖P≤k|Pw|2‖L2 . 23−α

2 k2α−1

2 (k−k1)‖w‖2U4S

(3.2.11)

for any k . k1 and spatially radial w ∈ U4α. To prove (3.2.11), let Vk = (qβ≤k+1)2.

Then, Vk ≥ 0 and we have the pointwise bound β≤k . β≤k+1 ∗ β≤k+1 . β≤k+2 on theFourier side, which implies

‖P≤k|Pw|2‖L2 . ‖Vk ∗ |Pw|2‖L2 . ‖P≤k+2|Pw|2‖L2 .

The quantity

n( f ) := ‖Φ( f )‖L4(R3), for Φ( f ) =(Vk ∗ |P f |2

) 12,

24


is subadditive, and ‖Vk ∗ |Pw|2‖L2 = ‖n(w(t))‖2L4

t. Indeed,

Φ2( f1 + f2)(x) =

∫R3

Vk(x − y)|P f1(y) + P f2(y)|2dy

≤

∫R3

Vk(x − y)|P f1(y) + P f2(y)||P f1(y)|dy

+

∫R3

Vk(x − y)|P f1(y) + P f2(y)||P f2(y)|dy

≤ Φ( f1 + f2)(x)Φ( f1)(x) + Φ( f1 + f2)(x)Φ( f2)(x)

by Cauchy-Schwarz, so Φ( f1 + f2) ≤ Φ( f1) + Φ( f2) and

n( f1 + f2) ≤ ‖Φ( f1) + Φ( f2)‖L4 ≤ n( f1) + n( f2)

follows. Also, we obviously have n(c f ) = |c|n( f ) for all c ∈ C. Due to (3.2.9) wehave

‖n(Sα(t)ϕ)‖L4t≤ C(k, k1)

12 ‖ϕ‖L2 (3.2.12)

for all radial ϕ ∈ L2(R3), where C(k, k1) denotes the constant in (3.2.9). Let w ∈ U4α

be radial with atomic decomposition

w =∑

j

c ja j, s.th.∑

j

|c j| ≤ 2‖w‖U4α, and radial U4

α-atoms a j.

We have‖n(w)‖L4

t≤

∑j

|c j|‖n(a j)‖L4t. C(k, k1)

12 ‖w‖U4

α, (3.2.13)

provided that for any U4α-atom a the estimate

‖n(a)‖L4t. C(k, k1)

12

holds true. Indeed, let a(t) =∑

m 1Im(t)Sα(t)ϕm, for some partition (Im) of R andradial ϕm ∈ L2(R3) satisfying

∑m ‖ϕm‖

4L2 ≤ 1. Then,

‖n(a)‖L4t≤

∥∥∥∥∑m

1Im(t)n(Sα(t)ϕm)∥∥∥∥

L4t

≤(∑

m

‖n(Sα(t)ϕm)‖4L4t

) 14

. C(k, k1)12(∑

m

‖ϕm‖4L2

) 14. C(k, k1)

12 ,

25


where we used (3.2.12) in the third inequality, which completes the proof of(3.2.13). This implies

‖P≤k|Pw|2‖L2(R1+3) . ‖n(w)‖2L4t. C(k, k1)‖w‖2U4

α, (3.2.14)

hence the claim (3.2.11).2. Step: Let v j := Pk ju j, j = 1, 2. We may assume ‖v j‖U4

α= 1. Let w± = v1±v2,

then w± = Pw± and ‖w±‖U4S. 1. Due to

Re(v1v2) =14

(|w+|

2 − |w−|2)

and Im(v1v2) = Re(−iv1v2). The estimate (3.2.11) yields

‖P≤k(v1v2)‖L2(R1+3) . ‖P≤k|Pw+|2‖L2(R1+3) + ‖P≤µ|Pw−|2‖L2(R1+3)

. C(k, k1)(‖w+‖

2U4α

+ ‖w−‖2U4α

). C(k, k1),

which implies

‖P≤k(v1v2)‖L2(R1+3) . C(k, k1)‖v1‖U4α‖v2‖U4

α. C(k, k1)‖v1‖V2

α‖v2‖V2

α,

where we used V2α ↪→ U4

α from Lemma 2.2.3. �

Proposition 3.2.6. For any k, k1, k2 ∈ Z and u, v ∈ U4α,

‖Pk(Pk1uPk2v)‖L2(R1+3) . min{2k, 2k1 , 2k2}12 2

2−α4 k12

2−α4 k2‖Pk1u‖U4

α‖Pk2v‖U4

α. (3.2.15)

Proof. The Bernstein inequality implies

‖Pk(Pk1uPk2v)‖L2(R1+3) . 2k2 ‖Pk1uPk2v‖

L2t L

32x

. 2k2 ‖Pk1u‖L4

t L3x‖Pk2v‖L4

t L3x

and Lemma 3.2.1 gives (3.2.15) if k ≤ k1, k2. Similarly,

‖Pk(Pk1uPk2v)‖L2(R1+3) . ‖Pk1u‖L4t L6

x‖Pk2v‖L4

t L3x. 2

k12 ‖Pk1u‖L4

t L3x‖Pk2v‖L4

t L3x.

This concludes the proof because we can interchange the roles of u and v. �

26


Remark 3.2.7. In the case 1 < α ≤ 2 this gives another simple proof of a re-sult which is slightly weaker than (3.2.10) but sufficient for our application: Theobvious U2

S-version of (3.2.10) (see Lemma 2.2.9) can now be interpolated with(3.2.15) in the case 2k . 2k1 ∼ 2k2 via Lemma 2.2.6, which gives

‖Pk(Pk1uPk2v)‖L2(R1+3) . 23−α

2 k2(k−k1)( α−12 −ε)‖Pk1u‖V2

α‖Pk2v‖V2

α,

for any fixed ε > 0.

For s ∈ R let‖u‖Xs :=

(∑k∈Z

22sk‖Pku‖2V2α

) 12. (3.2.16)

And we set X := {u ∈ X0 | u spatially radial}.

Corollary 3.2.8. Let 1 < α ≤ 2. For all u, v ∈ X, we have

‖(−∆)α−3

4 (uv)‖L2(R1+3) . ‖u‖X‖v‖X. (3.2.17)

Proof. We decompose

‖(−∆)α−3

4 (uv)‖L2(R1+3) .∑

k,k1,k2∈Z

2α−3

2 k‖Pk(Pk1uPk2v)‖L2(R1+3) . Σ1 + Σ2 + Σ3,

where Σ1 is the contribution of k1 � k2 ∼ k, Σ2 is the contribution of k2 � k1 ∼ k,and Σ3 is the contribution of k . k1 ∼ k2. From (3.2.15) and Cauchy-Schwarz weobtain

|Σ1| .∑

k1�k2

24−α

4 (k1−k2)‖Pk1u‖V2α‖Pk2v‖V2

α. ‖u‖X‖v‖X.

Similarly we prove|Σ2| . ‖u‖X‖v‖X.

Finally, using (3.2.10) we obtain

|Σ3| .∑

k.k1∼k2

2(k−k1) α−12 ‖Pk1u‖V2

α‖Pk2v‖V2

α. ‖u‖X‖v‖X,

where we have exploited the radiality. �

27


3.3 Proofs of the main results

3.3.1 Proof of Theorem 3.1.1

It suffices to consider positive times. We represent the solution of (1.1.1) on [0,∞)using the Duhamel’s formula

u(t) = 1[0,∞)(t)Sα(t)ϕ + iκNα(u, u, u)(t), (3.3.1)

Nα(u1, u2, u3)(t) = 1[0,∞)(t)∫ t

0Sα(t − t′)

[{| · |−α ∗ (u1u2)}u3

](t′)dt′. (3.3.2)

Let D(δ) be a complete metric space {u ∈ X∣∣∣ ‖u‖X ≤ δ} equipped with the metric

d(u, v) := ‖u − v‖X. We show that Ψ(u) = Sα(t)ϕ + iκNα(u) is a contraction onD(δ). First, for all ϕ ∈ H s

rad we immediately have

‖1[0,∞)Sϕ‖Xs . ‖ϕ‖Hs . (3.3.3)

Next, we study the nonlinear part. For all u1, u2, u3 ∈ X, we have∥∥∥Nα(u1, u2, u3)∥∥∥

X. ‖u1‖X‖u2‖X‖u3‖X. (3.3.4)

Indeed, we have

‖Nα(u1, u2, u3)‖X . supv∈X:‖v‖X≤1

∣∣∣∣" (−∆)α−3

2 (u1u2)u3(t)v(t)dtdx∣∣∣∣

by duality, see [17]. Further, we obtain∣∣∣∣" (−∆)α−3

2 (u1u2)u3(t)v(t)dtdx∣∣∣∣ . ‖(−∆)

α−34 (u1u2)‖L2‖(−∆)

α−34 (u3v)‖L2

. ‖u1‖X‖u2‖X‖u3‖X‖v‖X

by Corollary 3.2.8, which implies (3.3.4). If we take u1 = u − v, u2 = u, u3 = u, oru1 = v, u2 = u − v, u3 = u, or u1 = v, u2 = v, u3 = u − v, then we get

‖Nα(u, u, u) − Nα(v, v, v, )‖X . (‖u‖X + ‖v‖X)2‖u − v‖X. (3.3.5)

Thus we can find δ small enough for Ψ to be a contraction mapping onD+(δ).

28


Since Sα(−t)PNNα(u) and Sα(−t)PN∇SNα(u) are in V2−,rc(R; L2) from (4) of

Lemma 2.2.3, and∑N>0

N2sc(‖Sα(−t)PNNα(u)‖V2 + ‖Sα(−t)PN∇SNα(u)‖V2)2 < ∞

from (4.2.1), limt→+∞ Sα(−t)Nα(u) exists in L2rad(R3). Define a scattering state u+

withϕ+ := ϕ + lim

t→+∞Sα(−t)Nα(u).

By time symmetry we can argue in a similar way for the negative time. Thus weget the desired result.


Let Λ, r ≥ 1 be given. Recall that

Br,Λ :={ϕ ∈ L2

rad(R3) : ‖ϕ‖L2 ≤ r, ‖P>Λϕ‖L2 ≤ ηr−1},where the parameter 0 < η ≤ 1 will be determined below, and define

DR,ε :={u ∈ X : ‖u‖X ≤ R, ‖P>Λu‖X ≤ ε

},

for some 0 < ε ≤ R. We implicitely assume that all functions are supported in[0,T ]. Split Nα(u) = J1(u≤Λ, u>Λ) + J2(u≤Λ, u>Λ), where J1 is at least quadratic inu>Λ and J2 is at least quadratic in u≤Λ. For J1, we use (3.3.4) and obtain

‖J1(u≤Λ, u>Λ)‖X . Rε2,

for all u ∈ DR,ε . We turn to J2. First, we have

‖(−∆)α−3

2 (u≤Λu≤Λ)v‖L1t L2

x. Λα‖u≤Λ‖

2L2

t L2x‖v‖L∞t L2

x. TΛα‖u‖2X‖v‖X.

Second, we have

‖(−∆)α−3

2 (u≤Λv>4Λ)u≤Λ‖L1t L2

x. Λα−3‖u≤Λv>4Λ‖L1

t L2x‖u≤Λ‖L∞t L∞x

. TΛα−3‖v>4Λ‖L∞t L2x‖u≤Λ‖

2L∞t L∞x. TΛα‖u‖2X‖v‖X,

29


and we obtain

‖(−∆)α−3

2 (u≤Λv)u≤Λ‖L1t L2

x. TΛα‖u‖2X‖v‖X,

because the contribution of v≤4Λ can be treated as in the first estimate above. Weconclude that there exists a C ≥ 1, such that for all ϕ ∈ Br,Λ and all u ∈ DR,ε wehave

‖1[0,T )Sαϕ‖X ≤ Cr, ‖1[0,T )Sαϕ>Λ‖X ≤ Cηr−1, (3.3.6)

‖J1(u≤Λ, u>Λ)‖X ≤ CRε2, ‖J2(u≤Λ, u>Λ)‖X ≤ CTΛαR3, (3.3.7)

and similar estimates for differences. After choosing R = 4Cr, ε = 2−6C−2r−1,T = 2−18C−6r−4Λ−α, and η = 2−10C−3, one checks that

DR,ε → DR,ε , u 7→ 1[0,T )(Sαϕ +Nα(u))

is a strict contradiction, for given ϕ ∈ Br,Λ. The contraction mapping principleimplies Theorem 3.1.2.


We apply the abstract ill-posedness result established by Bejenaru and Tao in [1,Proposition 1]. For this, it suffices to show that the following inequality

supt∈[0,T ]

∥∥∥∥∥ ∫ t

0Sα(t − t′)(| · |−α ∗ |Sα(t′)ϕ|2Sα(t′)ϕ)dt′

∥∥∥∥∥Hs(R3)

. ‖ϕ‖3Hs(R3) (3.3.8)

fails to hold for some radial data ϕ ∈ H s(R3), if s < 0. We modify the constructionfor the case α = 1 from [21, Section 3] and define the annulus Aλ = {ξ ∈ R3 : λ ≤|ξ| ≤ 2λ}. Let ϕ be the inverse Fourier transform of the characteristic function 1Aλ .Clearly, ϕ is radial and ‖ϕ‖Hs(R3) ∼ λ

s+ 32 . With this choice of ϕ, let

Φ(t) :=∫ t

0Sα(t − t′)(| · |−α ∗ |Sα(t′)ϕ|2Sα(t′)ϕ)dt′.

30


We compute the spatial Fourier transform

Φ(t, ξ) =

∫ t

0

∫R3Sα(t − t′)

Fx(|Sα(t′)ϕ|2)(η)|η|3−α

FxSα(t′)ϕ(ξ − η)dηdt′

=

∫ t

0

"R3×R3

e−i(t−t′)|ξ|α e−it′ |η−σ|α1Aλ(η − σ)eit′ |σ|α1Aλ(σ)|η|3−α

e−it′ |ξ−η|α1Aλ(ξ − η)dσdηdt′

= e−it|ξ|α"R3×R3

∫ t

0eit′gα(ξ,η,σ)dt′

1Aλ(η)1Aλ(σ)1Aλ(ξ − η − σ)|η + σ|3−α

dσdη

wheregα(ξ, η, σ) = |ξ|α − |η|α + |σ|α − |ξ − η − σ|α.

Choose T = ελ−α with 0 < ε � 1. In the domain of integration we have

|tgα(ξ, η, σ)| . |tλα| � 1

and we get∣∣∣∣∣ ∫ t

0eit′gα(ξ,η,σ)dt′

∣∣∣∣∣ & ∫ t

0cos (t′gα(ξ, η, σ))dt′ & |t| for t < T. (3.3.9)

Thus, if ξ ∈ Aλ, we have

|Φ(t, ξ)| & |t|∫

Aλ

∫Aλ|η + σ|−3+αdηdσ & ελ−αλ3+α.

From this we easily obtain ‖Φ‖Hs(R3) & ελs+ 9

2 . In conclusion, these norm calcula-tions and (3.3.8) give

ελs+ 92 . ‖Φ(t)‖Hs . ‖ϕ‖3Hs ∼ λ3s+ 9

2 .

So if s < 0, we make (3.3.8) fail to hold by choosing λ sufficiently large.

31

Chapter 4

Scattering associated with angularderivative

4.1 Main results

In this chapter we continue to study the small-data scattering of Hartree typefractional Schrodinger equations in space dimension 2, 3 for mass-critical and su-pertricial case (α ≤ γ). Recall that the equation is scaling-critical in H sc withsc =

γ−α

2 . We extend the result for radial data in Chapter 2 when power of po-tential lies in γ ≥ 2d

d−1 . More precisely we show that the solution scatters in H sc,1,where H sc,1 is also a scaling critical space of Sobolev type taking in angular regu-larity with norm defined by ‖ϕ‖Hsc ,1 = ‖ϕ‖Hsc + ‖∇Sϕ‖Hsc . For this purpose we usethe recently developed Strichartz estimate which is L2

θ-averaged on the unit sphereSd−1 and utilize U p-V p space argument.

To state the main theorem let us introduce angularly regular Sobolev spaceH s,1. It is the set of all H s functions whose angular derivative is also in H s. Thenorm is defined by ‖ f ‖Hs,1 := ‖ f ‖Hs + ‖∇S f ‖Hs . Here ∇S is the gradient on the unitsphere and it can be represented as x × ∇. The angular momentum operator L is−i∇S.

Theorem 4.1.1. Let d = 2, 3, 1 < α < 2, α ≤ γ < 2 and γ > 2d2d−1 . Then there

32

CHAPTER 4. SCATTERING WITH ANGULAR DERIVATIVE

exists δ > 0 such that for any ϕ ∈ H sc,1 with ‖ϕ‖Hsc ,1 ≤ δ, (1.1.1) has a uniquesolution u ∈ (C ∩ L∞)(R; H sc,1) which scatters in H sc,1.

We show Theorem 4.1.1 essentially on the basis of the nonlinear estimates,which will be shown in Sections 4, 5, 6 below. The crucial part of the nonlinearestimate occurs in the case where high-high frequency interactions make low fre-quency outputs. Such case does not occur in the cubic problem. Thanks to theradial lemmas (Lemma 4.1.2 and Lemma 4.1.3 below) and endpoint Strichartzestimates ((4.1.1), (4.1.2) below) we can circumvent the singularity in the lowfrequency part in case that 2d

2d−1 < γ < 2. We hope to remove this restriction.This chapter is organized as follows: In Section 2 we give several preliminary

lemmas on the endpoint Strichartz estimates and radial functions. We prove The-orem 4.1.1 in Section 4.2 via nonlinear estimates. In Sections 4.2.1, 4.2.2, 4.2.3we provide a proof for the crucial nonlinear estimates.

4.1.1 Spherical Strichartz estimates and motivation

Let the pair (q, r) satisfy that 2 ≤ q, r ≤ ∞, 2q + d

r = d2 and (q, r, d) , (2,∞, 2).

Recall that from (3.2.1) we have for any 1 < α < 2 and k ∈ Z,

2−2−α

q k‖Sα(t)Pkϕ‖Lq

t Lrx(Rd) . ‖Pkϕ‖L2

x(Rd) (4.1.1)

The endpoint estimate can be extended to a wider range provided an angular av-erage is taken into account. More precisely, let d = 2, 3, 1 < α < 2, and let4d−22d−3 < r < 2d

d−2 , r∗ = 2 if d = 2 and 2 ≤ r∗ < 4r10−r if d = 3. Then there holds

2k(− d−α2 + d

r )‖Sα(t)Pkϕ‖L2t L

rρLr∗

θ. ‖Pkϕ‖L2

x. (4.1.2)

Here the norm LqtL

rρL

r∗θ with r < ∞ is defined by

‖u‖Lqt L

rρLr∗

θ=

∫R

∥∥∥∥∥( ∫Sd−1|u(t, ρθ)|r∗ dθ

) 1r∗

∥∥∥∥∥q

Lrρ(ρd−1dρ)

dt

1q

.

If r = ∞, then we define L∞ρ = L∞ρ . For (4.1.2) see Theorem 1.1 and Corollary 2.9of [16]. See also [6] for some general Strichartz estimates associated with angularregularity.

33


4.1.2 Some useful lemma

Lemma 4.1.2. Let ψ, f be smooth and let ψ be radially symmetric. Then

∇S(ψ ∗ f ) = ψ ∗ ∇S f .

Lemma 4.1.3 (Lemma 7.1 of [7]). If ψ(x) is radially symmetric, then

‖ψ ∗ f ‖LpρLq

θ≤ ‖ f ‖Lp1

ρ Lq1θ‖ψ‖Lp2

x,

for all p1, p2, p, q, q1 ∈ [1,∞] satisfying

1p1

+1p2− 1 =

1p,

1q1

+1p2− 1 ≤

1q.

The next is on the well-known convolution inequality for the sequence.

Lemma 4.1.4. For any f ∈ `1(Z) and g ∈ `2(Z) we have

‖ f ∗ g‖`2(Z) ≤ ‖ f ‖`1(Z)‖g‖`2(Z).

The final one is on the Sobolev inequality on the unit sphere.

Lemma 4.1.5. For any d − 1 < r < ∞

‖ f ‖L∞θ . ‖ f ‖W1,rθ

:= ‖ f ‖Lrθ+ ‖∇S f ‖Lr

θ,

‖ f ‖Lrθ. ‖ f ‖H1

θ:= ‖ f ‖L2

θ+ ‖∇S f ‖L2

θ.

For this see [11].

4.2 Proof of the main result

Let us define the Banach space Y s for s ∈ R by

Y s :={u : R→ H sc

∣∣∣∣ Pku,∇SPku ∈ U2α(R; L2

x) ∀N ≥ 1}

with the norm

‖u‖Y s =

∑k∈Z

N2sc‖Pku‖2U2,1α

12

, ‖w‖U2,1α

= ‖w‖U2α

+ ‖∇Sw‖U2α.

34


Clearly, ‖Sα(t)ϕ‖Y s . ‖ϕ‖Hs,1 for all ϕ ∈ H s,1. Thus as explained in Section 3.3.1,in order to prove Theorea 4.1.1 we suffices to show that

‖Nα(u1, u2, u3)‖Y sc . ‖u1‖Y sc ‖u2‖Y sc ‖u3‖Y sc , (4.2.1)

where Nα(u1, u2, u3)(t) = 1[0,∞)(t)∫ t

0Sα(t − t′)

[{| · |−α ∗ (u1u2)}u3

](t′)dt′. From the

duality form and Lemma 2.2.4 it follows that

‖PkNα(u1, u2, u3)‖U2α

= ‖Sα(t)PkNα(u1, u2, u3)‖U2

= sup‖v‖V2≤1

∣∣∣∣B(Sα(t)PkNα(u1, u2, u3), v

)∣∣∣∣. sup‖v‖V2≤1

∣∣∣∣ ∫R

∫Rd

(|x|−γ ∗ u1u2

)u3(t)Sα(t)Pkv(t) dxdt

∣∣∣∣. sup‖v‖V2

α≤1

∣∣∣∣ ∫R

∫Rd

(|x|−γ ∗ u1u2

)u3(t)Pkv(t) dxdt

∣∣∣∣and similarly

‖Pk∇SNα(u)‖U2α. sup‖v‖V2

α≤1

∣∣∣∣ ∫R

∫Rd∇S

[(|x|−γ ∗ u1u2

)u3

]Pkv(t) dxdt

∣∣∣∣.By Littlewood-Paley decomposition we get

‖Nα(u1, u2, u3)‖2Y sc

.∑k∈Z

22sck

sup‖v‖V2

α≤1

∑k1,k2,k3∈Z

∣∣∣∣ ∫ ∫|x|−γ ∗ (Pk1u1Pk2u2)Pk3u3(t)Pkv(t) dxdt

∣∣∣∣2

+ sup‖v‖V2

α≤1

∑k1,k2,k3∈Z

∣∣∣∣ ∫ ∫∇S

[|x|−γ ∗ (Pk1u1Pk2u2)Pk3u3(t)

]Pkv(t) dxdt

∣∣∣∣2 .(4.2.2)

35


To estimate (4.2.2) we consider more general situation: for any ui ∈ Y sc(i = 1, 2, 3)∑k>0

k2sck

sup‖v‖V2

α≤1

∑k1,k2,k3>0

∣∣∣∣ ∫ ∫|x|−γ ∗ (Pk1 u1Pk2 u2)Pk3 u3(t)Pkv(t) dxdt

∣∣∣∣2

+ sup‖v‖V2

α≤1

∑k1,k2,k3>0

∣∣∣∣ ∫ ∫∇S

[|x|−γ ∗ (Pk1 u1Pk2 u2)Pk3 u3(t)

]Pkv(t) dxdt

∣∣∣∣2

.∏

i=1,2,3

‖ui‖2Y sc ,

(4.2.3)

where ui denotes ui or ui.Now it remains to show (4.2.3). Let us split the summation of LHS of (4.2.3)

into three parts as follows:

LHS of (4.2.3) =: S1 +S2 +S3,

where

S1 =∑k3∼k

, S2 =∑k3�k

, S3 =∑k3�k

.

In the next three sections we show that Si .∏

i=1,2,3 ‖ui‖2Y sc .

4.2.1 Nonlinear estimate: S1

We split S1 = S11 +S12 +S13, where

S11 =∑

(1): k1∼k2k3∼k

, S12 =∑

(2): k1�k2k3∼k

, S13 =∑

(3): k1�k2k3∼k

.

Since the argument will not be affected by complex conjugation, we drop theconjugate symbol. Let us denote Pkiui by ui for i = 1, 2, 3 and Pkv by v.

We further decompose S1i using Lemma 4.1.2 as

S1i .∑

j=1,2,3

∑k∈Z

22sck sup‖v‖V2

α≤1

∑

(i)k3∼k

I j(k1, k2, k3, k)

2

=:∑

j=1,2,3

Sj1i,

36


where

I1 = |

"|x|−γ ∗ (u1u2)u3v dxdt|,

I2 = |

"|x|−γ ∗ (u1u2)(∇Su3)v dxdt|,

I3 = |

"|x|−γ ∗ (∇S(u1u2))u3v dxdt|.

ForS111 andS2

11 at first we have by Holder’s inequality and embedding lemmas(Lemma 2.2.3 and Lemma 2.2.8) that

I1 + I2 . B1,∞(u1u2) (‖u3‖L∞t L2x+ ‖∇Su3‖L∞t L2

x)‖v‖L∞t L2

x,

whereB1,∞(u1u2) :=

∑m.k1

‖Pm(|x|−γ ∗ (u1u2))‖L1t L∞x .

By choosing r such that 4d−22d−3 < r < 2d

d−γ (this is possible since γ > 2d2d−1 ) and then

applying Lemma 4.1.3 with p1 = r2 , p2 = r

r−2 and q1 = q = ∞ to the radial kernelof Pm, we get

B1,∞(u1u2) .∑m.k1

2−(d−γ)m‖Pm(u1u2)‖L1t L∞x

.∑m.k1

2m(−(d−γ)+ 2dr )

∏i=1,2

‖ui‖L2t Lr

x

. (2k12k2)−d−γ

2 + dr

∏i=1,2

(‖ui‖L2t L

rρL2

θ+ ‖∇Sui‖L2

t LrρL2

θ) (W1,2

θ ↪→ Lrθ).

(4.2.4)

Hence the endpoint Strichartz estimates (4.1.2) and Lemma 2.2.9 give us

S111 +S2

11 .∑

k

22sck

∑k1∼k2k3∼k

(2k12k2)sc∏

i=1,2,3

‖ui‖U2,1α

2

.∏

i=1,2,3

‖ui‖2Y sc .

We used Lemma 4.1.4 w.r.t. k1, k2, and k3, k.Next by Holder’s inequality we have for I3 of S3

11 that

I3 . B1,∞,d(u1u2) B∞,1,d

d−1 (u3v),

37


where

B1,∞,d(u1u2) =∑m.k1

2−m(d−γ)‖Pm(∇S(u1u2))‖L1t L∞ρ Ld

θ,

B∞,1,d

d−1 (u3v) = ‖u3v‖L∞t L

1ρL

dd−1θ

.

By Lemma 4.1.3 with p1 = r2 , p2 = r

r−2 with the same r as above and q = q1 = 2we estimate

B1,∞,d(u1u2)B∞,1,d

d−1 (u3v)

.∑m.k1

2m(−(d−γ)+ 2dr )‖∇S(u1u2)‖

L1t L

r2ρ Ld

θ

‖u3‖L∞t L

2ρL

2dd−2θ

‖v‖L∞t L2x

. (2k12k2)−d−γ

2 + dr

∏i=1,2

(‖ui‖L2t L

rρLd

θ+ ‖∇Sui‖L2

t LrρLd

θ)‖u3‖

L∞t L2ρL

2dd−2θ

.

Using the endpoint Strichartz estimate of (4.1.2) and Lemma 2.2.9 again we getby the same summation argument as above that

S311 .

∑k∈Z

22sck

∑k1∼k2k3∼k

(2k12k2)sc∏

i=1,2,3

‖ui‖U2,1α

2

.∏

i=1,2,3

‖ui‖2X+.

Now let us consider S j12. As for S1

12,S212, we have from the support condition

k1 � k2 and (4.2.4) with 2dd−γ < r < 2d

d−2 (in this case −d−γ2 + d

r < 0) that

I1 + I2 .∑m∼k2

2−m(d−γ)‖PM(u1u2)‖L1t L∞x (‖u3‖L∞t L2

x+ ‖∇Su3‖L∞t L2

x)‖v‖L∞t L2

x

.∑m∼k2

2m(−(d−γ)+ 2dr )‖u1u2‖

L1t L

r2ρ L∞θ

(‖u3‖L∞t L2x+ ‖∇Su3‖L∞t L2

x)‖v‖L∞t L2

x

. 2k2(−(d−γ)+ 2dr )‖u1‖L2

t LrρL∞θ‖u2‖L2

t LrρL∞θ

(‖u3‖L∞t L2x+ ‖∇Su3‖L∞t L2

x)‖v‖L∞t L2

x

38


and hence

S112 +S2

12 .∑k∈Z

22sck

∑

k1�k2k3∼k

2(k1−k2)( d−γ2 −

dr )(2k12k2)sc

∏i=1,2,3

‖ui‖U2,1α

2

.

∑k1�k2

2(k1−k2)( d−γ2 −

dr )(2k12k2)sc

∏i=1,2

‖ui‖U2,1α

2

‖u3‖2Y sc

.∏

i=1,2,3

‖ui‖2Y sc .

For the final inequality we first used Cauchy-Schwarz inequality in k1 and thenLemma 4.1.4 w.r.t k1, k2.

To treat I3 of S312 we use almost the same argument as I3 of S3

11 with the onlychange 2d

d−γ < r < 2dd−2 and by the same summation argument as above we get

I3 .∑m∼k2

2−m(d−γ)‖Pm(∇S(u1u2))‖L1t L∞ρ Ld

θ‖u3v‖

L∞t L1ρL

dd−1θ

. 2k2(−(d−γ)+ 2dr )‖∇S(u1u2)‖

L1t L

r2ρ Ld

θ

‖u3‖L∞t L

2ρL

2dd−2θ

‖v‖L∞t L2x.

and hence

S312 .

∑k∈Z

22sck

∑

k1�k2k3∼k

2k2(−(d−γ)+ 2dr )

∏i=1,2

(‖ui‖L2t L

rρLd

θ+ ‖∇Sui‖L2

t LrρLd

θ)‖u3‖

L∞t L2ρL

2dd−2θ

2

.∑

N

N2sc

∑N1�N2

2(k1−k2)( d−γ2 −

dr )(2k12k2)sc

∏i=1,2

‖ui‖U2,1α

2

‖u3‖2Y sc

.∏

i=1,2,3

‖ui‖2Y sc .

As for S13 we have only to change the role of u1 and u2. This finishes theproof for S1.

39



We split as S2 = S21 +S22 +S23, where

S21 =∑

(1): k1∼k2k3�k

, S22 =∑

(2): k1�k2k3�k

, S23 =∑

(3): k1�k2k3�k

.

Then for each i = 1, 2, 3 Plancherel’s theorem writes S2i =∑

j=1,2,3Sj2i, where

Sj2i = cd

∑k∈Z

22sck sup‖v‖V2

α≤1

∑

(i)k3�k

I j(k1, k2, k3, k)

2

,

where

I1 =∣∣∣∣ ∫ ∫

|∇|−d−γ

2 (u1u2)|∇|−d−γ

2 (u3v) dxdt∣∣∣∣,

I2 =∣∣∣∣ ∫ ∫

|∇|−d−γ

2 (u1u2)|∇|−d−γ

2 ((∇Su3)v) dxdt∣∣∣∣,

I3 =∣∣∣∣ ∫ ∫

|∇|−d−γ

2 [∇S(u1u2)]|∇|−d−γ

2 (u3v) dxdt∣∣∣∣.

Let us first consider S121. By Holder’s inequality we first estimate

I1 . B2,2, s2 (u1u2)B2,2, s

s−2 (u3v),

where 1 � s < ∞,

B2,2, s2 (u1u2) =

∑m.k1∼k2

‖Pm(|∇|−d−γ

2 (u1u2))‖L2

t L2ρL

s2θ

,

B2,2, ss−2 (u3v) =

∑m∼k

‖Pm(|∇|−d−γ

2 (u3v))‖L2

t L2ρL

ss−2θ

.

Note that m . k1 ∼ k2 from the support conditions.Applying Lemma 4.1.3 with p = 2, p1 = 2r

r+2 , p2 = r′, q = q1 = s2 for 4d−2

2d−3 <

r < 2dd−γ and p = 2, p1 = 2r

r+2 , p2 = r′, q = q1 = ss−2 for 2d

d−γ < r < 2dd−2 to u1u2 and

u3v, respectively, we get

B2,2, s2 (u1u2) . 2k1(− d−γ

2 + dr )‖u1u2‖

L2t L

2rr+2ρ L

s2θ

. 2k1(− d−γ2 + d

r )‖u1‖L2t L

rρLs

θ‖u2‖L∞t L

2ρLs

θ,

B2,2, ss−2 (u3v) . 2k(− d−γ

2 + dr )‖u3v‖

L2t L

2rr+2ρ L

ss−2θ

. 2k(− d−γ2 + d

r )‖u3‖L2

t LrρL

2ss−4θ

‖v‖L∞t L2x.

40


and hence by taking s with 2ss−4 <

4r10−r we have from Lemma 4.1.4 that

S121 .

( ∑k1∼k2

(2k12k2)sc∏i=1,2

‖ui‖U2,1α

)2 ∑k∈Z

∑k3�k

2(k3−k)( d−γ2 −

dr )2sck3‖u3‖U2

α

2

.∏

i=1,2,3

‖ui‖2Y sc .

By replacing u3 with ∇Su3 we also get S221 .

∏i=1,2,3 ‖ui‖

2Y sc .

As for I3 of S321 we take the same r, r as above and s � 1. Then since k1 ∼ k2,

we get

I3 . 2k1(− d−γ2 + d

r )‖∇S(u1u2)‖L2

t L2r

r+2ρ Ld

θ

2k(− d−γ2 + d

r )‖u3v‖L2

t L2r

r+2ρ L

dd−1θ

. 2k1(− d−γ2 + d

r )‖∇Su1‖L2

t LrρL

dss−dθ

‖u2‖L∞t L2ρLs

θ2k(− d−γ

2 + dr )‖u3‖

L2t L

rρL

2dd−2θ

‖v‖L∞t L2x

+ ‖u1‖L∞t L2ρLs

θ2k2(− d−γ

2 + dr )‖∇Su2‖

L2t L

rρL

dss−dθ

2k(− d−γ2 + d

r )‖u3‖L2

t LrρL

2dd−2θ

‖v‖L∞t L2x

and hence

S321 .

( ∑k1∼k2

(2k12k2)sc∏i=1,2

‖ui‖U2,1α

)2 ∑k

∑k3�k

2(k3−k)( d−γ2 −

dr )2k3 sc‖u3‖U2

α

2

.∏

i=1,2,3

‖ui‖2Y sc .

Now let us take onS j22. Note that k ∼ k2. For I1 ofS1

22 we carry out almost thesame argument for I1 of S1

21. The only difference is to take r with 2dd−γ < r < 2d

d−2

so that −d−γ2 + d

r < 0.

I1 .( ∑

m∼k2

‖Pm(D−d−γ

2 (u1u2))‖L2

t L2ρL

s2θ

)(∑m∼k

‖Pm(D−d−γ

2 (u3v))‖L2

t L2ρL

ss−2θ

). 2k2(− d−γ

2 + dr )‖u1u2‖

L2t L

2rr+2ρ L

s2θ

2k(− d−γ2 + d

r )‖u3v‖L2

t L2r

r+2ρ L

ss−2θ

. 2k2(− d−γ2 + d

r )‖u1‖L2t L

rρLs

θ‖u2‖L∞t L

2ρLs

θ2k(− d−γ

2 + dr )‖u3‖

L2t L

rρL

2ss−4θ

‖v‖L∞t L2x

41


and hence

S122 .

( ∑k1�k2

2(k1−k2)( d−γ2 −

dr )(2k12k2)sc

∏i=1,2

‖ui‖U2,1α

)2

×∑

k

( ∑k3�k

2(k3−k)( d−γ2 −

dr )2sck3‖u3‖U2

α

)2.

Then Lemma 4.1.4 givesS122 .

∏i=1,2,3 ‖ui‖

2Y sc . By replacing u3 with ∇Su3 we also

get S222 .

∏i=1,2,3 ‖ui‖

2Y sc .

As for I3 of S322 we take the same r, r as above and s � 1, and get

I3 . 2k2(− d−γ2 + d

r )‖∇S(u1u2)‖L2

t L2r

r+2ρ Ld

θ

2k(− d−γ2 + d

r )‖u3v‖L2

t L2r

r+2ρ L

dd−1θ

. 2k2(− d−γ2 + d

r )‖∇Su1‖L2

t LrρL

dss−dθ

‖u2‖L∞t L2ρLs

θ2k(− d−γ

2 + dr )‖u3‖

L2t L

rρL

2dd−2θ

‖v‖L∞t L2x

+ 2k2(− d−γ2 + d

r )‖u1‖L2t L

rρLs

θ‖∇Su2‖

L∞t LrρL

dss−dθ

2k(− d−γ2 + d

r )‖u3‖L2

t LrρL

2dd−2θ

‖v‖L∞t L2x

and hence

S322 .

( ∑k1�k2

2(k1−k2) d−γ2 −

dr (2k12k2)sc

∏i=1,2

‖ui‖U2,1α

)2

×∑

k

∑k3�k

2(k3−k)( d−γ2 −

dr )2sck3‖u3‖U2

α

2

.∏

i=1,2,3

‖ui‖2Y sc .

The final sum S23 can be treated by changing the role of u1 and u2 proves thecase. This proves the part S2.


We split as S3 . S31 +S32 +S33, where

S31 =∑

(1): k1∼k2k3�k

, S32 =∑

(2): k1�k2k3�k

, S23 =∑

(3): k1�k2k3�k

.

42


As previously we decompose S3i as S3i =∑

j=1,2,3Sj3i.

By Holder’s inequality we estimate with s � 1

S131 .

∑k∈Z

22sck sup‖v‖V2

α≤1

( ∑k1∼k2k3�k

( ∑m.k1∼k2

‖Pm(|∇|−d−γ

2 (u1u2))‖L2

t L2ρL

s2θ

)×

‖|∇|−d−γ

2 (u3v)‖L2

t L2ρL

ss−2θ

)2

.

Note that k3 . k1 ∼ k2.Let us first observe from Bernstein’s inequality and (4.1.1) that

‖|∇|−d−γ

2 (u3v)‖L2

t L2ρL

ss−2θ

. ‖|∇|−d−γ

2 (u3v)‖L2t L2

x. N−

d−γ2

3 ‖u3‖L4t L4

x‖v‖L4

t L4x

. 2(k−k3) d−α4 N sc

3 N−2−α

43 ‖u3‖

L4t L

2dd−1x

N−2−α

4 ‖v‖L4

t L2d

d−1x

. 2(k−k3) d−α4 N sc

3 ‖u3‖U4α‖v‖U4

α

. 2(k−k3) d−α4 N sc

3 ‖u3‖U2α‖v‖U4

α.

(4.2.5)

On the other hand, applying Lemma 4.1.3 for p = 2, p1 = 2rr+2 , p2 = r′, q = q1 =

ss−2 with r = 4d

d−α , we have

‖|∇|−d−γ

2 (u3v)‖L2

t L2ρL

ss−2θ

.∑m∼k3

‖Pm(|∇|−d−γ

2 (u3v))‖L2

t L2ρL

ss−2θ

. 2k3(− d−γ2 + d

r )‖u3v‖L2

t L2r

r+2ρ L

ss−2θ

. 2k3(− d−γ2 + d

r )‖u3‖L∞t L2x‖v‖

L2t L

rρL

2ss−4θ

.

If d = 3, then the Bernstein’s inequality yields

‖v‖L2

t LrρL

2ss−4θ

. 2k( 12−

3r )‖v‖L2

t L6x. 2k( 3−α

2 −3r )2k(− 2−α

2 )‖v‖L2t L6

x.

From the Strichartz estimate (4.1.1) and transfer principle it follows that

‖v‖L2

t LrρL

2ss−4θ

. 2k( 3−α2 −

3r )‖v‖U2

α.

43


If d = 2, then ‖v‖L2

t LrρL

2ss−4θ

. N2−α

2 −2r ‖v‖U2

α. This implies

‖|∇|−d−γ

2 (u3v)‖L2

t L2ρL

ss−2θ

. 2(k−k3)( d−α2 −

dr )2sck3‖u3‖U2

α‖v‖U2

α

. 2(k−k3) d−α4 2sck3‖u3‖U2

α‖v‖U2

α.

(4.2.6)

Putting (4.2.5) and (4.2.6) together through the V2-decomposition of Lemma 2.2.5with eit|∇|αv = w + z and K = 1, and then replacing v with e−it|∇|αw and e−it|∇|αz in(4.2.5) and (4.2.6), we get

‖|∇|−d−γ

2 (u3v)‖L2

t L2ρL

mm−2θ

. 2(k−k4) d−α4 2k3 sc‖u3‖U2

α‖v‖V2

α.

By the argument of S121 we already have

2ksc∑

m.k1∼k2

‖Pm(|∇|−d−γ

2 (u1u2))‖L2

t L2ρL

s2θ

.∑k1∼k2

(2k12k2)sc∏i=1,2

(‖ui‖U2α

+ ‖∇Sui‖U2α) .

∏i=1,2

‖ui‖2Y sc .

By Lemma 4.1.4 we conclude that

S131 .

∏i=1,2

‖ui‖2Y sc

∑k

∑k3�k

2(k−k3) d−α4 2k3 sc‖u3‖U2

α

2

.∏

i=1,2,3

‖ui‖2Y sc .

Since S231 contains ∇Su3 instead of u3, this bound also holds true for S2

31. Theargument of S3

21 can be applied to ∇S(u1u2). Therefore

S31 .∏

i=1,2,3

‖ui‖2Y sc .

Now we treat S32. For S132 and S2

32, by using S122 w.r.t. u1u2 and S1

31 w.r.t. u3vor ∇Su3v, and forS3

32 by usingS322 w.r.t. ∇S(u1u2) andS1

31 w.r.t. u3v or ∇Su3v, wereadily get S32 .

∏i=1,2,3 ‖ui‖

2Y sc . Changing the role of u1 and u2 proves the hexic

bound for S33. This is the end of estimate for S3.

44

Chapter 5

Modified scattering

5.1 Main result

We are concerned with the following Cauchy problem:i∂tu − |∇|αu = κ(|x|−1 ∗ |u|2

)u in R1+3,

u(0, x) = u0(x).(5.1.1)

By Duhamel’s formula (1.1.3) is written as for v(t, x) := eit|∇|αu(x)

v = u0(ξ) +

∫ t

0I(s, ξ) ds,

whereI(s, ξ) = ic0

∫R3

eisφα(ξ,η)|η|−2 |u|2(s, η)v(s, ξ − η) dηds. (5.1.2)

In this chapter we focus on asymptotic behavior of solution to (5.1.1) as timegoes to infinity. We say that the solution u scatters to a linear asymptotic state ifthe effect of the nonlinear term becomes negligible as time goes to infinity. Butour equation may not scatter even though the initial data is arbitrarily small [5].Instead, we can observe the phenomenon of “modified scattering” for small so-lutions by identifying a proper nonlinear logarithmic correction. Our main topiccan be stated as follows: For sufficiently small initial data u0 which is defined

45

CHAPTER 5. MODIFIED SCATTERING

in a weighted Sobolev space, there exist a global solution to (5.1.1) which de-cays in L∞ but behaves in nonlinear fashion over time. Our goal is to show thatthe global solution to (5.1.1) with the long-range (γ = 1) scatters in L∞ on thefrequency space. Heuristically speaking, the time decay of L2-norm of nonlinearterm, which is computed on a linear solution, is t−γ. For the details see the proofin [5, Theorem1.2]. In particular, the decay of nonlinear term is not integrable intime if γ = 1, which referred as “scattering-critical” case. The following is ourmain theorem.

Theorem 5.1.1 (Modified scattering). Let 1710 < α < 2 and N = 1500. Suppose u0

satisfies that

‖u0‖HN + ‖xu0‖H3 + ‖x2u0‖H2 + ‖〈ξ〉5u0‖L∞ ≤ ε0. (5.1.3)

Then there exists ε0 such that for all ε0 ≤ ε0, the Cauchy problem (5.1.1) has aunique global solution u(t, x) such that

supt>0〈t〉

32 ‖u(t)‖L∞ . ε0. (5.1.4)

Moreover, u satisfies the asymptotic behavior as follows: Let

Bα(t, ξ) := −κ

α(2π)3

∫ t

0

∫R3

∣∣∣∣∣ ξ

|ξ|2−α−

σ

|σ|2−α

∣∣∣∣∣−1

|u(ξ)|2dσϕ(s−θξ)1〈s〉

ds, (5.1.5)

where ϕ is a smooth compactly supported function and θ = 3α−540(α+1) . Then there

exist asymptotic state v+, such that for all t > 0

‖〈ξ〉5[e−iBα(t,ξ)v(t, ξ) − v+(ξ)]‖L∞ξ . 〈t〉−δ (5.1.6)

for some 0 < δ < min(2−α3α ,

1100 ). Similar result holds for t < 0.

Our approach is inspired by the work [28] of Pusateri developed to studysemi-relativistic equations. As stated in [28] we prefer to state asymptotic statein the frequency space because the formulae (5.1.5) and (5.1.6) appear explicitlyin the proof. We show the uniform norm estimate (5.1.4) and asymptotic behavior

46


(5.1.6) for sufficiently small initial data via refined time-decay and weighted en-ergy estimates in frequency space, which are rephrased as Proposition 5.3.1 andPropositions 5.4.2, 5.5.1, respectively.

Let us briefly give some intuition for the formula (5.1.5). For simplicity weassume that |ξ| ∼ 1 and |η| . 2n with n < 0. Applying the Taylor expansion to thephase function φ of (1.4.5), we approximate I by

ic1

"R3×R3

eiαs(η·z)|η|−2v(s, ξ + η)v(s, ξ + η + σ)v(s, ξ + σ)dηdσ

= ic1v(s, ξ)∫R3F −1(|η|−2)(sz)|v(s, σ)|2dσ + [err],

where z =ξ

|ξ|2−α− σ|σ|2−α

and c1F−1(|η|−2) = −κ(2π)−3|x|−1. This formula yields an

insight for (5.1.5) and (5.1.6). In Section 5 below [err] will turn out to be O(s−1−)as s → ∞. The contribution of remaining region |η| & 2n is shown to decay fasterthan s−1 by making integration by parts twice.

Similarly to the case of short-range potential with γ close to 1, we could notobtain a modified scattering in the whole range 1 < α < 2 for the present. This isdue to the lack of smoothness of |∇|α near zero frequency. This is the main differ-ence of fractional from the usual Schrodinger or semi-relativistic equations. Thedrawback can be overcome by the refined time-decay estimate. But it is inevitableto control at least the L2 norm of x2v for the requested time decay. Among theterms from taking ∇2

ξ to v and I in (5.1.2), the following is the most worst casewhen we consider small α:

c0

∫ t

0s∫R3

[∇ξ ⊗ ∇ξφα]eisφα(ξ,η)|η|−2 |u|2(η)v(ξ − η)dηds.

In this expression, twice differentiation of the phase function φα rises to singularitynear 0 of order α − 2, which makes a problem in bounding the low frequency partwhen α becomes closer to 1. In the derivation of the asymptotic correction term asabove on the region |η| . 2n, we require the integral of |η|−2+α over this region tobe O(s−2−). This condition also restricts the range of α. One of our next subjectswill be to remove the gap on α.

47


Remark 5.1.2. The indices α,N, δ, θ are not sharp and can be adjusted. For exam-ple, the range of α can be made slightly less than 17

10 by defining θ = 3α−51000(α+1) . See

conditions on the indices in Propositions 5.4.2 and 5.5.1.

Remark 5.1.3. One can obtain similar refined time-decay estimate for high dimen-sional case such that

‖e−it|∇|αv‖L∞(Rd) . 〈t〉− d

2 ‖〈ξ〉av‖L∞(Rd)

+ 〈t〉−32−δ

(‖x[ d

2 ]+1v‖L2(Rd) + ‖v‖HN (Rd)),

for some a(α, d) > 1 and δ(α, d) � 1 and for sufficiently large N(a, δ). It is highlyexpected to extend Theorem 5.1.1 to the high dimensional case (d ≥ 4) eventhough there will be much more complexity in frequency space analysis arisingfrom the [d/2] + 1-times differentiation of the phase function φ.

This paper is organized as follows: In Section 2, we deal with the global well-posedness and evolution of xv, x2v and 〈ξ〉5v. Section 3 is devoted to proving therefined time-decay estimate. In Section 4, we establish the weighted energy es-timate based on the Littlewood-Paley theory. Main effort is made to overcomethe singularity from differentiation. In Section 5 we move on to the last step forthe proof of modified scattering. In the last section we list lemmas for multiplierestimates and bounds for |z|.

5.2 Global well-posedness

In this section we establish a global theory.

Theorem 5.2.1. (1) Let 1 < α < 2 and N ≥ α2 . If u0 ∈ HN , then there exists a

unique solution u ∈ C(R; HN) satisfying mass and energy conservations.(2) Let 1 < α < 2 and N ≥ 5. Assume that

u0 ∈ HN , xu0 ∈ H3, x2u0 ∈ H2.

Then there exists a unique solution u to (5.1.1) such that

u ∈ C(R; HN), ∂tu ∈ C(R; HN−α), xv ∈ C(R; H3), x2v ∈ C(R; H2).

48


Moreover, if we further assume that 〈ξ〉5u0 ∈ L∞, then 〈ξ〉5v ∈ C(R; Cb(R3)).

Proof. For the proof of the global well-posedness in HN and conservations, werefer the readers to Theorem 3.3 of [4]. The control norm of well-posedness is‖u‖

H12. That is, if ‖u‖

H12

is finite at some time, then the solution evolves beyondthe time in HN . Regardless of the sign of κ, one can show that the control norm isuniformly bounded in time due to the subcritical nature of (5.1.1). In fact, if κ < 0,then by interpolation and conservation laws we have

‖u‖H

12≤ ‖u‖

α−1α

L2 ‖|∇|α2 u‖

1α

L2 ≤ 21

2α m(u0)α−12α

(|E(u0)| − V(u)

) 12α ,

where V(u) = − κ4

!|x − y|−1|u(x)|2|u(y)|2 dxdy. Hardy-Sobolev inequality gives

−V(u) ≤ Cm(u0)‖u‖2H

12

and hence

‖u‖H

12≤ 2

12α m(u0)

α−12α |E(u0)|

12α + (2C)

12α m(u0)

12 ‖u‖

1α

H12.

Now Young’s inequality yields ‖u‖H

12≤ C(m(u0), E(u0)).

By direct calculation we get xv = αt|∇|α−2∇u + eit∇|α xu and

x2v = αt((α − 2)|∇|α−4∇2 + |∇|α−2I)u + αt|∇|α−2∇ ⊗ xu + eit|∇|α x2u.

Since ‖|∇|α−2u‖L2 . ‖|x|2−αu‖L2 by Hardy-Sobolev inequality, to show that xv ∈C(R; H3), x2v ∈ C(R; H2) we have only to take into account xu(t) ∈ C(R; H3),x2u(t) ∈ C(R; H2). But this can be done by the standard approximation withψε(x) = e−ε|x|

2. We treat this part in Lemma 5.2.2 below.

Lastly, we show that 〈ξ〉5v ∈ C(R; Cb(R3)). Since x2v ∈ L2, v ∈ Cb(R3). Weshow the time continuity on [0,∞). The continuity on the negative time interval

49


can be shown by symmetry. From (5.1.2) it follows that

‖〈ξ〉5v(t)‖L∞ξ

. ‖〈ξ〉5u0‖L∞ξ +

∫ t

0

∥∥∥〈ξ〉5 ∫R3

eis(|ξ|α−|ξ−η|α)|η|−2 |u|2(η)v(ξ − η)dη∥∥∥

L∞ξds

. ‖〈ξ〉5u0‖L∞ξ +

∫ t

0

∥∥∥ ∫R3|η|−2|η|5 |u|2(η)v(ξ − η)dη

∥∥∥L∞ξ

ds

+

∫ t

0

∥∥∥ ∫R3|η|−2 |u|2(η)〈ξ − η〉5v(ξ − η)dη

∥∥∥L∞ξ

ds

. ‖〈ξ〉5u0‖L∞ξ +

∫ t

0‖v‖L2

ξ‖∇3|u|2‖L2

ξds +

∫ t

0‖〈ξ〉5v‖L∞ξ

∥∥∥|η|−2 |u|2(η)∥∥∥

L1ηds.

Now for the second integral we use the estimate∫|η|−2||u|2| dη =

∫|η|≤1

+

∫|η|>1. ‖u‖2L2 + ‖u‖2L4

and get

‖〈ξ〉5v(t)‖L∞ξ . ‖〈ξ〉5u0‖L∞ξ +

∫ t

0‖u‖3H3 ds +

∫ t

0‖〈ξ〉5v‖L∞ξ ‖u‖

2H1 ds. (5.2.1)

Therefore by Gronwal’s inequality we obtain that for each t > 0

‖〈ξ〉5v(t)‖L∞ξ .(‖〈ξ〉5u0‖L∞ξ +

∫ t

0‖u‖3H3 ds

)eC

∫ t0 ‖u‖

2H1 ds

< ∞.

Then time continuity readily follows from considering the inequality

‖〈ξ〉5(v(t) − v(t′))‖L∞ .∫ t

t′

(‖u‖3H3 + ‖〈ξ〉5v‖L∞ξ ‖u‖

2H1

)ds

for any 0 ≤ t′ < t. This completes the proof of Theorem 5.2.1.�

Lemma 5.2.2. Let u be the solution to (5.1.1) belong to C(R; H5) with ∂tu ∈C(R; H5−α) and initial data u0 such that xu0 ∈ H3, x2u0 ∈ H2. Then xu ∈ C(R; H3),x2u ∈ C(R; H2).

50


Proof of Lemma 5.2.2. Let us set ψε(x) = e−ε|x|2. Let u = ∇`u for 0 ≤ ` ≤ 3, and

M`,ε(t) =

∫u(t) : |x|2ψ2

εu(t) dx.

From the regularity of the solution u it follows that

ddtM`,ε(t) = 2Im

∫u : [|∇|α, |x|2ψ2

ε]u dx

+ 2κIm∫|x|ψεu : |x|ψε∇`((|x|−1 ∗ |u|2)u) dx

=: 2( A` + B` ).

(5.2.2)

We rewrite A` as

A` = Im∫|x|ψεu : [|∇|α(1 − ∆)−1, |x|ψε](1 − ∆)u dx

+ Im∫|∇|α(1 − ∆)−1(|x|ψεu) : [1 − ∆, |x|ψε]u dx =: A`,1 + A`,2.

Here [T, S ] denotes the commutator TS − S T . By the kernel representation of|∇|α(1 − ∆)−1, we have

|[||∇|α(1 − ∆)−1, |x|ψε|](1 − ∆)u(x)|

≤

∫K(x − y)||x|ψε(x) − |y|ψε(y)||(1 − ∆)uu(y)| dy

.

∫K(x − y)|x − y||(1 − ∆)u(y)| dy.

Since (1 + |x|)N K is integrable for all N ≥ 1 (see [?], Cauchy-Schwarz inequalitygives

A`,1 .√M`,ε ‖u‖H`+2 .

As for A`,2 we have

A`,2 = −Im∫|∇|α(1 − ∆)−1(|x|ψεu) :

(∆(|x|ψε)u + 2∇(|x|ψε) · ∇u

)dx

.√M`,ε ‖u‖H`+1 .

51


Now we proceed to estimate B. If u = u, then B = 0. For the case u = ∇`uwith ` > 0 let us observe that

B` .√M`,ε

∑1≤`′≤`

‖|x|∇`′

(|x|−1 ∗ |u|2)‖L∞‖∇`−`′

u‖L2

.√M`,ε

∑1≤`′≤`

‖|x|(|x|−1 ∗ ∇`′

(|u|2)‖L∞‖∇`−`′

u‖L2 .

By Young’s and Hardy-Sobolev inequalities we have∣∣∣|x|(|x|−1 ∗ ∇`′

(|u|2))∣∣∣ ≤ ∫

∇`′

(|u|2) dy +

∫|x − y|−1|y|∇`

′

(|u|2) dy

. ‖xu‖2H`′−1 + ‖u‖2H`′+1 .

By integrating (5.2.2) over [0, t] we have√M0,ε . ‖xu0‖L2 +

∫ t

0‖u‖H2 ds,√

M`,ε . ‖xu0‖H` +

∫ t

0(‖u‖H`+2 + ‖u‖3H`+1 + ‖xu‖2H`−1‖u‖H`−1) ds,

Fatou’s lemma and induction lead us to xu(t) ∈ L∞(K; H3) for any compact inter-val K ⊂ R provided xu0 ∈ H3. By using this fact and the equation (5.1.1) we canconclude that xu ∈ C(R; H3).

Let us move onto the proof of x2u ∈ C(R; H2). Let us set u = (1 − ∆)u and

Mε(t) =

∫u(t)|x|4ψ2

εu(t) dx.

Differentiating w.r.t t we have as above thatddtMε(t) = 2Im

∫u[|∇|α, |x|4ψ2

ε ]u dx

+ 2κIm∫|x|2ψεu|x|2ψε(1 − ∆)((|x|−1 ∗ |u|2)u) dx

=: 2( A + B ).

A is written as

A = Im∫|x|2ψεu[|∇|α(1 − ∆)−1, |x|2ψε](1 − ∆)u

⟩+ Im

∫|∇|α(1 − ∆)−2(|x|2ψεu)[1 − ∆, |x|2ψε]u dx =: A1 + A2.

52


Then we have

|[||∇|α(1 − ∆)−1, |x|2ψε|](1 − ∆)u(x)|

≤

∫K(x − y)||x|2ψε(x) − |y|2ψε(y)||(1 − ∆)u(y)| dy

.

∫K(x − y)|x − y|(|x − y| + 2|y|)|(1 − ∆)u(y)| dy

and hence

A1 .

√Mε (‖u‖H4 + ‖xu‖H2).

As for A2 we have

A2 = −Im∫|∇|α(1 − ∆)−1(|x|2ψεu)

(∆(|x|2ψε)v + 2∇(|x|2ψε) · ∇u

)dx

.

√Mε (‖u‖H2 + ‖xu‖H3).

B can be treated similarly to B` as follows:

B .√Mε

(‖|x|(1 − ∆)(|x|−1 ∗ |u|2)‖L∞‖|x|u‖L2

+ ‖|x|∇(|x|−1 ∗ |u|2)‖L∞‖|x|∇u‖L2)

.

√Mε ‖xu‖2H1‖u‖H2 .

Combining these estimates and the previous argument, we deduce that x2u ∈C([0,∞); H2) since u ∈ C([0,∞); H5) and xu ∈ C([0,∞); H3). At the negativetime, we can carry out the same argument. This completes the proof of Lemma5.2.2. �

5.3 Time decay

Proposition 5.3.1. Let 1 < α < 2, t ∈ R and N0 > 4. Then we have

‖e−it|∇|αv‖L∞ . 〈t〉−32 ‖〈ξ〉4−2αv‖L∞ + 〈t〉−

32−( 1

4−1

N0)(‖x2v‖L2 + ‖v‖HN

),

if N ≥ 72 N0(2 − α) + 2α − 5

2 .

53


Proof of Proposition 5.3.1. We assume that t ≥ 0.Since ‖e−it|∇|αv‖L∞ . ‖v‖HN for N > 3

2 , we may assume t > 1. We write bydyadic decomposition in the Fourier side

‖e−it|∇|αv‖L∞ . supx

∑k∈Z

Ik(t, x),

where

Ik(t, x) =1

(2π)3

∣∣∣ ∫ eiφt,x(ξ)βk(ξ)v(ξ)dξ∣∣∣, φt,x(ξ) = −t |ξ|α + x · ξ. (5.3.1)

For the low frequency part of summation we estimate∑2k≤〈t〉−

12

Ik(t, x) .∑

2k≤〈t〉−12

‖βk‖L1 ‖v‖L∞ . 〈t〉−32 ‖v‖L∞ .

And for the high frequency part we have that∑2k≥〈t〉

12N0(2−α)

Ik(t, x) .∑

2k≥〈t〉1

2N0(2−α)

23k2 ‖βkv‖L2

.∑

2k≥〈t〉1

2N0(2−α)

23k2 2−Nk‖v‖HN

. 〈t〉−1

2N0(2−α) (N− 32 )‖v‖HN .

If N ≥ 72 N0(2 − α) + 2α − 5

2 , then∑2k≥〈t〉

12N0(2−α)

Ik(t, x) . 〈t〉−32−( 1

4−1

N0)‖v‖HN . (5.3.2)

Now, let us bound the remaining part:∑〈t〉−

12 ≤2k≤〈t〉

12N0(2−α)

∣∣∣∣∣∫ ei(−t|ξ|α+x·ξ)βk(ξ)v(ξ)dξ∣∣∣∣∣ . (5.3.3)

First, we consider the non-stationary case. We write the phase function as

∇φt,x(ξ) = −αt(ξ

|ξ|2−α−

ξ0

|ξ0|2−α ), where ξ0 =

(|x|αt

) 1α−1 x|x|. (5.3.4)

54


For ξ0 in (5.3.4), there exist k0 ∈ Z such that |ξ0| ∼ 2k0 . Then by Lemma 2.3.3, wesee that if |k − k0| ≥ 4

|∇φt,x(ξ)| & |t|2(α−1)k for |ξ| ∼ 2k.

By taking integration by parts twice in the expression (5.3.3), we get∑k∈[Non−S tat]

Ik(t, x) . A + B + C,

where [Non − S tat] denotes the set {k : 〈t〉−12 ≤ 2k ≤ 〈t〉

12N0(2−α) , |k − k0| > 4},

A =∑

k∈[Non−S tat]

t−22−2αk‖βkv‖L1 ,

B =∑

k∈[Non−S tat]

t−22(−2α+1)k‖∇(βkv)‖L1 ,

C =∑

k∈[Non−S tat]

t−22(−2α+2)k‖∇2(βkv)‖L1 .

We denote briefly vk = Pkv, then it holds βkv = βkvk. Holder’s and Bernstein’sinequalities give us that

A .∑〈t〉−

12 ≤2k

t−22−2αk‖βk‖L1‖vk‖L∞ .∑〈t〉−

12 ≤2k

t−22−2αk23k‖vk‖L∞

.∑〈t〉−

12 ≤2k

t−22−k‖〈ξ〉4−2αv‖L∞ . t−32 ‖〈ξ〉4−2αv‖L∞ .

By Sobolev embedding and Plancherel’s theorem we have

‖∇v‖L6 . ‖∇2v‖L2 . ‖x2v‖L2 . (5.3.5)

55


Using this inequality, we see that

B .∑

k∈[Non−S tat]

t−22(−2α+1)k(2−k‖βkv‖L1 + ‖βk∇v‖L1

).

∑k∈[Non−S tat]

t−22(−2α+1)k(22k‖vk‖L∞ + 2

5k2 ‖∇v‖L6

).


t−22(−2α+1)k(22k‖vk‖L∞ + 2

5k2 ‖x2v‖L2

).

∑〈t〉−

12 ≤2k

t−22−k2(−2α+4)k‖vk‖L∞ +∑

k∈[Non−S tat]

t−22(−2α+ 72 )k‖x2v‖L2 .

The first sum is bounded by t−32 ‖〈ξ〉4−2αv‖L∞ . The second sum can be estimated

case by case w.r.t. α as follows:∑k∈[Non−S tat]

t−22(−2α+ 72 )k‖x2v‖L2

.

t−2+(−2α+ 7

2 ) 12N0(2−α) , if 1 < α < 7

4 ,

t−2 ln(1 + t), if α = 74 ,

t−2+ 12 (2α− 7

2 ), if 74 < α < 2

‖x2v‖L2

. t−74 ‖x2v‖L2 .

Lastly, we see that

C . t−2∑

k∈[Non−S tat]

(2(−2α+3)k‖vk‖L∞ + 2(−2α+2)k2

3k2 ‖x2v‖L2

).

This can be treated similarly to B. In conclusion, we have the bound for non-stationary case:


Ik(t, x) . t−32 ‖〈ξ〉4−2αv‖L∞ + t−

74 ‖x2v‖L2 .

Now, we move onto stationary case:

k ∈ [S tat] := {k : 〈t〉−12 ≤ 2k ≤ 〈t〉

12N0(2−α) , |k − k0| ≤ 4}.

56


Let n0 denote the smallest integer such that 2n0 ∼ t−12 . We further decompose the

integral Ik in (5.3.1) dyadically around ξ0 as

∑k∈[S tat]

Ik(t, x) .∑

k∈[S tat]

k0+4∑n≥n0

Ik,n(t, x), (5.3.6)

whereIk,n(t, x) =

∣∣∣∣∣∫R3

eiφt,x(ξ)βk(ξ)β(n0)n (ξ − ξ0)v(ξ)dξ

∣∣∣∣∣and

β(n0)n (ξ − ξ0) =

βn(ξ − ξ0), if n > n0,

ϕ( ξ−ξ02n0 ), if n = n0

(5.3.7)

for a fixed ϕ ∈ C∞0 which cut off near the origin. If n = n0, it is easy to check

Ik,n(t, x) . ‖β(n0)n0‖L1‖βkv‖L∞ . t−

32 ‖v‖L∞ .

If n > n0, Lemma 2.3.3 yields

|∇φt,x(ξ)| & |t|2n2(α−2)k for |ξ − ξ0| ∼ 2n.

By taking integration by parts twice, we estimate

Ik,n . t−2∑

0≤`≤2

2−(4−l)n22(2−α)k‖∇`(βkβ(n0)n v)‖L1 . Ak,n + Bk,n + Ck,n,

where

Ak,n = t−22−4n22(2−α)k‖βkβ(n0)n v‖L1 ,

Bk,n = t−22−3n22(2−α)k‖βkβ(n0)n ∇v‖L1 ,

Ck,n = t−22−2n22(2−α)k‖βkβ(n0)n ∇

2v‖L1 .

We bound∑

Ak,n as

∑k∈[S tat]

k+4∑n>n0

Ak,n . t−2∑|k−k0 |≤4

k+4∑n>n0

2−4n22(2−α)k‖β(n0)n ‖L1‖vk‖L∞

. t−2t12 ‖〈ξ〉4−2αv‖L∞ .

57


For∑

Bk,n, let us invoke that 2k ≤ 〈t〉1

2N0(2−α) . Then by using (5.3.5) we estimate

∑k∈[S tat]

k+4∑n>n0

Bk,n . t−2∑|k−k0 |≤4

22(2−α)kk0+8∑n>n0

2−n2 ‖x2v‖L2

. t−2t14 t

1N0 ‖x2v‖L2 .

The sum∑

Ck,n is bounded as

∑k∈[S tat]

k+4∑n>n0

Ck,n . t−2∑|k−k0 |≤4

k+4∑n>n0

2−2n22(2−α)k23n2 ‖x2v‖L2

. t−2t14 t

1N0 ‖x2v‖L2 .

In conclusion, we obtain∑k∈[S tat]

Ik(t, x) . t−32 ‖〈ξ〉4−2αv‖L∞ + t−

32 t−( 1

4−1

N0)‖x2v‖L2 .

Comparing the bound obtained by non-stationary case with that by stationary caseand (5.3.2), we get the desired result.

�

Corollary 5.3.2. Let N0 > 4 and N ≥ 72 N0(2 − α) + 2α − 5

2 . Then we have

‖e−it|∇|αv‖W2,∞ . 〈t〉−32 ‖〈ξ〉6−2αv‖L∞ + 〈t〉−

32−( 1

4−1

N0)(‖v‖HN+2 + ‖x2v‖H2

).

Proof. By definition we estimate

‖e−it|∇|αv‖W2,∞ =∑

0≤`≤2

‖e−it|∇|α∇`v‖L∞

.∑

0≤`≤2

〈t〉−32 ‖〈ξ〉4−2α∇`v‖L∞ + 〈t〉−

32−( 1

4−1

N0)(‖x2(∇`v)‖L2 + ‖∇`v‖HN )

. 〈t〉−32 ‖〈ξ〉6−2αv‖L∞ + 〈t〉−

32−( 1

4−1

N0)(‖x2v‖H2 + ‖v‖HN+2

).

�

58


5.4 Weighted energy estimates

We prove Theorem 5.1.1 in Section 5.4 through Section 5.5. All the estimates inboth sections are implemented for the positive time.

To investigate the asymptotic behavior of the solution u at time infinity weintroduce a control norm.

Definition 5.4.1. Let N, δ0,T > 0. Then we define the norm ‖·‖ΣT for the functionsu ∈ C([0,T ]; HN) with xv ∈ C([0,T ]; H3), x2v ∈ C([0,T ]; H2) by

‖u‖ΣT := sup[0,T ]

[〈t〉−δ0‖u(t)‖HN + 〈t〉−δ0‖xv(t)‖H3 + 〈t〉−2δ0‖x2v(t)‖H2

+ ‖〈ξ〉5v(t)‖L∞ξ].

As observed in the proof of Lemma 5.2.2, the role of ‖xv(t)‖H3 is crucial tocontrol of ‖x2v(t)‖H2 . So, it is taking part in the definition of ‖ · ‖ΣT .

Proposition 5.4.2 (Weighted energy estimate). Let 53 < α < 2, N ≥ 10. And let

δ0 > 0 be such tat

max(0,

1712−

5α6

)≤ δ0 ≤

136.

Suppose that u is the global solution to (5.1.1) such that ‖u‖ΣT ≤ ε1 for someε1,T > 0 and u0 satisfies (5.1.3) with ε0 ≤ ε1. Then we have

supt∈[0,T ]

〈t〉−δ0‖u‖HN ≤ ε0 + Cε31 , (5.4.1)

supt∈[0,T ]

〈t〉−δ0‖xv‖H3 ≤ ε0 + Cε31 , (5.4.2)

supt∈[0,T ]

〈t〉−2δ0‖x2v‖H2 ≤ ε0 + Cε31 . (5.4.3)

In this section, we prove Proposition 5.4.2. We emphasize again that 53 < α <

2, N ≥ 10 and max(0, 1712 −

5α6 ) ≤ δ0 ≤

136 . If we choose N0 = 6 in Corollary 5.3.2,

then since N ≥ 10 we have

〈t〉32 ‖e−it|∇|αv‖W2,∞ . ‖〈ξ〉5v‖L∞ + 〈t〉−2δ0

(‖x2v‖H2 + ‖v‖HN

). (5.4.4)

59


Therefore if ‖u‖ΣT ≤ ε1, then by the definition of ΣT and (5.4.4) we have

‖u(t)‖W2,∞ . 〈t〉−32 ε1. (5.4.5)

We will see in Section 5.4.3 why the conditions α > 53 and δ0 ≥

1712 −

5α6 are

necessary.

5.4.1 Proof of (5.4.1)

From (1.1.3) and Leibniz rule it follows that for any t > 0

‖u(t)‖HN

≤ ‖u0‖HN + C∫ t

0

∥∥∥(|x|−1 ∗ |u|2)u∥∥∥

HN ds

≤ ‖u0‖HN + C∫ t

0(‖|x|−1 ∗ |u|2‖L∞‖u‖HN + ‖|x|−1 ∗ |u|2‖HN

6‖u‖L3) ds.

Using the estimate ‖|x|−1 ∗ |u|2‖L∞ . ‖u‖L2‖u‖L6 and Leibniz rule once more,

‖u‖HN ≤ ‖u0‖HN + C∫ t

0(‖u‖L2‖u‖L6 + ‖u‖2L3)‖u‖HN ds

≤ ‖u0‖HN + C∫ t

0‖u‖L2‖u‖L6‖u‖HN ds

≤ ‖u0‖HN + C∫ t

0‖u‖

43

L2‖u‖23L∞‖u‖HN ds

≤ ε0 + Cε531 ‖u0‖

43

L2

∫ t

0〈s〉−1+δ0 ds

≤ ε0 + C〈t〉δ0ε31 .

5.4.2 Proof of (5.4.2)

In order to prove (5.4.2), we need to establish the following.

Lemma 5.4.3. Let u satisfy the condition of Proposition5.4.2. Then we have

‖Pk(|u|2)(s)‖L∞ . min(〈2k〉−2〈s〉−3, 23k)ε21 , (5.4.6)

60


‖Pk(|u|2)(s)‖L2 . min(23k2 , 2

3k2 〈2k〉−5)ε2

1 , (5.4.7)

‖Pk(xv)(s)‖L2 . min(2(2−α)k〈s〉(3−α)δ0 , 〈2k〉−3〈s〉δ0). (5.4.8)

Proof of Lemma 5.4.3. We first show (5.4.6). Using the fact ‖F −1(βk〈ξ〉−2)‖L1 .

〈2k〉−2 and also (5.4.5), we obtain

‖Pk(|u|2)‖L∞ = supx

∣∣∣ ∫ eix·ξβk(ξ)|u|2(ξ) dξ∣∣∣

= supx|

∫eix·ξβk(ξ)〈ξ〉−2 (1 − ∆)(|u|2)(ξ) dξ|

. ‖F −1(βk〈ξ〉−2) ∗ (1 − ∆)(|u|2)‖L∞

. ‖F −1(βk〈ξ〉−2)‖L1‖(1 − ∆)(|u|2)‖L∞

. 〈2k〉−2‖u‖2W2,∞ . 〈2k〉−2〈s〉−3ε21 .

On the other hand, we see that

‖Pk(|u|2)‖L∞ . ‖βk |u|2‖L1 . ‖βk‖L1‖|u|2‖L∞ . 23k‖u‖2L2 . 23kε21 .

We consider (5.4.7). One can easily check

‖Pk(|u|2)‖L2 . ‖βk‖L2‖|u|2‖L∞ . 23k2 ε2

1 .

Plancherel’s theorem gives us

‖Pk(|u|2)‖2L2 .

∫R3βk(ξ)|u ∗ u|2(ξ)dξ

.

∫R3βk(ξ)‖〈ξ〉5u‖4L∞

∣∣∣ ∫R3

(1 + |y|)−5(1 + |y − ξ|)−5dy∣∣∣2dξ

. ε41

∫R3βk(ξ)|

∫R3

(1 + |y|)−5(1 + |y − ξ|)−5dy|2dξ

. ε4123k〈2k〉−10.

For (5.4.8) we use Bernstein’s inequality to get

‖Pk(xv)‖L2 . 2k(2−α)‖xv‖L

67−2α

.

61


Using Holder’s inequality, for any R > 0 we have

‖xv‖6

7−2α

L6

7−2α.

3∑j=1

(∫|x|≤R|x j f (x)|

67−2α dx +

∫|x|≥R|x j f (x)|

67−2α dx

). R

6(2−α)7−2α ‖xv‖

67−2α

L2 + R6(1−α)7−2α ‖x2v‖

67−2α

L2 .

The optimization by taking R = ‖x2v‖‖xv‖−1 leads us to the estimate

‖xv‖L

67−2α. ‖x2v‖2−αL2 · ‖xv‖α−1

L2 . 〈s〉(3−α)δ0ε1.

Also, we estimate in another manner

‖Pk(xv)‖L2 . 〈2k〉−3‖xv‖H3 . 〈2k〉−3〈s〉δ0ε1.

This completes the proof of Lemma 5.4.3.�

Note that Plancherel’s theorem yields

‖xv‖H3 ∼ ‖ 〈ξ〉3 xv‖L2 = ‖ 〈ξ〉3 ∇ξv‖L2 . (5.4.9)

Differentiating the both sides of (5.1.2) w.r.t. ξ, we have

(∇v)(t, ξ) = (∇u0)(ξ) + I1(t, ξ) + I2(t, ξ),

I1(t, ξ) = ic0

∫ t

0

∫R3

eisφα(ξ,η)|η|−2 |u|2(η)(∇v)(ξ − η) dηds,

I2(t, ξ) = ic0

∫ t

0s∫R3

m(ξ, η)eisφα(ξ,η)|η|−2 |u|2(η)v(ξ − η) dηds,

where

m(ξ, η) := ∇ξ[φα(ξ, η)] = α(|ξ|α−2ξ − |ξ − η|α−2(ξ − η)

). (5.4.10)

Then the estimate (5.4.2) follows from

‖〈ξ〉3I1(t, ξ)‖L2 + ‖〈ξ〉3I2(t, ξ)‖L2 . 〈t〉δ0ε31 .

We consider this in the next two subsections.

62


Estimate of I1(s, ξ)

We rewrite I1 as

I1(t, ξ) = ic0

∫ t

0

∫R3

eis|ξ|α |η|−2 |u|2(η)e−is|ξ−η|α(∇v)(ξ − η)dηds

= κ

∫ t

0F

{eis|∇|α(| · |−1 ∗ |u|2)(e−is|∇|α xv)

}ds.

From the Sobolev estimates and ‖|x|−1 ∗ |u|2‖H3 . ‖∇|u|2‖L2 , it follows that

‖ 〈ξ〉3I1(t, ξ)‖L2

.

∫ t

0‖eis|∇|α(| · |−1 ∗ |u|2)eis|∇|α(xv)‖H3 ds

.

∫ t

0‖| · |−1 ∗ |u|2‖L∞‖xv‖H3 ds +

∫ t

0‖| · |−1 ∗ |u|2‖H3‖xv‖L∞ ds

.

∫ t

0‖u‖L2‖u‖L6‖xv‖H3ds +

∫ t

0‖∇|u|2‖L2‖xv‖H3 ds.

(5.4.11)

Since ‖u‖ΣT ≤ ε1, (5.4.5) implies ‖u(s)‖L∞ . 〈s〉−32 ε1, and by mass conservation

‖u(s)‖L2 = ‖u0‖L2 ≤ ε1. Hence we have

‖u‖L6 ≤ ‖u‖13

L2‖u‖23L∞ . 〈s〉

−1ε1

‖∇|u|2‖L2 ≤ ‖u‖L∞‖∇u‖L2 . 〈s〉−32 +δ0ε2

1 .(5.4.12)

These give us

‖ 〈ξ〉3 I1(t, ξ)‖L2 . ε31

∫ t

0〈s〉−1

〈s〉δ0 ds . 〈t〉δ0ε31 .

63


Estimate of I2(t, ξ)

By the dyadic decomposition, we have∥∥∥ 〈ξ〉3 I2(t, ξ)∥∥∥

L2

.

∫ t

0s

∑k1,k2,k∈Z

〈2k〉3∥∥∥∥∥ ∫R3

eisφα(ξ,η)m(ξ, η)

× βk(ξ)βk1(η − ξ)βk2(η)|η|−2 Pk2(|u|2)(η) Pk1(v)(ξ − η)dη∥∥∥∥∥

L2ξ

ds

.

∫ t

0s

∑k1,k2,k∈Z

Ik,k1,k22 (s) ds,

where

Ik,k1,k22 (s) = 〈2k〉3

∥∥∥∥∥∫ mk,k1,k2(ξ, η) Pk2(|u|2)(η)e−is|ξ−η|α Pk1(v)(ξ − η)dη∥∥∥∥∥

L2ξ

and

mk,k1,k2(ξ, η) = m(ξ, η)|η|−2βk(ξ)βk1(η − ξ)βk2(η). (5.4.13)

Thus it suffices to prove that∑k1,k2,k∈Z

Ik,k1,k22 (s) . 〈s〉−2+δ0ε3

1 . (5.4.14)

In view of the Fourier support condition we can divide the sum over k, k1, k2

into three possible cases:∑k1,k2,k∈Z

Ik,k1,k22 (s) .

( ∑k∼k1≥k2

+∑

k�k1∼k2

+∑

k∼k2�k1

)Ik,k1,k22 (s).

In the first two cases, we use the following inequality: If k . k1, then for anypositive integers `1, `2∣∣∣∣∇`1

ξ ∇`2η mk,k1,k2(ξ, η)

∣∣∣∣. max(2k, 2k1 , 2k2)α−22−k22−k1`12−k2`2 βk(ξ)βk1(η − ξ)βk2(η).

(5.4.15)

64


This can be readily shown by direct calculation. We omit the details. This leads usto the bound for Lemma 2.3.1.∥∥∥∥∥"

R3×R3mk,k1,k2(ξ, η)eix·ξeiy·η dξdη

∥∥∥∥∥L1

x,y

. max(2k, 2k1 , 2k2)α−22−k2 . (5.4.16)

In the last case (k1 � k ∼ k2), even though mk,k1,k2 fails to satisfy (5.4.15), theestimate (5.4.16)) turns out to hold. Indeed, by making change of variables, onecan verify that∥∥∥∥∥"

R3×R3mk,k1,k2(ξ, η)eix·ξeiy·η dξdη

∥∥∥∥∥L1

x,y

=

∥∥∥∥∥"R3×R3

mk,k1,k2(ξ, η)eix·ξeiy·η dξdη∥∥∥∥∥

L1x,y

,

where

mk,k1,k2(ξ, η) = α(|ξ|α−2ξ − |η|α−2η

)|ξ − η|−2βk(ξ)βk1(η)βk2(ξ − η).

Then mk,k1,k2(ξ, η) satisfies that∣∣∣∣∇l1ξ ∇

l2η mk,k1,k2(ξ, η)

∣∣∣∣. max(2k, 2k1 , 2k2)α−22−k22−kl12−k1l2 βk(ξ)βk1(η)βk2(ξ − η)

and hence the claim follows.(Case: k ∼ k1 ≥ k2) Applying the Lemma 2.3.1 whose A(mk,k1,k2) is the RHS

of (5.4.16) and then using (5.4.6), we estimate∑k2.k∼k1

Ik,k1,k22 (s)

.∑

k2.k∼k1

〈2k〉3 max(2k, 2k1)α−22−k2‖Pk2(|u|2)‖L∞‖Pk1(v)‖L2

.∑

k2.k∼k1

〈2k〉3 max(2k, 2k1)α−22−k2

×min(〈2k2〉−2〈s〉−3, 23k2

)〈2k1〉−52

3k12 ε3

1

.( ∑

2k2≤〈s〉−1

22k2 +∑

2k2≥〈s〉−1

〈s〉−32−k2)∑

k

〈2k〉−22k(α− 12 )ε3

1

. 〈s〉−2+δ0ε31 .

65


(Case: k � k1 ∼ k2) The sum over k with 2k ≤ 〈s〉−2 can be easily dealt withusing the pointwise bound (5.4.15) as follows.∑

k�k1∼k22k≤〈s〉−2

Ik,k1,k22 (s)

.∑

k�k1∼k22k≤〈s〉−2

〈2k〉3‖mk,k1,k2(ξ, η)‖L∞‖βk‖L2‖Pk2(|u|2)‖L2‖

Pk1(v)‖L2

.∑

k�k1∼k22k≤〈s〉−2

〈2k〉32k1(α−2)2−k223k2 〈2k2〉−52

3k22 〈2k1〉−52

3k12 ε3

1

.∑k�k2

2k≤〈s〉−2

〈2k〉323k2 2k2α〈2k2〉−10ε3

1

.∑

2k≤〈s〉−2

23k2 ε3

1

∑k2

2k2α〈2k2〉−7

. 〈s〉−3ε31 .

The remaining case can be estimated by applying Lemma 2.3.1 with the boundA(mk,k1,k2) from (5.4.16), as follows.∑

k�k1∼k22k≥〈s〉−2

Ik,k1,k22 (s) .

∑k�k1∼k22k≥〈s〉−2

〈2k〉32k1(α−2)2−k2‖Pk2(|u|2)‖L∞‖Pk1(v)‖L2

.∑k�k2

2k≥〈s〉−2

〈2k〉32k2(α− 32 ) min(〈2k2〉−2〈s〉−3, 23k2)〈2k2〉−5ε3

1

.∑

2k2≤〈s〉−1

2k2(α+ 32 )ε3

1

∑〈s〉−2≤2k≤〈s〉−1

1

+∑

2k2≥〈s〉−1

〈s〉−32k2(α− 32 )〈2k2〉−7

∑k≤k2

2k≥〈s〉−2

〈2k〉3ε31

. 〈s〉−α−32 +δ0ε3

1 + 〈s〉−52 ε3

1

∑2k2≥〈s〉−1

2k2(α−1)〈2k2〉−3∑k≤k2

2k≥〈s〉−2

〈2k〉−1

. 〈s〉−2+δ0ε31 .

66


(Case: k1 � k ∼ k2) In this case we use Lemma 2.3.1 with A(mk,k1,k2) of(5.4.16) and get that∑

k1≤k∼k1

Ik,k1,k22 (s)

.∑

k1�k∼k2

〈2k〉32k(α−1)2−2k2‖Pk2(|u|2)‖L∞‖Pk1(v)‖L2

.∑k1�k

〈2k〉32k(α−3) min(〈2k〉−2〈s〉−3, 23k)〈2k1〉−523k1

2 ε31

. ε31

∑2k≤〈s〉−1

2αk∑k1≤k

232 k1 + ε3

1

∑2k≥〈s〉−1

〈s〉−32k(α−2)∑k1≤k

〈2k1〉−52k12

. 〈s〉−α−1ε31 .

5.4.3 Proof of (5.4.3)

By Plancherel’s theorem we have

‖x2v‖H2 ∼ ‖ 〈ξ〉2 x2v‖L2 = ‖ 〈ξ〉2 ∇2v‖L2 .

Then the second derivative of v can be written as

∇2v(t, ξ) = J1(t, ξ) + J2(t, ξ) + J3(t, ξ) + J4(t, ξ).

Here Ji are defined by

J1(t, ξ) = ic0

∫ t

0

∫R3

eisφα(ξ,η)|η|−2 |u|2(η)(∇2v)(ξ − η)dηds,

J2(t, ξ) = 2c0

∫ t

0s∫R3

m(ξ, η) ⊗ [eisφα(ξ,η)|η|−2 |u|2(η)(∇v)(ξ − η)]dηds,

J3(t, ξ) = c0

∫ t

0s∫R3

[∇ξm(ξ, η)]eisφα(ξ,η)|η|−2 |u|2(η)v(ξ − η)dηds,

J4(t, ξ) = −ic0

∫ t

0s2

∫R3

m(ξ, η) ⊗m(ξ, η)eisφα(ξ,η)|η|−2 |u|2(η)v(ξ − η)dηds,

where m is defined as in (5.4.10).

67


Estimate of J1

By (5.4.11) and (5.4.12) one can readily get

‖〈ξ〉2J1(t, ξ)‖L2 .

∫ t

0‖eis|∇|α(| · |−1 ∗ |u|2)eis|∇|α(x2v)‖H2 ds

. 〈t〉2δ0ε31 .

Estimate of J2

By dyadic decomposition in ξ, η and ξ − η, we write

〈ξ〉2J2(t, ξ)

= 2c0

∑k,k1,k2∈Z

〈ξ〉2∫ t

0s∫R3

mk,k1,k22 (ξ, η)

⊗ [eisφα(ξ,η) Pk2(|u|2)(η)˜Pk1(xv)(ξ − η)]dηds,

where mk,k1,k22 = mk,k1,k2 as in (5.4.13). So, it suffices to show that∑

k,k1,k2∈Z

〈2k〉2J2k,k1,k2(s) . 〈s〉−2+2δ0ε3

1 ,

where

J2k,k1,k2(s) =

∥∥∥∥∥ ∫R3

mk,k1,k22 (ξ, η) ⊗ [ Pk2 |u|2(η)e−is|ξ−η|α ˜Pk1(xv)(ξ − η)]dη

∥∥∥∥∥L2ξ

.

We then divide the sum into three possible cases as above.(Case: k ∼ k1 ≥ k2) We proceed with the same method as in Section 5.4.2.

68


Using Lemma 2.3.1 with A(mk,k1,k22 ) of (5.4.16) and the Lemma 5.4.3, we have∑

k∼k1≥k2

〈2k〉2J2k,k1,k2(s) .

∑k∼k1≥k2

〈2k〉22k(α−2)2−k2‖Pk2 |u|2‖L∞‖Pk1(xv)‖L2

.∑

k∼k1≥k2

〈2k〉22k(α−2)2−k2 min(〈2k2〉−2〈s〉−3, 23k2)

×min(2k1(2−α)〈s〉(3−α)δ0 , 〈2k1〉−3〈s〉δ0)ε31

. ε31

∑2k2≤〈s〉−1

22k2∑k≥k2

〈2k〉22k(α−2) min(2k(2−α)〈s〉(3−α)δ0 , 〈2k〉−3〈s〉δ0)

+ 〈s〉−3ε31

∑2k2≥〈s〉−1

〈2k2〉−22−k2∑k≥k2

〈2k〉22k(α−2)

×min(2k(2−α)〈s〉(3−α)δ0 , 〈2k〉−3〈s〉δ0)

. 〈s〉−2+2δ0ε31 .

(Case: k � k1 ∼ k2) At first, we deal with the case 2k ≤ 〈s〉−2. By (5.4.15)) wehave ∑

k�k1∼k22k≤〈s〉−2

〈2k〉2J2k,k1,k2(s)

.∑

k�k1∼k22k≤〈s〉−2

〈2k〉22k1(α−3)‖βk‖L2‖Pk2(|u|2)‖L2‖Pk1(xv)‖L2

.∑

2k≤〈s〉−2

23k2

∑k1�k

2k1(α− 32 )〈2k1〉−2

×min(2k1(2−α)〈s〉(3−α)δ0 , 〈2k1〉−3〈s〉δ0)ε31

. 〈s〉−3+δ0ε31( ∑

2k1≤〈s〉−δ0

2k12 〈s〉(2−α)δ0 +

∑2k1≥〈s〉−δ0

〈2k1〉−52k1(α− 32 ))

. 〈s〉−2+2δ0ε31 .

69


For the case 2k ≥ 〈s〉−2, we use Lemma 2.3.1 as follows.∑k�k1∼k22k≥〈s〉−2

〈2k〉2J2k,k1,k2(s) .

∑k�k1∼k22k≥〈s〉−2

〈2k〉22k1(α−3)‖Pk2(|u|2)‖L∞‖Pk1(xv)‖L2

.∑

2k1≤〈s〉−1

22k1〈s〉δ0(3−α)ε31

∑〈s〉−2≤2k≤〈s〉−1

1

+∑

2k1≥〈s〉−1

〈s〉−32k1(α−3)〈2k1〉−2ε31

×min(2k1(2−α)〈s〉(3−α)δ0 , 〈2k1〉−3〈s〉δ0)∑k≤k1

〈2k〉2

. 〈s〉−2+2δ0ε31 .

(Case: k1 � k ∼ k2) Using Lemma 2.3.1 as above, we get∑k1�k∼k2

〈2k〉2J2k,k1,k2(s) .

∑k1�k∼k2

〈2k〉22k2(α−3)‖Pk2(|u|2)‖L∞‖Pk1(xv)‖L2

. ε31

∑2k2≤〈s〉−1

2αk2∑k1≤k2

2k1(2−α)〈s〉−δ0(3−α)

+ 〈s〉−3+δ0ε31

∑2k2≥〈s〉−1

2k2(α−3)∑k1≤k2

min(2k1(2−α)〈s〉(2−α)δ0 , 〈2k1〉−3)

. 〈s〉−2+2δ0ε31 .

Estimate of J3

Through the dyadic decomposition in ξ, η and ξ − η, it suffices to show that∑k,k1,k2∈Z

〈2k〉2J3k,k1,k2(s) . 〈s〉−2+2δ0ε3

1 ,

where

J3k,k1,k2(s) =

∥∥∥∥∥ ∫R3

mk,k1,k23 (ξ, η) Pk2(|u|2)(η)e−is|ξ−η|α Pk1(v)(ξ − η)dη

∥∥∥∥∥L2ξ

70


andmk,k1,k2

3 (ξ, η) = [∇ξm(ξ, η)]|η|−2βk(ξ)βk1(ξ − η)βk2(η).

Direct calculation gives: If k . k1, then for any positive integers `1, `2∣∣∣∣∇`1ξ ∇

`2η mk,k1,k2

3 (ξ, η)∣∣∣∣ . min(2k, 2k1)α−2 max(2k, 2k1)−12−k22−`1k2−`2k2

× βk(ξ)βk1(ξ − η)βk2(η).(5.4.17)

And also from this we have∥∥∥∥∥"R3×R3

eix·ξeiy·ηmk,k1,k23 (ξ, η)dξdη

∥∥∥∥∥L1

x,y

. min(2k, 2k1)α−2 max(2k, 2k1)−12−k2 .

(5.4.18)

Now we divide the sum into three parts:

∑k,k1,k2∈Z

〈2k〉2J3k,k1,k2(s) .

∑k∼k1≥k2

+∑

k�k1∼k2

+∑

k1�k∼k2

〈2k〉3J3k,k1,k2(s).

(Case: k ∼ k1 ≥ k2) If α ≥ 32 , we can obtain the desired bound by applying

Lemma 2.3.1 with A(mk,k1,k23 ) of (5.4.18). On the other hand, if α < 3

2 , the sumover 2k ≤ 〈s〉−2 can be estimated by using just the pointwise bound (5.4.17) asbefore, and the remaining case can be treated by applying the Lemma 2.3.1.

(Case: k � k1 ∼ k2) In this case the condition δ0 ≥1712−

5α6 is necessary. Taking

the sum over 2k ≤ 〈s〉−53 , we estimate∑

k�k1∼k2

2k≤〈s〉−53

〈2k〉2J3k,k1,k2(s)

.∑

k�k1∼k2

2k≤〈s〉−53

‖mk,k1,k23 ‖L∞‖βk‖L2‖Pk2(|u|

2)‖L2‖Pk1(v)‖L2

.∑

2k≤〈s〉−53

2k(α− 12 )

∑k1�k

2k1〈2k1〉−10ε31 .

. 〈s〉−53 (α− 1

2 )ε31 .

71


And the sum over 2k ≥ 〈s〉−53 follows from Lemma 2.3.1 with A(mk,k1,l2

3 ) of (5.4.18):∑k�k1∼k2

2k≥〈s〉−53

〈2k〉2J3k,k1,k2(s)

.∑

k�k1∼k2

2k≥〈s〉−53

〈2k〉22k(α−2)2−2k1‖Pk2(|u|2)‖L∞‖Pk1(v)‖L2

.∑

2k1≤〈s〉−1

25k1

2

∑〈s〉−

53 ≤2k≤〈s〉−1

2k(α−2)ε31

+ 〈s〉−3∑

2k1≥〈s〉−1

2−k12 〈2k1〉−5

∑2k≥〈s〉−

53

2k(α−2)ε31

. 〈s〉−52 + 5

3 (2−α)ε31 .

If δ0 ≥1712 −

5α6 , then the last two terms of the above estimates have the desired

bound.(Case: k1 � k ∼ k2) By using Lemma 2.3.1 again, we have∑

k1�k∼k2

〈2k〉2J3k,k1,k2(s) . 〈s〉−

12−α+δ0ε3

1 .

Therefore, if α ≥ 32 , then we get the desired bound.

Estimate of J4

By dyadic decomposition 〈ξ〉2J4(s, ξ) can be written as

〈ξ〉2J4(s, ξ) = −ic0

∑k,k1,k2∈Z

〈ξ〉3∫ t

0s2

∫R3

mk,k1,k24 (ξ, η)

× eisφα(ξ,η) Pk2(|u|2)(η) Pk1(v)(ξ − η)dηds,

wheremk,k1,k2

4 = m(ξ, η) ⊗mk,k1,k2(ξ, η).

It is also enough to show the following estimate∑k,k1,k2∈Z

〈2k〉3J4k,k1,k2(s) . 〈s〉−3+δ0ε3

1 ,

72


where

J4k,k1,k2(s) =

∥∥∥∥∥ ∫R3

mk,k1,k24

Pk2(|u|2)(η)e−is|ξ−η|α Pk1(v)(ξ − η)dη∥∥∥∥∥

L2ξ

.

From (5.4.15) and Leibniz rule, it follows that if k . k1, then for any positiveintegers `1, `2∣∣∣∣∇`1

ξ ∇`2η mk,k1,k2

4 (ξ, η)∣∣∣∣ . max(2k, 2k1 , 2k2)2α−42−`1k2−`2k2

× βk(ξ)βk1(ξ − η)βk2(η).(5.4.19)

Then one can easily observe from this that∥∥∥∥∥"R3×R3

eix·ξeiy·ηmk,k1,k24 (ξ, η)dξdη

∥∥∥∥∥L1

x,y

. max(2k, 2k1 , 2k2)2α−4. (5.4.20)

Now dividing the sum into three parts, we have∑k,k1,k2∈Z

〈2k〉2J4k,k1,k2(s) .

( ∑k∼k1≤k2

+∑

k�k1∼k2

+∑

k1�k∼k2

)〈2k〉2J4

k,k1,k2(s).

(Case: k ∼ k1 ≥ k2) We can easily obtain the bound applying the Lemma 2.3.1with A(mk,k1,k2

4 ) of (5.4.20).∑k∼k1≤k2

〈2k〉2J4k,k1,k2(s) .

∑2k2≤〈s〉−1

23k2∑k≥k2

〈2k〉−22k(2α− 52 )ε3

1

+ 〈s〉−3∑

2k2≥〈s〉−1

〈2k2〉−2∑k≥k2

〈2k〉−22k(2α− 52 )ε3

1

. 〈s〉−3+δ0ε31 .

(Case: k � k1 ∼ k2) Taking the sum over 2k ≤ 〈s〉−2, by the pointwise estimatewe have ∑

k�k1∼k22k≤〈s〉−1

〈2k〉2J4k,k1,k2(s) .

∑2k≤〈s〉−2

23k2

∑k1≥k

2k1(2α−1)〈2k1〉−5ε31

. 〈s〉−3ε31 .

73


The remaining case can also be estimated by Lemma 2.3.1 as∑k�k1∼k22k≥〈s〉−2

〈2k〉2J4k,k1,k2(s)

. ε31

∑2k1≤〈s〉−1

2(2α−1)k1∑k≤k1

2k≥〈s〉−2

23k2

+ 〈s〉−3ε31

∑2k1≥〈s〉−1

〈2k1〉−42k1(2α− 52 )

∑k≤k1

2k≥〈s〉−2

〈2k1〉−1

. 〈s〉−3+δ0ε31 .

(Case: k1 � k ∼ k2) By Lemma 2.3.1 with (5.4.20) we finally have∑k1�k∼k2

〈2k〉2J4k,k1,k2(s) . ε3

1

∑2k≤〈s〉−1

2(2α−1)k∑k1≤k

23k1

2

+ 〈s〉−3ε31

∑2k≥〈s〉−1

2k(2α−4)∑k1≤k

23k1

2 〈2k1〉−5

. 〈s〉−2α− 12 +δ0ε3

1

. 〈s〉−3+δ0ε31 , if α ≥

54.

5.5 Modified scattering

We consider L∞-control for 〈ξ〉5v and show the modified scattering Theorem 5.1.1.At first we have the following.

Proposition 5.5.1. Let 53 < α < 2, N > 75α+35

3α−5 and

0 < δ0 < min(

3α − 5100(α + 1)

(N − 5) −35,

2 − α3α

,1

100

).

74


Suppose that u is the global solution to (5.1.1) such that ‖u‖ΣT ≤ ε1 for someε1,T > 0 and u0 satisfies (5.1.3) with ε0 ≤ ε1. Then we have

supt∈[0,T ]

‖〈ξ〉5v‖L∞ ≤ ε0 + Cε31 . (5.5.1)

Proof. If T ≤ 2, (5.5.1) follows from (5.2.1). The case where T ≥ 2 followsstraightforwardly from Proposition 5.5.2 below by considering t1 = 1 and t2 =

T . �

Given a solution u of (5.1.1) satisfying the priori bounds ‖u‖ΣT ≤ ε1, we definefor any t ∈ [0,T ] and ξ ∈ R3

Bα(t, ξ) =c0

4απ

∫ t

0

∫R3

∣∣∣∣∣ ξ

|ξ|2−α−

σ

|σ|2−α

∣∣∣∣∣−1 ∣∣∣v(s, σ)∣∣∣2 dσϕs(ξ)

1〈s〉

ds,

whereϕs(ξ) = ϕ

(s−θξ

), θ =

(3α − 5)40(α + 1)

.

for a smooth compactly supported function ϕ. We also define the modified profile

w(t, ξ) := e−iBα(t,ξ)v(t, ξ).

From Lemma 2.3.3 we have the following a priori estimate.

|Bα| .

∫ t

0

∫(|σ|−(α−1) + |ξ − σ|−1|σ|2−α)|u(s, σ)|2 dσϕs(ξ)

ds1 + s

. (5.5.2)

For the inner integral we have∫(|σ|−(α−1) + |σ|2−α |ξ − σ|−1)|u(s, σ)|2 dσ

. ‖|x|α−1

2 v‖2L2 + ‖u‖2H

2−α2

+ ‖| · |2−α

2 u‖2L∞

. ‖xv‖2H3 + ‖u‖2HN + ‖〈·〉5u‖2L∞ < ∞

(5.5.3)

Therefore Bα is well-defined and real-valued.The main goal of this section is to show the following.

75


Proposition 5.5.2. Let 53 < α < 2. Assume that u ∈ C([0,T ]; HN) satisfies a priori

bound ‖u‖ΣT ≤ ε1 for N, δ0 > 0 as in Proposition 5.5.1. Then there holds

supt1≤t2∈[1,T ]

〈t1〉δ0‖〈ξ〉5

(w(t2, ξ) − w(t1, ξ)

)‖L∞ξ . ε

31 . (5.5.4)

Remark 5.5.3. The conditions of Theorem 5.1.1 for α,N, δ0 satisfy simultaneouslythose in Proposition 5.4.2 and Proposition 5.5.1. If we choose ε0, ε1 such thatε1 = 3ε0 and Cε3

1 ≤ ε0, then by combining Proposition 5.5.1 with Proposition5.4.2 and by continuity argument we deduce that ‖u‖ΣT ≤ 3ε0 for all T > 0. Inparticular, we can choose ε0 as in Theorem 5.1.1 as

ε0 =1

3√

3C.

Let us set ‖u‖Σ∞ := supT>0 ‖u‖ΣT . If ε0 < ε0 as mentioned in Remark 5.5.3, thenclearly

‖u‖Σ∞ ≤ ε1.

From this we deduce the following.

Corollary 5.5.4. Let 53 < α < 2. Assume that u ∈ C([0,∞); HN) satisfies a priori

bound ‖u‖Σ∞ ≤ ε1 for N, δ0 > 0 as in Proposition 5.5.1. Then there exist asymptoticstate v+, such that for all t > 0

‖〈ξ〉5[w(t, ξ) − v+(ξ)]‖L∞ξ . ε31〈t〉

−δ0 (5.5.5)

Proof. Letting t2 → ∞ in (5.5.4) implies the modified scattering (5.5.5) once wedefine

v+ := limt2→∞

w(t2, ξ).

Here the limit has been taken in 〈ξ〉−5L∞ξ . �

Proof of Proposition 5.5.2. For (5.5.4), it suffices to show that if t1 ≤ t2 ∈ [2m −

1, 2m+1] ∩ [0,T ] for some positive integer m, then

‖〈ξ〉5(w(t2, ξ) − w(t1, ξ)

)‖L∞ξ . 2−δ0mε3

1 . (5.5.6)

76


Indeed, Applying (5.5.6) we get for any 1 ≤ t1 ≤ t2 ∈ [0,T ],

‖〈ξ〉5(w(t2, ξ) − w(t, ξ)

)‖L∞ξ ≤

‖〈ξ〉5(w(t, ξ) − w(2m, ξ)

)‖L∞ξ +

m′−1∑j=0

‖〈ξ〉5(w(2m+ j, ξ) − w(2m+ j+1, ξ)

)‖L∞ξ

+ ‖〈ξ〉5(w(2m+m′ , ξ) − w(t2, ξ)

)‖L∞ξ

.m′−1∑j=0

2−δ0(m+ j)ε31 . t−δ0ε3

1 , (5.5.7)

where t ∈ [2m − 1, 2m+1] and t2 ∈ [2m+m′ − 1, 2m+m′+1]. Since s ≥ 1, we assumethat s ∼ 2m. Making change of variables, the nonlinear term I as in (1.4.5) can bewritten as

I(s, ξ) = ic1

"R3×R3

eisφ(ξ,η,σ)|η|−2v(s, ξ + σ)v(s, ξ + η + σ)v(s, ξ + η)dηdσ,

where

φ(ξ, η, σ) = |ξ|α − |ξ + η|α − |ξ + σ|α + |ξ + η + σ|α. (5.5.8)

Let n0 = n0(α,m) ∈ Z be the largest integer satisfying

n0 <

(−2 − 2δ0

α + 1−

10θ3

)m. (5.5.9)

Then, it is clear that 2−n0 ∼ 2( 2+2δ0α+1 + 10θ

3 )m.Let us now invoke the cut-off function β(n0)

n1 in as (5.3.7) replaced with n = n1.Then the time derivative of v can be decomposed as

∂sv(s, ξ) = ∂svn0(s, ξ) +∑

n1>n0,n1∈Z

∂svn1(s, ξ), (5.5.10)

where

∂svn0(s, ξ) = ic1

"R3×R3

eisφ(ξ,η σ)|η|−2β(n0)n0

(η)

× v(s, ξ + η)v(s, ξ + η + σ)v(s, ξ + σ)dηdσ,

∂svn1(s, ξ) = ic1

"R3×R3

eisφ(ξ,η σ)|η|−2β(n0)n1

(η)

× v(s, ξ + η)v(s, ξ + η + σ)v(s, ξ + σ)dηdσ.

77


By (5.5.10) we have

‖〈ξ〉5(w(t1, ξ) − w(t2, ξ))‖L∞ξ =∥∥∥〈ξ〉5 ∫ t2

t1∂sw(s, ξ)ds

∥∥∥L∞ξ

=∥∥∥〈ξ〉5 ∫ t2

t1e−iBα(s,ξ)(∂sv(s, ξ) − i∂sBα(s, ξ)v(s, ξ))ds

∥∥∥L∞ξ

.∥∥∥〈ξ〉5 ∫ t2

t1e−iBα(s,ξ)

(∂svn0(s, ξ) − i∂s[Bα(s, ξ)]v(s, ξ)

+∑

n1>n0,n1∈Z

∂svn1(s, ξ))ds

∥∥∥L∞ξ

. ‖〈ξ〉5∫ t2

t1e−iBα(s,ξ)

(∂svn0(s, ξ) − i∂s[Bα(s, ξ)]v(s, ξ)

)ds‖L∞ξ

+ ‖〈ξ〉5∫ t2

t1

∑n1>n0

e−iBα(s,ξ)∂svn1(s, ξ)ds‖L∞ξ

.∥∥∥〈ξ〉5 ∫ t2

t1e−iBα(s,ξ)(∂svn0(s, ξ)

(1 − ϕs(ξ)

))ds

∥∥∥L∞ξ

+∥∥∥〈ξ〉5 ∫ t2

t1e−iBα(s,ξ)(∂svn0(s, ξ)ϕs(ξ) − i∂s[Bα(s, ξ)]v(s, ξ)

)ds

∥∥∥L∞ξ

+∥∥∥〈ξ〉5 ∫ t2

t1

∑n1>n0

e−iBα(s,ξ)∂svn1(s, ξ)ds∥∥∥

L∞ξ.

So, it suffices to show that the following three estimates hold.

|∂svn0(s, ξ)(1 − ϕs(ξ))| . 2(−1−δ0)m〈ξ〉−5ε31 , (5.5.11)

|∂svn0(s, ξ)ϕs(ξ) − i∂sBα(s, ξ)v(s, ξ)| . 2(−1−δ0)m〈ξ〉−5ε31 , (5.5.12)∑

n1>n0

|∂svn1(s, ξ)| . 2(−1−δ0)m〈ξ〉−5ε31 . (5.5.13)

�

5.5.1 Proof of (5.5.11) (high frequency part)

Decomposing of (5.5.11) as in (5.5.10), we get

|∂svn0(s, ξ)(1 − ϕs(ξ))| .∑

n1<n0+10

|∂svn1(s, ξ)|.

78


In the case of high frequency |ξ| � |η|, i.e. |ξ| ∼ |ξ + η|, we estimate

|∂svn1(s, ξ)| . 2−2n1〈ξ〉−N‖βn1‖L2‖(1 + |ξ + ·|)N v(ξ + ·)v ∗ v‖L2

. 2−n12 〈ξ〉−N‖u‖HN ‖v‖2L2

. 2−n12 〈ξ〉−N〈s〉δ0ε3

1 .

(5.5.14)

And we also have

|∂svn1(s, ξ)| . 2−2n1〈ξ〉−5‖β‖L1‖(1 + |ξ + ·|)5v(ξ + ·)v ∗ v‖L∞

. 2n1〈ξ〉−5‖(1 + | · |)5v‖L∞ ‖v‖2L2

. 2n1〈ξ〉−5ε31 .

(5.5.15)

Then by dividing the sum w.r.t. n1 into two parts and using 〈ξ〉 & 〈s〉θ ∼ 2θm weget ∑

n1<n0+10

|∂svn1(s, ξ)|

.∑

n1≤(−1−δ0)m

2n1〈ξ〉−5ε31 +

∑(−1−δ0)m≤n1≤n0+10

2−n12 〈ξ〉−N〈s〉δ0ε3

1

. 〈ξ〉−5ε31

(2(−1−δ0)m + 2−θ(N−5)m+δ0m

∑(−1−δ0)m≤n1≤n0+10

2−n12)

. 2(−1−δ0)m〈ξ〉−5ε31 ,

since δ0 ≤3α−5

100(α+1) (N − 5) − 35 . This proves (5.5.11).

5.5.2 Proof of (5.5.12)

Phase approximation

Define

v0,1(s, ξ) = ic1

"R3×R3

eisφ(ξ,η,σ)|η|−2β(n0)n0

(η)

× v(s, ξ + η)v(s, ξ + η + σ)v(s, ξ + σ)dηdσ,

79


where

φ(ξ, η, σ) = α

(ξ · η

|ξ|2−α−

(ξ + σ) · η|ξ + σ|2−α

). (5.5.16)

By Lemma 2.3.4 we have

|φ(ξ, η, σ) − φ(ξ, η, σ)| . |η|α.

Now, we have

|∂svn0(s, ξ) − v0,1(s, ξ)|

. s"R3×R3

|η|α|η|−2β(n0)n0

(η)∣∣∣∣v(s, ξ + η)v(s, ξ + η + σ)v(s, ξ + σ)

∣∣∣∣ dηdσ

. s〈ξ〉−5"R3×R3

|η|α−2β(n0)n0

(η)〈max(|ξ + η + σ|, |ξ + η|, |ξ + σ|)〉5

× |v(s, ξ + η)v(s, ξ + η + σ)v(s, ξ + σ)|dηdσ

. s〈ξ〉−5‖〈ξ〉5v‖L∞‖| · |α−2β(n0)n0‖L1 ‖v‖L2 ‖v‖L2

. s〈ξ〉−52n0(α+1)ε31

. 2m2−(2+2δ0+ 112 (3α−5))m〈ξ〉−5ε3

1

. 2(−1−δ0)m〈ξ〉−5ε31 for any δ0 > 0.

Profile approximation

We further approximate v0,1(s, ξ) by

v0,2(s, ξ) = ic1

"R3×R3


(η)v(s, ξ)v(s, ξ + σ)v(s, ξ + σ)dηdσ.

In order to do this, we define with the notation in (5.3.7)

v≤J(x) := β(J)J (x)v(x) and v>J(x) := v(x) − v≤J(x) for J ≥ 0.

We see that for |η| . 2n0 ,

|v(ρ + η) − v(ρ)| . |v>J(ρ + η) − v>J(ρ)| + |v≤J(ρ + η) − v≤J(ρ)|

. 2‖v>J‖L∞ + ‖∇v≤J‖L∞ · 2n0

. ‖(|x|−2)>J‖L2‖x2v>J‖L2 + ‖(|x|−1)≤J‖L2‖x2v≤J‖L22n0

. (2−J2 + 2

J2 2n0)〈s〉2δ0ε1.

80


Choosing J = −n0 we obtain

|v(ρ + η) − v(ρ)| . 2n02 〈s〉2δ0ε1.

For the low frequency part, i.e. |ξ| . sθ we see that

|v0,1(s, ξ) − v0,2(s, ξ)|

. 2n02 〈s〉2δ0ε1

( ∫R3

∫R3|v(s, σ)||v(s, η + σ)|dσ|η|−2β(n0)

n0(η)dη

+

∫R3

∫R3|v(s, ξ + σ)||v(s, ξ)|dσ|η|−2β(n0)

n0(η)dη

). 2

n02 〈s〉2δ0ε1‖| · |

−2β(n0)n0‖L1

(‖v ∗ v‖L∞ + 〈2k〉−5‖〈·〉5v‖L∞ ‖v‖L1

). 2

3n02 22δ0mε3

1

. 2(− 3+3δ0α+1 −5θ+2δ0)m25θm〈ξ〉−5ε3

1

. 2(−1−δ0)m〈ξ〉−5ε31 ,

provided δ0 ≤2−α3α .

Final approximation

We need to show that∣∣∣∣∣v0,2(s, ξ) − ic0

4π

∫R3|z|−1 |v(s, σ)|2dσ

1〈s〉

v(s, ξ)∣∣∣∣∣

. 2(−1−δ0)m〈ξ〉−5ε31 ,

(5.5.17)

wherez =

ξ

|ξ|2−α−

σ

|σ|2−α.

Observe that since the Fourier transform of |η|−2 is 2π2

|x| ,∣∣∣∣ ∫ eiη·x|η|−2β(n0)n0

(η)dη −2π2

|x|

∣∣∣∣ . |x|−22−n0 .

After change of variable and applying above inequality, (5.5.17) can be reducedto

2−n0

∫|sz|−2 |v(s, σ)|2dσ . 2(−1−δ0)mε2

1. (5.5.18)

81


Since |z| & min(|σ|α−1, |ξ−σ||σ|2−α

) from Lemma 2.3.3, we see that∫R3|z|−2 |v(s, σ)|2dσ . ‖〈·〉5v‖2L∞ .

Plugging this into (5.5.18), we obtain (5.5.17) provided α > 53 .

5.5.3 Proof of (5.5.13)

We aim to prove that ∑n1>n0

|∂svn1(s, ξ)| . ε312(−1−δ0)m〈ξ〉−5. (5.5.19)

Decomposing all the profiles dyadically, we have

∂svn1(s, ξ) =∑

k1,k2,k3∈Z

Ik1,k2,k3n1

(s, ξ), (5.5.20)

where

Ik1,k2,k3n1

(s, ξ)

= ic1

"R3×R3


(η)βk1(ξ + η)βk2(σ + ξ + η)βk3(ξ + σ)

× vk1(s, ξ + η)vk2(s, σ + ξ + η)vk3(s, ξ + σ)dηdσ.

By vk j we denote Pk j(v) for simplicity. Now Young’s convolution inequality yields

|Ik1,k2,k3n1

(s, ξ)| . 2−n12 ‖vk1‖L2‖vk2‖L2‖vk3‖L2 .

Since ‖vk‖L2 . min(〈2k〉−52

3k2 ε1, 〈2k〉−N sδ0ε1

), we observe that∑

k∈Z

‖vk(s)‖L2 . min(1, 〈2k〉−5sδ0)ε1. (5.5.21)

First, we consider the sum in (5.5.20) over k1, k2, k3 with min(k1, k2, k3) ≤ −23 (α+2

α+1 (1+

2δ0) + 53θ)m. We estimate the sum over k1 ≤ k2 ≤ k3 using (5.5.21), then the others

82


can be treated similarly. Since max(k1, k2, k3) ∼ k, we estimate∑n1>n0

k1≤−23 ( α+2

α+1 (1+2δ0)+ 53 θ)m

|Ik1,k2,k3n1

(s, ξ)|

.∑

n1>n0k1≤−

23 ( α+2

α+1 (1+2δ0)+ 53 θ)m

2−2l1‖β(n0)n1‖L2‖vk1‖L2

∑k2≥k1

‖vk2‖L2

∑k3≥k2

‖vk3‖L2

. 2−n02

∑k1≤−

23 ( α+2

α+1 (1+2δ0)+ 53 θ)m

‖vk1‖L2

∑k2≥k1

‖vk2‖L2〈2k〉−5sδ0ε1

. 2( 1+δ0α+1 + 5

3 θ+δ0)m〈2k〉−5ε31

∑k1≤−

23 ( α+2

α+1 (1+2δ0)+ 53 θ)m

23k1

2

. ε31〈2

k〉−5 2(−1−δ0)m.

Now we estimate the sum over max(k1, k2, k3) ≥ 2mN−5 . We also treat only the case

where max(k1, k2, k3) = k1 ∼ k.∑n1>n0

k1≥2m

N−5

|Ik1,k2,k3n1

(s, ξ)| .∑

n1>n0k1>

2mN−5

2−n12 ‖vk1‖L2

∑k2,k3≤k1

‖vk2‖L2‖vk3‖L2

. 2−n02

∑k1>

2mN−5

2(−N+5)k1 sδ0〈ξ〉−5ε31

. ε31〈ξ〉

−52(−1−δ0)m.

Let us consider the remaining case:

n1 > n0, −23(α + 2α + 1

(1 + 2δ0) +53θ)m ≤ k1, k2, k3 ≤

2mN − 5

. (5.5.22)

Since n1 ≤ max(k2, k3) + 10 ≤ 2mN−5 + 10, the remaining indexes in the sum are

O(m4). Thus, to prove (5.5.19) it suffices to show that

|Ik1,k2,k3n1

(s, ξ)| . ε312(−1−2δ0)m〈ξ〉−5

for each n1, k1, k2, k3 satisfying (5.5.22).Let us further decompose

Ik1,k2,k3n1

=∑n2∈Z

Ik1,k2,k3n1,n2

(s, ξ), (5.5.23)

83


where

Ik1,k2,k3n1,n2

(s, ξ)

= ic1

"R3×R3


(η)βn2(σ)βk1(ξ + η)βk2(ξ + η + σ)

× vk1(s, ξ + η)vk2(s, ξ + η + σ)vk3(s, ξ + σ)dηdσ.

The above terms are zero if n2 ≥2m

N−5 + 10. Moreover, we can estimate

|Ik1,k2,k3n1,n2

(s, ξ)| . 2−2n1〈ξ〉−5‖〈ξ〉5v‖3L∞‖β(n0)n1‖L1‖βn2‖L1

. ε31〈ξ〉

−52n123n2 .

This shows that ∑{n2:3n2+n1≤(−1−2δ0)m}

|Ik1,k2,k3n1,n2

(s, ξ)| . ε31〈ξ〉

−52(−1−2δ0)m. (5.5.24)

We are then left again with a summation over n2 with only O(m) terms. Therefore,(5.5.19) will be a consequence of the following.

Proposition 5.5.5. Let Ik1,k2,k3n1,n2 be defined by (5.5.23), and assume that ‖u‖ΣT≤ ε1.

Then one has|Ik1,k2,k3

n1,n2(s, ξ)| . ε3

1〈ξ〉−52(−1−3δ0)m,

whenever

−23(α + 2α + 1

(1 + 2δ0) +53θ)m ≤ k1, k2, k3 ≤

2mN − 5

,

n1 > n0 and n1 + 3n2 ≥ (−1 − 2δ0)m.(5.5.25)

Case: max(k1, k2) ≤ n1

Recall that

Ik1,k2,k3n1,n2

(s, ξ) = ic1

"R3×R3

mk1,k2,k3n1,n2

(η, σ)eisφ(ξ,η,σ)

× vk1(s, ξ + η)vk2(s, ξ + η + σ)vk3(s, ξ + σ)dηdσ,

84


wheremk1,k2,k3

n1,n2(η, σ) = |η|−2βn1(η)βn2(σ)βk1(ξ + η)βk2(ξ + η + σ).

It can be readily checked that mk1,k2,k3n1,n2 verifies the assumption of Lemma 2.3.1 with

A(mk1,k2,k3n1,n2 ) = 2−2n1 . Since |ξ| ≤ |ξ + η + σ| + |ξ + η| + |ξ + σ| and hence 〈ξ〉5 . 2

10mN−5

by (5.5.25), we can estimate

|Ik1,k2,k3n1,n2

(s, ξ)| . 2−2n1‖vk1‖L2‖vk2‖L2‖Pk3(u)‖L∞

. s−32 2−2 max(k1,k2)2

3k12 〈2k1〉−52

3k22 〈2k2〉−5ε3

1

. 2−32 m2

12mN−5 〈ξ〉−5ε3

1

. 2(−1−3δ0)m〈ξ〉−4ε31 ,

provided δ0 ≤16 −

4N−5 .

Case: max(k1, k2) ≥ n1, |k1 − k2| ≥ 10

Lemma 5.5.6. Suppose f satisfies the condition of Proposition 5.4.2. Then wehave

‖∇vk(s)‖L2 . min(2k2 , 〈2k〉−3)s2δ0ε1,

‖∇2vk(s)‖L2 . 2−k〈2k〉−2s2δ0ε1.

Proof of Lemma 5.5.6. By Sobolev inequality we have

‖βk xv(s)‖L2 . ‖βk‖L3‖xv(s)‖L6 . 2k‖x2v(s)‖L2 . 2ks2δ0ε1.

Also it is easy to see that

‖βk xv(s)‖L2 . 〈2k〉−3‖xv(s)‖H3 . 〈2k〉−3〈s〉δ0ε1.

Then, we have

‖∇vk(s)‖L2 . 2k2 ‖vk‖L∞ + ‖βk xv(s)‖L2

. 2k2 〈2k〉−5‖〈ξ〉5v‖L∞ + min(2k, 〈2k〉−3)s2δ0ε1

. min(2k2 , 〈2k〉−3)s2δ0ε1.

85


And we estimate

‖∇2vk(s)‖L2 . 2−k2 ‖vk‖L∞ + ‖∇βk ⊗ ∇v‖L2 + ‖βk∇

2v‖L2

. 2−k2 〈2k〉−5ε1 + 2−k〈2k〉−2‖xv‖H2 + 〈2k〉−2‖x2v‖H2

. 2−k〈2k〉−2s2δ0ε1.

�

Here it should be noticed that in this range the following additional conditionholds:

2k3 . 2max(k1,k2). (5.5.26)

This leads us to2k ∼ 2max(k1,k2) . 2

2mN−5 . (5.5.27)

In order to make an efficient integration by parts w.r.t. η in the expression(5.5.23) we introduce the identity

eisφ(ξ,η,σ) =1is

P(ξ, η, σ) · ∇η[eisφ(ξ,η,σ)], P(ξ, η, σ) =∇η[φ(ξ, η, σ)]|∇η[φ(ξ, η, σ)]|2

.

Then we writeIk1,k2,k3n1,n2

(s, ξ) = −I1(s, ξ) − I2(s, ξ),

where

I1(s, ξ) =c1

s

"R3×R3

eisφ(ξ,η,σ)Q(ξ, η, σ)·∇η[vk1(s, ξ + η)vk2(s, ξ + η + σ)

]× vk3(v)(s, ξ + σ)dηdσ,

I2(s, ξ) =c1

s

"R3×R3

eisφ(ξ,η,σ)∇η · [Q(ξ, η, σ)]vk1(s, ξ + η)vk2(s, ξ + η + σ)

× vk3(v)(s, ξ + σ)dηdσ

and

Q(ξ, η, σ) = P(ξ, η, σ)|η|−2β(n0)n1

(η)βn2(σ)βk1(ξ + η)βk2(ξ + η + σ).

We first estimate I1.

86


Lemma 5.5.7. Consider

|k1 − k2| ≥ 10, max(k1, k2) ≥ n1 (5.5.28)

and let

Q`1γ (ξ, η, σ) = ∇`1

η ;[P(ξ, η, σ)|η|−γβn0

n1(η)βn2(σ)βk1(η + ξ)βk2(η + σ + ξ)

]for nonnegative integer γ and `1. Then Q`1

γ satisfies the assumption of Lemma 2.3.1with

‖F −1(Q`1γ )‖L1(R6) . 2−max(k1,k2)(α−1)2−γn12−`1 min(k1,n1). (5.5.29)

Proof of Lemma5.5.7. Performing integration by parts and then change of vari-ables, we have∥∥∥∥∥"

R3×R3Q`1γ (ξ, η, σ)eixηeiyσdηdσ

∥∥∥∥∥L1

x,y

∼

∥∥∥∥∥x`1

"R3×R3

Q0γ(ξ, η, σ)eixηeiyσdσdη

∥∥∥∥∥L1

x,y

∼

∥∥∥∥∥x`1

"R3×R3

Q0γ(ξ, η, σ)eixηeiyσdηdσ

∥∥∥∥∥L1

x,y

,

where

Q0γ(ξ, η, σ) =

|σ|α−2σ − |η|α−2η∣∣∣|σ|α−2σ − |η|α−2η∣∣∣2 |η − ξ|−γβn1(η − ξ)βn2(σ − η)βk1(η)βk2(σ).

In the range of (5.5.28), we see that Q0γ verifies the following inequality: for any

positive integers ˜1, ˜2

|∇˜1η ∇

˜2σ Q0

γ(ξ, η, σ)| . 2−max(k1,k2)(α−1)2−γn12−min(k1,n1 )`12−k2˜2

× βn1(η − ξ)βn2(σ − η)βk1(η)βk2(σ).

This gives us the desired result. �

87


Now applying Lemma 2.3.1 to I1 with (5.5.29) and Lemma 5.5.6, we estimate

|I1(s, ξ)| . s−12−max(k1,k2)(α−1)2−2n1

×(‖∇vk1‖L2‖vk2‖L2 + ‖vk1‖L2‖∇vk2‖L2

)‖uk3(s)‖L∞

. 2−m2−max(k1,k2)(α−1)2−2n1(〈2k1〉−32

k12 22mδ02

3k22 〈2k2〉−5

+ 23k1

2 〈2k1〉−5〈2k2〉−32k22 22mδ0

)2−

3m2 ε3

1

. 2(− 52 +2δ0)m2−2n0〈2k〉−3ε3

1 (n1 > n0)

. 2(− 52 +2δ0)m2( 4+4δ0

α+1 + 203 θ)m2

4mN−5 〈2k〉−5ε3

1

. 2(−1−3δ0)m〈2k〉−5ε31 ,

(5.5.30)

since α > 5/3 and 0 < δ0 ≤ δ3 := 15α+9 (α − 5

3 −4(α+1)

N−5 ). We have used (5.5.27) forthe third and last inequality. From now on, we will use this technique repeatedlyto derive 〈2k〉−5 term without mentioning it.

Let us move onto I2. Due to the additional term 2−min(k1,n1) in (5.5.29), wecannot apply the Lemma 2.3.1 to I2 as before. So, we perform an additional inte-gration by parts in I2

I2(s, ξ) = J1(s, ξ) + J2(s, ξ), (5.5.31)

J1(s, ξ) = ic1

s2

"R3×R3

eisφ(ξ,η,σ)[∇η ·Q(ξ, η, σ)]

× P(ξ, η, σ) · ∇η(vk1(s, ξ + η)vk2(s, ξ + η + σ)

)vk3(s, σ + ξ)dσdη,

J2(s, ξ) = ic1

s2

"R3×R3

eisφ(ξ,η,σ)∇η · [((∇η ·Q)P)(ξ, η, σ)]

× vk1(s, ξ + η)vk2(s, ξ + η + σ)vk3(s, σ + ξ)dσdη.

By Lemma 5.5.7 and Lemma 2.3.2 we have

‖F −1((∇η ·Q)P)‖L1(R6) . ‖F−1(Q1

2 ⊗Q00)‖L1(R6)

. 2−2 max(k1,k2)(α−1)2−2n12−min(k1,n1).

88


Applying Lemma 2.3.1 to J1 with above bound and using Lemma 5.5.6, we obtain

|J1(s, ξ)| . s−22−2 max(k1,k2)(α−1)2−2n12−min(k1,n1)

×[‖∇vk1‖L2‖vk2‖L2 + ‖vk1‖L2‖∇vk2‖L2

]‖uk3‖L∞

. 2−72 m+2δ0m〈2k〉−32−2n12−min(k1,n1)ε3

1

. s−72 m+2δ0m〈2k〉−52m

(4

N−5 +4+4δ0α+1 + 20θ

3 + 23

(α+2α+1 (1+2δ0)+ 5θ

3

))ε3

1 .

If α > 53 , then

23(α + 2α + 1

(1 + 2δ0) +5θ3

)< 1.

And hence we get as in (5.5.30) that

|J1(s, ξ)| . 2(−1−3δ0)m〈2k〉−5ε31 .

To estimate J2 in (5.5.31), we only use the pointwise bound

|∇η · ((∇η ·Q)P)(ξ, η, σ)| . 2−2 max(k1,k2)(α−1)2−2n12−2 min(k1,k2,n1),

and then we see that

|J2(s, ξ)| .1s2 ‖∇η · ((∇η ·Q)P)‖L∞η,σ‖βn1 vk1(ξ + ·)‖L1‖vk2‖L2‖vk3‖L2

. s−22−2 max(k1,k2)(α−1)2−2n12−2 min(k1,k2,n1) min(23n1 , 23k1)

× 〈2k1〉−523k2

2 〈2k2〉−523k3

2 〈2k3〉−5ε31 .

Using (5.5.26), we see that

|J2(s, ξ)| .

ε312−2m〈2k〉−52−

k22 , if min(k1, k2, n1) = k2,

ε312−2m〈2k〉−52−n1 , otherwise.

And both cases give the desired bound of J2 because of (5.5.25).

Case: max(k1, k2) ≥ n1, |k1 − k2| ≤ 10

First, let us observe that

2k3 . 2k ∼ 2k1 ∼ 2k2 . 22m

N−5 .

89


Since |ξ + η| ∼ 2k1 , |ξ + η + σ| ∼ 2k2 with k1 ∼ k2, |η| ∼ 2n1 and |σ| ∼ 2n2 , one has

|∇`1η ∇

`2σP(ξ, η, σ)| . 2k1(2−α)2−n22−n1`12−n2`2 ,

|∇`1η ∇

`2σQ(ξ, η, σ)| . 2k1(2−α)2−n22−2n12−n1`12−n2`2

(5.5.32)

for any positive integers `1, `2. These can be induced from mean-value theorem,Lemma 2.3.3 and the observation that n2 ≤ max(k1, k2) + 10. As a consequence,

‖F −1(P ⊗Q)‖L1(R6) . 2k1(2−α)2−2n12−2n2 , (5.5.33)

‖F −1((∇η ·Q)P)‖L1(R6) . 22k1(2−α)2−3n12−2n2 . (5.5.34)

Now if we try to estimate I1 applying the Lemma 2.3.1 with (5.5.33) as before,then due to 2−n2 term we cannot obtain the desired result in this case. So, weperform an integration by parts twice in the expression for Ik1,k2,k3

n1,n2 in (5.5.23).With the previous calculation in (5.5.31), we write

|Ik1,k2,k3n1,n2

(s, ξ)| . |J1(s, ξ)| + |J2(s, ξ)| + |J3(s, ξ)|,

J3(s, ξ) = ic1

s2

"R3×R3

eisφ(ξ,η,σ)(P ⊗Q)(ξ, η, σ);

∇2η

(vk1(s, ξ + η)vk2(s, ξ + η + σ)

)vk3(s, σ + ξ)dσdη.

First applying Lemma 2.3.1 to J3 with (5.5.33), and then Lemma 5.5.6, we haveas in (5.5.30) that

|J3(s, ξ)| . s−222k1(2−α)2−2n12−2n2‖u3‖L∞

×(‖∇2vk1‖L2‖vk2‖L2 + ‖∇vk1‖L2‖∇vk2‖L2 + ‖vk1‖L2‖∇2vk2‖L2

). 2(− 7

2 +4δ0)m2−2n12−2n2〈2k〉−6ε31

. 2(− 72 +4δ0)m2−

23 (n1+3n2)2−

4n13 〈2k〉−5ε3

1

. 2(− 72 +4δ0)m2

23 (1+2δ0)m2−

4n13 〈2k〉−5ε3

1

. 2(−1−3δ0)m〈2k〉−5ε31 ,

90


provided δ0 ≤1

100 . In the third inequality, we have used n1 > n0 and n1 + 3n2 ≥

−(1 + 2δ0)m.To estimate J1, we also apply Lemma 2.3.1 with (5.5.34):

|J1(s, ξ)|

. s−222k1(2−α)2−3n12−2n2(‖∇vk1‖L2‖vk2‖L2 + ‖vk1‖L2‖∇vk2‖L2

)‖u3‖L∞

. 2−7m2 22δ0m2−

23 (n1+3n2)2−

7n13 〈2k〉−5ε3

1

. 2(− 72 +2δ0)m2( 2

3 +4δ0

3 )m2( 14+14δ03(α+1) + 21α−35

36(α+1) )m〈2k〉−5ε3

1

. 2(−1−3δ0)m〈2k〉−5ε31

for δ0 ≤1

100 . J2 can be treated with the pointwise bound for ∇η((∇η · Q)P). From(5.5.32), it follows that

|∇η((∇η ·Q)P)(ξ, η, σ)| . 2−2n22(4−2α)k12−4n1 .

Note that in this range we have 2n2 . 2max(k1,k2). Thus if δ0 ≤1

100 , then we see that

|J2(s, ξ)| .1s2 ‖∇η((∇η ·Q)P)‖L∞η,σ‖β

(n0)n1‖L1‖βn2‖L1

3∏i=1

‖vki‖L∞

. s−22(4−2α)k12n22−n1〈2k1〉−5〈2k2〉−5〈2k3〉−5ε31

. s−22(4−2α)k12max(k1,k2)2−n1〈2k1〉−5〈2k2〉−5〈2k3〉−5ε31

. 2−2m2−n1〈2k〉−5ε31

. 2−2m2( 2+2δ0α+1 + 3α−5

12(α+1) )〈2k〉−5ε3

1

. 2(−1−3δ0)m〈2k〉−5ε31 .

91

Appendix A

Ill-posedness

In this section we pose the ill-posedness of following fractional Schrodinger equa-tion −i∂tu + (−∆)

α2 u = κ(| · |−γ ∗ |u|2)u

u(0, ·) = ϕ ∈ H s(Rd), d ≥ 2.(A.0.1)

In [4] local well-posedness of (A.0.1) is established for s ≥ γ(2−α)4 . We now

claim that this result is optimal. The main idea of proof is similar to the radialcase, Theorem 3.1.3 in Section 3.3.3.

Theorem A.0.8. If there exists a time T > 0 such that a unique solution to (A.0.1)exists on the interval [−T,T ] and the flow map ϕ 7→ u is C3 at origin from H s(Rd)to C([0,T ] : H s(Rd)), then s ≥ γ(2−α)

4 .

Proof. Since the flow map is C3 from H s(Rd) to C([0,T ] : H s(Rd)), it holds

supt∈[0,T ]

∥∥∥∥∥ ∫ t

0Sα(t − τ)(| · |−γ ∗ |Sα(τ)ϕ|2Sα(τ)ϕ)dτ

∥∥∥∥∥Hs(Rd)

. ‖ϕ‖3Hs(Rd), (A.0.2)

for all ϕ ∈ H s(Rd).Let 1 < µ � λ. We will choose µ = µ(λ) = δλ

2−α2 for 0 < δ � 1. Define the

cube

W±λ = {(ξ1, · · · , ξd) ∈ Rd : |ξ1 ∓ λ| ≤ µ, |ξi| ≤ µ, for i = 2, · · · , d}

92

APPENDIX A. ILL-POSEDNESS

Let ϕ be the inverse fourier transform of the characteristic function χW+λ. Obvi-

ously, ‖ϕ‖Hs(Rd) ∼ µd2λs. Next, we consider

Ft(ξ) := Fx

( ∫ t

0Sα(t − τ)(| · |−γ ∗ |Sα(τ)ϕ|2Sα(τ)ϕ)dτ

).

Our goal is to show that for 0 < t � 1 and all ξ ∈ W+λ ,

|Ft(ξ)| & |t|µd+γ. (A.0.3)

Assuming that (A.0.3) holds, we have from (A.0.2)

|t|λsµ3d2 +γ . ‖〈ξ〉sFt(ξ)‖L2(Rd) . µ

3d2 λ3s

which for µ = δλ2−α

2 is equivalent to

|t|δγ . λ2s− 2−α2 γ

which can hold only if s ≥ 2−α4 γ. Hence, it suffices to show (A.0.3).

We compute

Ft(ξ) = eit|ξ|α"Rd×Rd

∫ t

0eiτrα(ξ,η,σ)dτ

χW+λ(η)χW−λ (σ)χW+

λ(ξ − η − σ)

|η + σ|d−γdηdσ,

gα(ξ, η, σ) = −|ξ|α + |η|α − |σ|α + |ξ − η − σ|α.

We notice that in the domain of integration we have by mean-value theorem∣∣∣ − |ξ|α + |η|α − |σ|α + |ξ − η − σ|α∣∣∣ . µ2λα−2.

Hence, |gα(ξ, η, σ)| � 1 (by choosing δ > 0 small enough). Then, it is clear that∣∣∣∣∣ ∫ t

0eiτgα(ξ,η,σ)dτ

∣∣∣∣∣ & ∫ t

0cos (τgα(ξ, η, σ))dτ & |t|.

Therefore, if ξ ∈ W+λ , we have

|Ft(ξ)| & |t|∫

W+λ

∫W−λ

|η + σ|−d+γdηdσ & |t|µd+γ.

�

93

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97

국문초록

이논문에서는삼차하트리비선형분수차슈뢰딩거방정식의초깃값문

제에관한세가지주제를다룬다.첫번째와두번째주제는낮은정칙성조건에서해의존재성,유일성과산

란성에관한연구이다.각에대한대칭성또는각에대한정칙성이초깃값에주어지고 크기가 충분히 작은 경우, 질량 임계 방정식의 해의 존재성과 산란성을척도불변함수공간에서보였다.초깃값이척도불변함수공간보다낮은 정칙성을 갖는 경우에 대해서는 해가 존재 하지 않는 다는 것을 보임

으로써, 우리가 얻은 결과가 최적이라는 것을 보였다. 이를 증명하기 위해선형,이중선형부등식과이차변동이유계인함수들의공간에관한이론이적용되었다.세번째주제는퍼텐셜이쿨롬형태인방정식의수정된산란성에관한연

구이다.이런형태의방정식은 ”임계산란성”문제라고불린다.우리는해의비선형적인 점근적 행동을 분석하고 적절히 수정함으로써, 시간이 흐름에따른 해의 산란성을 묘사하였다. 이를 보이기 위해 시간에 따른 감소 비율부등식과가중부등식이정밀하게계산되었다.이부등식을적용하고승수가 0이 되는 구조를 분석함으로써, 해에 관한 푸리에 공간에서의 점근적인해석이이루어졌다.

주요어휘:분수차슈뢰딩거방정식,하트리비선형,낮은정칙성문제,산란성,선형과이중선형부등식학번: 2012-20249

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