The ``Finite time' Euler equation: An Introduction to the ...€¦ · The \Finite time" Euler equation: An Introduction to the Workshop Claude Bardos Retired, Laboratoire Jacques

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The “Finite time” Euler equation: An Introduction tothe Workshop

Claude Bardos

Retired, Laboratoire Jacques Louis Lions, Universite Pierre et Marie

Analysis and Computation of Incompressible Fluid Flow IMA 2010.

Claude Bardos The “Finite time” Euler equation: An Introduction to the Workshop

Introduction: The Euler and Navier-Stokes equations

The incompressible Navier-Stokes and Euler equation.

∂tuν + uν · ∇uν − ν∆uν +∇pν = 0

∇ · uν =∑

1≤i≤d

∂xi (uν)i = 0 , uν · ∇uν =∑

1≤i≤d

(uν)i∂xi uν .

ν ≥ 0 is the viscosity the pressure ∇p is the Lagrange multiplier of theconstraint ∇ · u = 0 ν > 0⇒ Navier-Stokes, ν = 0⇒ Euler . Abstract form

(no external force, no coupling with the temperature, no boundary pure fluid

effect).

Derivation of Euler equation 1755. Application of Newton laws to continuous

media.

D’Alembert, 1759 (for a Prize Problem of the Berlin Academy on flow drag): “It

seems to me that the theory (potential flow), developed in all possible rigor,

gives, at least in several cases, a strictly vanishing resistance, a singular paradox

which I leave to future Geometers to elucidate”. The d’Alembert paradox which

would imply that birds cannot fly. The resolution lies in the introduction of the

viscosity and the boundary effects. Navier and Stokes (1822), (1845)Claude Bardos The “Finite time” Euler equation: An Introduction to the Workshop

Introduction: The Euler and Navier-Stokes equations

The incompressible Navier-Stokes and Euler equation.

∂tuν + uν · ∇uν − ν∆uν +∇pν = 0

∇ · uν =∑

1≤i≤d

∂xi (uν)i = 0 , uν · ∇uν =∑

1≤i≤d

(uν)i∂xi uν .

ν ≥ 0 is the viscosity the pressure ∇p is the Lagrange multiplier of theconstraint ∇ · u = 0 ν > 0⇒ Navier-Stokes, ν = 0⇒ Euler . Abstract form

(no external force, no coupling with the temperature, no boundary pure fluid

effect).

Derivation of Euler equation 1755. Application of Newton laws to continuous

media.

D’Alembert, 1759 (for a Prize Problem of the Berlin Academy on flow drag): “It

seems to me that the theory (potential flow), developed in all possible rigor,

gives, at least in several cases, a strictly vanishing resistance, a singular paradox

which I leave to future Geometers to elucidate”. The d’Alembert paradox which

would imply that birds cannot fly. The resolution lies in the introduction of the

viscosity and the boundary effects. Navier and Stokes (1822), (1845)Claude Bardos The “Finite time” Euler equation: An Introduction to the Workshop

Introduction: The Euler equation

The Euler equation too abstract to describe physical situations. Moreover

Scheffer, Shnirelman and De Lellis and Szekelyhidi ⇒ solutions that would

correspond to instantaneous creation or extinction of energy and as such would

solve the energy crisis. In many cases (most of the cases) viscosity is small. Its

study is essential for the “physical” and mathematical structure of the problem.

Figure: Euler, D’Alembert, Navier and Stokes


















Introduction: The Euler equation and Statistical Theory ofTurbulence

When the viscosity ν goes to zero the fluid may become turbulent anddescribed by a “random flow ”⇒The statistical theory of turbulence inparticular one has the Kolmogorow law:

〈|u(x + r)− u(x)|2〉12 ∼ (ν〈|∇u|2〉)

23 |r |

13

What can be deduced for individual solutions from the statistical theory?..Claude Bardos The “Finite time” Euler equation: An Introduction to the Workshop

Introduction: The Euler equation and Statistical Theory ofTurbulence

When the viscosity ν goes to zero the fluid may become turbulent anddescribed by a “random flow ”⇒The statistical theory of turbulence inparticular one has the Kolmogorow law:

〈|u(x + r)− u(x)|2〉12 ∼ (ν〈|∇u|2〉)

23 |r |

13

What can be deduced for individual solutions from the statistical theory?..Claude Bardos The “Finite time” Euler equation: An Introduction to the Workshop

Basic estimates Euler equation energy and vorticity

∂tu +∇ · (u ⊗ u) +∇p = 0 , ∇ · u = 0 , in Ω .

u · ~n = 0 on ∂Ω ,

∂tω + u · ∇ω = ω · ∇u in Ω

∇ · u = 0 ,∇∧ u = ω in Ω , and u · ~n = 0 on ∂Ω

Formal Energy conservationd

dt

∫Ω

|u(x , t)|2

2dx = 0

Formal 2d Vorticity conservation ∂tω + u · ∇ω = 0 ,

x(s) = u(x(s), s) ,D

Dtω =

d

dtω(x(t), t) = 0.


Content

1 The Shear flow.

2 Cauchy Problem Instabilities and criticallity of C 1

3 Weak solutions existence, uniqueness regularity and conservation ofenergy. There are singular solutions that conserve the energy.Onsager conjecture : For weak solutions equation energy decay isrelated to some loss of regularity. 1/3 appears to be a critical value ofsuch regularity. What is proven is that any solution more regular than

the Besov space B133,∞ conserve energy Constantin E Titi; Cheskidov

Constantin Friedlander.

4 Weak limit of oscillating solutions may not be solutions of the Eulerequation and the corresponding “turbulent ” tensor is different fromwhat would be predicted by statistical theory of turbulence.


The Shear flow and the basic issues

5 There exists solutions of the Navier-Stokes equations with viscosityν → 0 which converge to rough solutions of the Euler equations withno Energy dissipation.

6 Conjecture: New sophisticated estimates on Navier-Stokes (Besov,BMO−1 ) are all viscosity dependent with may be the exception of L3

7 More remarks about energy dissipation.

8 Boundary effects and energy dissipation.


The Shear Flow

u(x , t) = (u1(x2), 0, u3(x2, x1 − tu1(x2)) (1)

∇ · u = 0 and ∂tu +∇(u ⊗ u) = −∇p with p ≡ 0 (2)

d

dt

∫ ∫ ∫|u(x1, x2, x3, t)|2dx1dx2dx3 = 0 (3)

2 (in the sense of distributions) and 3 (on the torus ) are true under theonly hypothesis that

u1 ∈ L2x2

and u3 ∈ L2x1,x2

In most of the examples u3(x1 − tu1(x2))


The Shear Flow

The shear flow is a special example introduced in our community by DiPerna and Majda. It does not fit in the statistical theory of turbulencebecause

It is not turbulent...

In a statistical theory it corresponds to a very “special class of events”

I use it to shows that there is no hope to prove statistical results bydeterministic functional analysis.


Instability of Cauchy Problem, Loss of regularity

For initial data in C 1,α the Euler equation has a unique local in timesolution in C 1,α

∂t ||ω||0,α ≤ ||ω · ∇u||0,α + ||ω||0,α||u||lip

The original proof of Lichtenstein (1925) based on

∂tω + u · ∇ω = ω · ∇u

u(x , t) =

∫G (x , y)ω(y)dy ,G (x , y) = O(

1

|x − y |d−1)

Same result later for initial data in Hs , s > 52 .


Stability- Blow up ??-Weak solutions

Stability persistence of regularity as long as

Beale Kato Majda

∫ t

0||ω(t)||L∞ <∞

Constantin, Fefferman, Majda u(t) ∈ L∞ , | ω(x)

|ω(x)|− ω(y)

|ω(y)|| ≤ |x − y |α

Global in 2d Youdovitch-Wolibner..


W 1,p Instability

Theorem

Di Perna - Lions For every p ≥ 1, T > 0 and M > 0 there exists a smoothshear flow solution for which ‖u(x , 0)‖W 1,p = 1 and ‖u(x ,T )‖W 1,p > M.

Proof (Bardos-Titi version)

∂x2u3(x2, x1 − tu1(x2)) ∼ −t∂x2u1(x2)∂x1u3(x2, x1 − tu1(x2))

u(0) ∈W 1,p does not implies u(t) ∈W 1,p .


C 1 Criticallity

Theorem

In the Holder spaces C 1 is the critical space for local in time wellposedness

1 For (u1(x), u3(x)) ∈ C 1,α , 0 ≤ α < 1 the shear flow solution is alwaysin C 1,α ,

2 For (u1(x), u3(x)) ∈ C 0,α the shear flow solution is always in C 0,α2,

There exists shear flow solutions which for t = 0 belong to C 0,α andwich for t 6= 0 are not in C 0,β for β > α2.


Proof

Regularity results concern only the component u3

|u3(x1 − tu1(x2 + h))− u3(x1 − tu1(x2))|hα2 =

|u3(x1 − tu1(x2 + h))−u3(x1 − tu1(x2))||tu1(x2 + h)−tu1(x2)|α

(|tu1(x2 + h)−tu1(x2)|

hα

)α≤ |t|α||u3||0,α(||u1||0,α)α .

Introduces two periodic functions u1 and u3 which near the point x = 0coincide with |x |α . Then the for t given and x1 and x2 small enoughu3(x1 − tu1(x2)) coincides with

|x1 − t|x2|α|α

For (x1, x2, x3) = (0, x2, x3) one has

u3(x1 − tu1(x2)) = |t|α|x2|α2

and the conclusion follows.Claude Bardos The “Finite time” Euler equation: An Introduction to the Workshop

Other spaces and optimal spaces

The Besov spaces:

Bsp,q = f \

∑j∈Z||2js∆j f ||qLp <∞

C 1,α = B1+α∞,∞ ⊂ B1

∞,1 ⊂ C 1 ⊂ F 1∞,2 ⊂ B1

∞,∞ ⊂ Bα∞,∞ = C 0,α .

Theorem

The 3d Euler equation is well posed in B1∞,1(Pak and Park). It is not well

posed in B1∞,∞ or in the Triebel-Lizorkin space ⊂ F 1

∞,2


Proof

B1∞,∞ is the Zygmund class ie bounded functions with

supx∈R,h∈R

|f (x + h) + f (x − h)− 2f (x)||h|

<∞

They are not Lipschitz but log-Lipschitz

|f (x + h)− f (x)| ≤ C |h| log1

|h|

Now v(y) smooth outside 0 with

v(y) ∼ y log1

|y |near 0

is in the Zygmund class. Then with u1(y) = u3(y) = v(y) and x1 = 0

|u3(−tu1(h))− u3(−tu1(0))| ∼ th(log h)2!!

Same proof for ⊂ F 1∞,2. More delicate: Construction of a log lipschitz

function in this space.Claude Bardos The “Finite time” Euler equation: An Introduction to the Workshop

Weak solutions

To be considered, if the smooth solution lose regularity, if the initial dataare not regular, for the limit of the zero viscosity Navier-Stokes equation

∂tu +∇ · (u ⊗ u) +∇p = 0

Require at least u ∈ L2

In 2d with initial data ω0 ∈ Lp , 1 < p ≤ ∞ existence of weak solutionfor the Cauchy problem. Ok also for signed measures or measureswith change of sign by symmetries.

Non uniqueness for a set of initial data (no explicit construction butlarge: residual set) Scheffer, Shnirelman, DeLellis and Szekelyhidi.

Huge gap in 2d between the requirement for uniqueness ω0 ∈ L∞ andfor existence ω0 ∈ L1 .


Weak solutions


∂tu +∇ · (u ⊗ u) +∇p = 0






Weak solutions


∂tu +∇ · (u ⊗ u) +∇p = 0






Weak solutions


∂tu +∇ · (u ⊗ u) +∇p = 0






Weak solutions

Try to define a critical threshold for uniqueness. The construction ofDe Lellis and Szekelyhidi bears many similarities with the theory ofisometric imbedding. For any 0 < r < 1 there is a C 1 isometricimbedding of Sn(1) into Bn+1(r) and there is no C 2 imbedding.

May not conserve energy. What is proven is that any solution more

regular than the Besov space B133,∞ conserve energy Constantin E Titi;

Cheskidov Constantin Friedlander. It is also believed that in theaverage conservation of energy should imply a better regularity than“ 1

3 ”. “Not-explicit examples” of singular solutions conserving energygiven by De Lellis and Szekelyhidi. Explicit examples with the shearflow on the torus. It conserves the energy with L2 regularity!

Construction by weak or zero viscosity limit (done in 2d withoutboundary open problem in 3d)


Weak solutions








Weak solutions








Wigner Measure-Reynold stress tensor

uε ∈ L∞(L2) ∇ · uε = 0

∂tuε +∇ · (uε ⊗ uε) +∇pε = 0

∇ · u = 0 in Ω , u · ~n = 0 on ∂Ω ,

∂tu +∇ · (u ⊗ u) +∇ · RT (uε) +∇p = 0 in Ω ,

RT (uε)(x , t) = limε→0

((uε − u)⊗ (uε − u)) =

lim(uε ⊗ uε)− (lim uε ⊗ lim uε) = 0 modulo ∇q ????



Assume that the sequence uε is ε-oscillating

ε2

∫ t

0

∫|∇uε|2dxdt ≤ C

Then modulo a subsequence RT is given by a Wigner Measure

W (x , k , t) = lim1

(2π)d

∫Rd

e−ik·r uε(x + εr

2, t)⊗ u(x − ε r

2, t)dr

RT (x , t) =

∫(W (x , k , t)− (u ⊗ u)(x , t)δk)dk



There is some similarity between weak convergence and statistical theoryof turbulence with random fluctuations of mean value 0. u = U + u

∂tU +∇ · (U ⊗ U) +∇ · 〈u ⊗ u〉+∇p = 0

〈u ⊗ u〉 =

∫W (x , t, k)dk

W (x , t, k)dk =1

(2π)d

∫Rd

e−ik·r 〈u(x +r

2, ·)⊗ u(x − r

2, ·)〉dr

In the statistical theory of turbulence this spectra has the followingproperties: homogeneity isotropy and decay

|k|2Trace(W (x , t, k)) ' |k |−53


Weak limits of shear flows

Original Di Perna-Majda example: Sequence of weak solutions ε oscillatingand with energy estimate:

∇ · u = 0 , ∂tu +∇ · (u ⊗ u) +∇ · RT (uε) +∇p = 0 ,

RT (uε)(x , t) = limε→0

((uε − u)⊗ (uε − u)) 6= 0

uε(x , t) = (u1(x2

ε), 0, u3(x1 − tu1(

x2

ε))

∫ 1

0u1(s)ds = 0

limε→0

weak uε = (0, 0, u3) , u3 =

∫ 1

0u3(x1 − tu1(s))ds

∂tu3 + limε→0∇ · uε ⊗ uε3 = 0

limε→0∇ · uε ⊗ uε3 = ∂x1

∫ 1

0u1(s)u3(x1 − tu1(s))ds 6= ∂x1 · u1 ⊗ u3 = 0

In this example no istropy no rate of decay for the Turbulent spectra! Tooparticular??


Viscosity limit, estimates and energy dissipation.

The viscous limit of Leray solution of 3d Navier-Stokes is an openproblem. It is also an open problem in 2d in the presence of no slipboundary. It is also a common, belief that the appearance of roughsolution of Euler equation is related to energy dissipationThe only 3d general result is the fact that the limit is a dissipative solutionin the sense of P.L. Lions.However DeLellis and Szekelyhidi have shown that the notion of dissipativesolution is not a criteria for stability and uniqueness!


Viscosity limit, estimates and energy dissipation.

The viscous limit of Leray solution of 3d Navier-Stokes is an openproblem. It is also an open problem in 2d in the presence of no slipboundary. It is also a common, belief that the appearance of roughsolution of Euler equation is related to energy dissipationThe only 3d general result is the fact that the limit is a dissipative solutionin the sense of P.L. Lions.However DeLellis and Szekelyhidi have shown that the notion of dissipativesolution is not a criteria for stability and uniqueness!


Viscosity limit, estimates and energy dissipation

∂tuν + uν · ∇uν − ν∆uν +∇pν = 0 , ∇ · uν = 0 ,

1

2|uν(t)|2 + ε(t) ≤ 1

2|uν(0)|2 , ε(t) = ν

∫ t

0

∫|∇uν(x , t)|2dxdt .

For the shear flow the solution is given by

uν(x1, x2, x3) = ((uν)1(x2, t), 0, (uν)3(x1, x2, t))

∂t(uν)1(x2, t))− ν∂2x2

(uν)1(x2, t)) = 0 ,

∂t(uν)3 + (uν)1(x2, t)∂x1(uν)3 − ν(∂2x2

+ ∂2x2

)(uν)3 = 0 .

Proposition

With L2((R/Z)3) initial data (uν)1(x2, t) converges strongly (inC (R+

t ; L2((R/Z))) to u2(x2, 0) and uν converges in to the shear flowsolution.


Viscosity limit.

In the Torus (R/Z)3:

Proposition

For ν > 0 and u0 = u(x , 0) ∈ X with

X ∈ H12 ⊂ L3 ⊂ B

−1+ 1p

1,∞ (1 ≤ p <∞) ⊂ BMO−1 (4)

there exists a time < Tν(u0)∗ ≤ ∞ and a constant C (ν, u0) such that:there exist a unique solution uν(x , t) ∈ C (0,T ∗; X ) of theν−Navier-Stokes equation which satisfies the estimate:

for u0 ∈ X and 0 ≤ t ≤ T ∗ν ||u(x , t)||X ≤ C (ν, u0) (5)

The previous examples and computations on the shear flow show that for

X = H12 the statement of the above proposition has no chance to be true

with u0 ∈ X and T ∗ν ,C (ν, u0) ν-independent. Conjecture same result forother spaces.


Viscosity limit.

In the Torus (R/Z)3:

Proposition

For ν > 0 and u0 = u(x , 0) ∈ X with

X ∈ H12 ⊂ L3 ⊂ B

−1+ 1p

1,∞ (1 ≤ p <∞) ⊂ BMO−1 (4)

there exists a time < Tν(u0)∗ ≤ ∞ and a constant C (ν, u0) such that:there exist a unique solution uν(x , t) ∈ C (0,T ∗; X ) of theν−Navier-Stokes equation which satisfies the estimate:

for u0 ∈ X and 0 ≤ t ≤ T ∗ν ||u(x , t)||X ≤ C (ν, u0) (5)

The previous examples and computations on the shear flow show that for

X = H12 the statement of the above proposition has no chance to be true

with u0 ∈ X and T ∗ν ,C (ν, u0) ν-independent. Conjecture same result forother spaces.


Energy conservation.

In the Torus (R/Z)3: the solution uν of

uν(x1, x2, x3) = ((uν)1(x2, t), 0, (uν)3(x1, x2, t))

∂t(uν)1(x2, t))− ν∂2x2

(uν)1(x2, t)) = 0 ,

∂t(uν)3 + (uν)1(x2, t)∂x1(uν)3 − ν(∂2x2

+ ∂2x2

)(uν)3 = 0 .

converges to the shear flow solution in C (R+t ; L2

weak((R/Z)3). Since thelimit conserves the energy the convergence is strong and the energydissipation goes to zero. A weak solution with only L2 regularity, viscouslimit of Leray solutions with vanishing energy dissipation and energyconservation




uν(x1, x2, x3) = ((uν)1(x2, t), 0, (uν)3(x1, x2, t))

∂t(uν)1(x2, t))− ν∂2x2

(uν)1(x2, t)) = 0 ,

∂t(uν)3 + (uν)1(x2, t)∂x1(uν)3 − ν(∂2x2

+ ∂2x2

)(uν)3 = 0 .






uν(x1, x2, x3) = ((uν)1(x2, t), 0, (uν)3(x1, x2, t))

∂t(uν)1(x2, t))− ν∂2x2

(uν)1(x2, t)) = 0 ,

∂t(uν)3 + (uν)1(x2, t)∂x1(uν)3 − ν(∂2x2

+ ∂2x2

)(uν)3 = 0 .




Summary and more comments about energy dissipation.

Any solution with C 0,α α > 13 regularity conserves energy.

There exist solutions which preserves the energy and are much lessregular The shear flow and some solutions constructed by DeLelis-Szekelyhidi.

There exist solutions with energy decay (or energy increase) ShefferShnirelman, they do not seem “physical” and do not seem to be limitof viscosity solution.

To the best of my knowledge there is no solution in C 0,α α < 13 with

energy decay there is an “ almost example” due to Eyink.

I do believe that energy decay really appears in problems with no slipboundary condition


































Boundary effect and energy dissipation.

With boundary (for instance an obstacle) and no slip boundary conditionthe problem of the zero viscosity limit is almost completely open even in2d It is naturally related to the issue of energy dissipation:

∂tuν − ν∆uν +∇ · (uν ⊗ uν) +∇pν = 0 , uν(x , t) = 0 on ∂Ω ,

∂tu +∇ · (u ⊗ u) +∇ = 0 , u · ~n = 0 on ∂Ω , uν(x , 0) = u(x , 0) in Ω

1

2

∫Ω|uν(x ,T )|2dx + ν

∫ T

0

∫Ω|∇uν(x , t)|2dxdt =

1

2

∫Ω|uν(x , 0)|2dx


The 1983 Kato Theorem

Theorem

The following facts are equivalent.

(i) limν→0

ν

∫ T

0

∫∂Ω

(∇∧ uν) · (~n ∧ u)dσdt = 0 (6)

(ii) uν(t)→ u(t) in L2(Ω) uniformly in t ∈ [0,T ] (7)

(iii) uν(t)→ u(t) weakly in L2(Ω) for each t ∈ [0,T ] (8)

(iv) limν→0

ν

∫ T

0

∫Ω|∇uν(x , t)|2dxdt = 0 (9)

(v) limν→0

ν

∫ T

0

∫Ω∩d(x ,∂Ω)<ν

|∇uν(x , t)|2dxdt = 0 . (10)



Conclusions.

The relation between dissipation of energy and loss of regularity is anessential issue in the statistical theory of turbulence in relation with theKolmogorov Obukhov law. It has been shown in the deterministicframework that a regularity of this type implies conservation of energy. Wehave shown that there is no hope for a converse statement even in thecase of solutions singular on a slit as Shvydkoy. This observation may notinvalidate the physical belief because the Kolmogorov Oboukov lawbelongs to the statistical theory of turbulence were results are true in someaveraged sense . On the other hand our family of examples are notturbulent and particular enough to be of measure zero with respect to anyensemble measure compatible with the statistical theory of turbulence(such measure has not been constructed, up to now, with fullmathematical rigor). However it is also important to notice that theexamples of deduced from De Lellis, L. Szekelyhidi form a “dense set”(Baire theorem)..... may be of zero measure ?Eventually turbulent behavior with energy dissipation may requireboundary effect.


Conclusions.



Conclusions.



Conclusions.



THANK YOU FOR YOUR ATTENTION


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The ``Finite time' Euler equation: An Introduction to the ...€¦ · The \Finite time" Euler equation: An Introduction to the Workshop Claude Bardos Retired, Laboratoire Jacques