Introduction - UniTrento · Introduction: The Euler and Navier-Stokes equations The incompressible...

Introduction

Claude Bardos

Retired, Laboratoire Jacques Louis Lions, Universite Pierre et Marie Curie

Levico Termine July 2010

Claude Bardos Introduction

Content

The first lecture is devoted to a short introduction on weak and strongsolutions of the Euler and Navier Stokes equations...

In the second and third lectures I will explain with more details the recentcontributions of De Lellis and Szekelyhidi on “wild solutions” of the Eulerequation.


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 .

u(x , t) is a vector field the velocity of the fluid at the point x and at thetimet, p is the pressure. With the incompressibility it is the Lagrangemultiplier of the constraint ∇ · u = 0. Eventually ν ≥ 0 is the viscosity ofthe media. To lead to relevant information the variable and unknown haveto be adapted:Characteristic lenght scale L, Characteristic velocity, U , Characteristictime scale T = L/U and Reynolds number < = UL

ν . which corresponds toratio between the strenght of the nonlinear effects and the strenght of thelinear viscous effects. This number is always “big” because U is big and νis small ( for a bicycle!! ≈ 102, for a car ≈ 6× 105 and for an airplane≈ 2× 107) or L is large (oceanography).


The equations take the nondimensional form:

x ′ =x

L, t ′ =

t

Tand u′ =

u′

U

∂tu′ +∇x ′ · (u′ ⊗ u′)− 1

<∆x ′u

′ +∇x ′p′ = 0 , ∇ · u′ = 0 .

This is the equation to be considered in the sequel omitting the ′ andreturning to the notation ν for <−1 . When physical boundary are presentthe problem is considered in the open set Ω ⊂ Rd , d = 2 , d = 3 withpiecewise smooth boundary ∂Ω and with ~n denoting the outward normalon this boundary.


The Euler equation

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”. This is the d’Alembert paradox

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

the viscosity, ( Saint Venant, Navier and Stokes (1822), (1845)), the boundary

effects in particular the no slip boundary condition and finally the conjunction of

these effects on the stationary regime (t →∞).


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. Purpose of the next lectures. Howevere since viscosity is

small its study is essential for the “physical” and mathematical structure of the

problem with ν > 0.

Figure: Euler, D’Alembert, Navier and StokesClaude Bardos Introduction

















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〉)

13 |r |

13

What can be deduced for individual solutions from the statistical theory?..Claude Bardos Introduction

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〉)

13 |r |

13

What can be deduced for individual solutions from the statistical theory?..Claude Bardos Introduction

Basic estimates, weak solutions...

∂tuν +∇ · (uν ⊗ uν)− ν∆uν +∇pν = 0 ,

∇ · u = 0 , in Ω .

on ∂Ω for ν > 0 , uν = 0 and for ν = 0 u · ~n = 0 ,

∂tω + u · ∇ω − ν∆ω = ω · ∇u in Ω

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


Basic estimates, weak solutions...

Formal Energy conservation, Energy dissipation and 2d Vorticityconservation:

∂t|uν |2

2+∇(uν(

|uν |2

2) + puν)− ν∆

|uν |2

2+ ν|∇u|2 = 0 ,

∂t

∫Ω

|uν |2

2dx + ν

∫Ω|∇u|2dx = 0

∂tω + u · ∇ω = 0 ,

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

Dtω =

d

dtω(x(t), t) = 0.

A weak solution of Euler is a divergence free field

u ∈ C (Rt ; L2w (Ω))∇ · u = 0 , in Ω , u · ~n = 0 on ∂Ω

∀φ ∈ C∞c (Rt × Ω) ,∇φ = 0 ,

∫Rn×Rt

(u · ∂tφ+ (u ⊗ u) : ∇φ)dxdt = 0

(u, p)∀φ ∈ C∞c (Rt × Ω)

∫Rn×Rt

(u · (∂tφ+ (u ⊗ u) : ∇φ)dxdt =〈p,∇ · φ〉 .


Remarks on weak solutions...

• With the property ∇ · u = 0 the trace of the normal component of u onthe boundary is well defined.• Also form the equation one deduces that any weak solution is continuouswith value in L2 weak. The initial value and the Cauchy problem are welldefined.• The hypothesis u ∈ L∞(L2) is enough (and almost necessary) to definethe tensor u ⊗ u in the sense of distributions. However one of the maindifficulty is the fact that for function un converging in L2 weak one mayhave:

lim(un ⊗ un) 6= (lim un)⊗ (lim un)


Remarks on weak solutions and dissipative solutions

((u · ∇u)− v∇v), u − v) = (u∇(u − v), u − v) + ((u − v)S(v)(u − v))) = ((u − v)S(v)(u − v)))

S(v) =1

2(∇v +∇v t)

Let w(x , t) divergence free test functions, w · ~n = 0 on ∂Ω .

E (w) = ∂tw + P(w · ∇w) , with P the Leray-Helmholtz projector .

For any smooth, divergence free, solution of the Euler equations u(x , t) inΩ , with boundary condition u · ~n = 0 on ∂Ω ,

∂tu +∇ · (u ⊗ u) +∇p = 0

∂tw +∇ · (w ⊗ w) +∇q = E (w)

d |u − w |2

dt+ 2(S(w)(u − w), (u − w)) = 2(E (w), u − w)

|u(t)− w(t)|2 ≤ eR t

0 2||S(w)(s)||∞ds |u(0)− w(0)|2

+2

∫ t

0e

R ts 2||S(w)(τ)||∞(τ)dτ (E (w)(s), (u − w)(s))ds .


Weak solutions and Dissipative solution

u ∈ w − C (Rt ; (L2(Ω))d) , u · ~n = 0 on ∂Ω

• Any classical solution u of the Euler equations is a dissipative solution.• Any dissipative solution which coincide for t = 0 with a smooth solutioncoincides with it as long as it remains smooth (always in 2d).• For weak solution the formal conservation of energy is not always true(see below) however Any weak solution which satisifies the weak energydecay inequality is a dissipative solution.• For weak solutions equation energy decay is related to some loss ofregularity. 1/3 appears to be a critical value of such regularity. Any

solution more regular than the Besov space B133,∞ conserves energy

Constantin E Titi; Cheskidov Constantin Friedlander.


• There exist solutions u ∈ C (Rt ; L2w ) ∩ C ([0,T ]; L2) with non constant

energy.• There exists an infinite set (residual) of initial data u0 ∈ L2(Rn) forwhich there exist infinitely many solution (with decaying energy) ofcorresponding Cauchy problem.• In the absence of boundary any Leray solution of Navier- Stokes, in 2and 3 dimension converges modulo subsequences to a dissipative solutionand the dissipation of energy goes to zero. However what about such limitsince uniqueness does not holds.• In the presence of boundary this may not be true in particular in 2d andthe only “general result” is the theorem of Kato.


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 (1)

(ii) uν(t)→ u(t) in L2(Ω) uniformly in t ∈ [0,T ] (2)

(iii) uν(t)→ u(t) weakly in L2(Ω) for each t ∈ [0,T ] (3)

(iv) limν→0

ν

∫ T

0

∫Ω|∇uν(x , t)|2dxdt = 0 (4)

(v) limν→0

ν

∫ T

0

∫Ω∩d(x ,∂Ω)<ν

|∇uν(x , t)|2dxdt = 0 . (5)



The Shear Flow One of the classical examples.

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... and in a statistical theory it corresponds to avery “special class of events” I use it also to shows that there is no hopeto prove statistical results by deterministic functional analysis.

u(x , t) = (u1(x2), 0, u3(x2, x1 − tu1(x2)) (6)

∇ · u = 0 and ∂tu +∇(u ⊗ u) = −∇p with p ≡ 0 (7)

d

dt

∫ ∫ ∫|u(x1, x2, x3, t)|2dx1dx2dx3 = 0 (8)

7 (in the sense of distributions) and 8 (on the torus ) are true under theonly hypothesis that

u1 ∈ L2x2

and u3 ∈ L2x1,x2

Consider also the boundary value problem inΩ = Rx1 × 0 < x2 < L× Rx3In most of the examples u3(x1 − tu1(x2))


Viscosity limit, estimates and energy dissipation

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

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(∂2x1

+ ∂2x2

)(uν)3 = 0 .

(uν)1(0, t) = (uν)1(L, t) = (uν)3(x1, 0, t) = (uν)3(x1, L, t) = 0

Proposition

With L2 initial data (uν)1(x2, t) converges strongly (in C (R+t ; L2) to

u2(x2, 0) and uν converges in C (R+t ; L2

weak) to the shear flow solution.


Since the limit conserves the energy the convergence is strong and theenergy dissipation goes to zero. A weak solution with only L2 regularity,viscous limit of Leray solutions with vanishing energy dissipation andenergy conservation converges to a non regular solution of the Eulererquation. No dissipation of energy in spite of the existence of boundary.


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∫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.The proof is not trivial uses L∞ conservation of the vorticity.


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 Introduction

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

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 Introduction

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

Typical example being the weak limit of a sequence of solutions of theNavier-Stokes equations with no slip boundary condition and ε = ν.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

Here no isotropy no rate of decay for the Turbulent spectra! Tooparticular??


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Introduction - UniTrento · Introduction: The Euler and Navier-Stokes equations The incompressible...

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