Onsager Conjecture, the Kolmogorv 1/3 law and the 1984 Kato Criteria in bounded domains with boundaries: In progress with Edriss Titi and E. Wiedemann. Workshop on kinetic and fluid Partial Differential Equations Claude Bardos Em´erite-LJLL- Universit´e Denis Diderot. https://www.ljll.math.upmc.fr/ bardos/

Claude Bardos (Uni. Denis-Diderot)

Onsager Conjecture, and the Kolmogorov 1/3 law

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Onsager, Kolmogorov and Kato

Claude Bardos (Uni. Denis-Diderot)

Onsager Conjecture, and the Kolmogorov 1/3 law

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Introduction

Several of my recent contributions , with Edriss Titi, Emile Wiedemann and others were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity , in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence. I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation. Give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary. Give sufficient condition for the global conservation of energy in a domain with boundary and show how this imply the absence of anomalous energy dissipation. Give several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations. Insisting that in such case the absence of anomalous energy dissipation is equivalent to the persistence of regularity in the zero viscosity limit. Claude Bardos (Uni. Denis-Diderot)

Onsager Conjecture, and the Kolmogorov 1/3 law

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The magic number 13

u(x, t) ∈ Cweak ([0, T ]; L2 (Ω)) weak solution of the incompressible Euler equations Ω ⊂ Rd with C 2 boundary ∂Ω and exterior normal n~ In D0 ((0, T ) × Ω)

∂t u + ∇x · (u ⊗ u + pI ) = 0

and

∇ · u = 0,

On ∂Ω × [0, T ] u · n~ = 0 .

(1)

If u is a smooth solution say Lipschitz one has: Z Z ∇p · udx = − p∇ · udx = 0 , Ω Ω Z Z Z X |u|2 |u|2 dx = ∇ · (u )dx = 0 ∇(u ⊗ u) : udx = uj ∂xj 2 2 Ω Ω Ω j

And this implies the conservation of energy Z d |u(t, x)|2 dx = 0 . dt Ω 2 Claude Bardos (Uni. Denis-Diderot)

Onsager Conjecture, and the Kolmogorov 1/3 law

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The magic number 13

Onsager gave a semi formal proof of the conjecture that by now carries its name: Any weak solution which belongs to the space C 0,α with α > 13 conserves the energy. The “formal ” proof goes as follow: The term to control is 1

1

1

h∇(u ⊗ u)ui ' ((∇ 3 u ⊗ ∇ 3 u) : ∇ 3 u) 1

hence appears the quantity k∇ 3 ukL3 (Ω×[0,T ]) . This observation is the origin of serious proofs: Eyink , Constantin , E, Titi. in 1994. Recent papers Buckmaster and als.. have shown, for α < 31 , the existence of wild solutions in C 0,α ((0, T ) × T3 ) .

Claude Bardos (Uni. Denis-Diderot)

Onsager Conjecture, and the Kolmogorov 1/3 law

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The magic number 13

Under weak convergence or in statistical theory, with Navier-Stokes equation loss of regularity anomalous energy dissipation are related: Z tZ 2 kuν (t)kL2 (Ω) + 2ν |∇uν (x, s)|2 ds = kuν (0)k2L2 (Ω) 0

(2)

Ω

If uν converge weakly to a solution of the Euler equation which conserves the energy there is no anomalous energy dissipation. In statistical theory one has the Kolmogorov law: h

uν (. + l, t) − uν (., t)

Claude Bardos (Uni. Denis-Diderot)

|l|

1 3

1

i ' (hν|∇uν (., t)|2 i) 3 .

Onsager Conjecture, and the Kolmogorov 1/3 law

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Local Energy equation

Theorem I Let (u, p) ∈ L∞ (0, T ; L2 (Ω)) × D0 ((0, T ) × Ω) a weak solution: ∂t u + ∇ : (u ⊗ u) + ∇p = 0 , ∇ · u = 0 (3) ˜ satisfies: which in a subopen set U = (t1 , t2 ) × Ω ˜ < γ} there exists a For any small enough γ > 0 and Vγ = {x, d(x, ∂ Ω) β(V ) > 0 such that : p ∈ C ((t1 , t2 ); H −β(Vγ ) (Vγ ) ≤ M0 (V ) < ∞ Z t2 2 ku(., t)k3C 0,α (Ω) ˜ dt ≤ M(U) < ∞ .

1

(4a) (4b)

t1

˜ the local energy conservation: Then (u, p) satisfies in (t1 , t2 ) × Ω    |u|2 |u|2 ˜ + ∇x · u +p = 0 in D0 ((t1 , t2 ) × Ω) ∂t 2 2  2  d |u|2 |u| ˜ ⇔∀φ ∈ D(Ω) hφ, i − h∇x φ, u + p i = 0 in D0 (t1 , t2 ). dt 2 2 Claude Bardos (Uni. Denis-Diderot)

Onsager Conjecture, and the Kolmogorov 1/3 law

(5)

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Proof of Theorem I : Localisation and Elliptic Estimates

˜ hence with a support in an open (5) is proven with any given φ ∈ D(Ω) ˜ set Sφ such that Sφ ⊂⊂ Ω . Introduce η > 0 small enough, Ω3η , Ω2η , Ωη such that ˜ ⊂⊂ Ω Sφ ⊂⊂ Ω3η ⊂⊂ Ω2η ⊂⊂ Ωη ⊂⊂ Ω ˜ =η d(Ω3η , ∂Ω2η ) = d(Ω2η , ∂Ωη ) = d(Ωη , ∂ Ω) ˜ equal to 1 in Ωη and θ ∈ D(Ω) X − ∆θp = −θ( ∂xi ∂xj ui uj ) − (∇p · ∇θ + p∆θ)

(6)

i,j

Then with elliptic theory and X 2 ˜ ∂xi ∂xj ui uj ∈ L 3 ((t1 , t2 ); C 0,α (Ω)) i,j

Claude Bardos (Uni. Denis-Diderot)

Onsager Conjecture, and the Kolmogorov 1/3 law

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Proof of Theorem I : Localisation and Elliptic Estimates

one has: Z t2 2 kpkC3 0,α (Ω3η ) (., t)dt ≤ C (U) t1

⇒ ∂t u|Ω

3η

The formula

3

∈ L 2 ((t1 , t2 ); H −1 (Ω3η )) . (7)   2  |u|2 |u| ∂t + ∇x · u +p =0 2 2

= −∇ · (u|Ω

3η

⊗ u|Ω ) − ∇p|Ω 3η

3η

is well defined on (t1 , t2 ) × Ω3η Moreover: ∂t u|Ω

3η

= −∇ · (u|Ω

Claude Bardos (Uni. Denis-Diderot)

3η

⊗ u|Ω ) − ∇p|Ω 3η

3η

3

∈ L 2 ((t1 , t2 ); H −1 (Ω3η ))

Onsager Conjecture, and the Kolmogorov 1/3 law

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Proof of Theorem: Regularization

˜ , v denotes its extension by 0 outside Ω ˜. ∀v ∈ L1 (Ω) x 7→ ρ(x) ∈ C ∞ (Rn ) a mollifier, ρ(x) ≥ 0, with support in |x| ≤ 1, Z ρ(x)dx = 1 with Rn

For