Domain continuity for the Euler and Navier-Stokes equations (Based - PowerPoint PPT Presentation

Domain continuity for the Euler and Navier-Stokes equations (Based on joint works with C. Lacave and A.L. Dalibard) David Gérard-Varet IMJ, Université Paris 7 1 / 19

1. Domain continuity Starting point: Fluid in a domain Ω n , parametrized by n ≫ 1. Assumption: Ω n → Ω in a suitable topology Question: Does the fluid have an asymptotic behavior ? Namely: ◮ Does the fluid velocity field u n → u in a suitable sense ? ◮ Does u satisfy the same equations as the u n ’s ? ◮ Does u satisfy the same conditions at ∂ Ω as the u n ’s at ∂ Ω n ? Plan: To adress this kind of questions, for Euler and Navier-Stokes. 2 / 19

2. The 2D Euler equation in non-smooth sets Let Ω ⊂ R 2 an open set.  ∂ t u + u · ∇ u + ∇ p = 0 ,    div u = 0 , (E)   u | t = 0 = u 0 , u · ν | ∂ Ω = 0  Aim: To solve (E) with minimal requirements on Ω . We focus on the construction of weak solutions, inspired by the whole space case. Here: weak solutions with vorticity in L p (Ω) , p > 1. Problem: Global weak solutions for general Ω ? Until recently, limited results. 3 / 19

◮ Most works deal with C 1 , 1 boundaries . [Wolibner,33], [Yudovitch,63], [Kato,67], [Bardos,72] ... Reason: Use of the Biot and Savart law: u = ∇ ⊥ ∆ − 1 ω. Regularizing effect of ∇ ⊥ ∆ − 1 weakens in non-smooth sets. ◮ Convex domains [Taylor,00] → ∆ − 1 : L 2 (Ω) �→ H 2 (Ω) . − → weak solutions with vorticity in L p (Ω) , p > 2. − ◮ Exterior of a smooth Jordan curve [Lacave, 09] Relies on an explicit Biot and Savart law, through conformal mapping. The smoothness of the curve is needed. Objective: To go beyond such specific cases. 4 / 19

Bounded open sets Ω := ˜ Ω \ ∪ k i = 1 C i , k ∈ N , with (A1) (Connectedness): - ˜ Ω bdd simply connected domain. - C i ’s connected compact subsets. (A2) (Positive H 1 capacity): for all i = 1 , . . . , k , cap ( C i ) > 0. Definition: E ⊂ R N : cap ( E ) := inf { � v � H 1 ( R N )) , v ≥ 1 a.e. in a neighborhood of E } Very roughly: cap ( E ) ≈ Leb ( E ) + ( n − 1 ) -dimensional "measure" of ∂ E . 5 / 19

Remarks: 1. The C i ’s can be of positive measure, smooth curves ... 2. k = 0: any bounded simply connected domain. Theorem: Let Ω satisfying (A1)-(A2), p > 1 , and � u 0 ∈ L 2 (Ω) , ω 0 ∈ L p (Ω) , ∀ ψ 0 ∈ C 1 c ( R 2 ) . with u 0 ·∇ ψ 0 = 0 , Ω Then, there exists u = u ( t , x ) such that u ∈ L ∞ ( R + ; L 2 (Ω)) , ω = curl u ∈ L ∞ ( R + ; L p (Ω)) � � ∀ ψ ∈ D ([ 0 , + ∞ [; C 1 c ( R 2 )) , with u · ∇ ψ = 0 , R + Ω and s.t. for all ϕ ∈ D ([ 0 , + ∞ [ × Ω) with div ϕ = 0 , � � � ( u · ∂ t ϕ + u ⊗ u : ∇ ϕ ) = − u 0 · ϕ ( 0 , · ) . R + Ω Ω 6 / 19

Ideas of proof Basic idea: Smoothing procedure ◮ Approximate Ω by smooth Ω n , u 0 by smooth u n 0 . ◮ u n , solution of Euler in Ω n " → " u , solution of Euler in Ω . 7 / 19

Ideas of proof Basic idea: Smoothing procedure ◮ Approximate Ω by smooth Ω n , u 0 by smooth u n 0 . ◮ u n , solution of Euler in Ω n " → " u , solution of Euler in Ω . a) Approximation of Ω : � O i , n � of smooth � ˜ Ω n � Lemma: There exists sequences and Jordan domains such that Ω n = ˜ Ω n \ ∪ k i , n H Ω n , Ω = lim i = 1 O Definition: Let (Ω n ) be a sequence of confined open sets in R N , B a compact set with Ω n ⊂ B for all n. H Ω n d H ( B \ Ω n , B \ Ω) → 0 . − → Ω if 8 / 19

Sketch of Proof . ◮ Approximation of ˜ Ω by ˜ Ω n : use conformal mapping. ◮ Approximation of C i by O i , n : - One approximates C i by a finite union of disks. - One makes slits in this union of disks to get it simply connected. 9 / 19

Sketch of Proof . ◮ Approximation of ˜ Ω by ˜ Ω n : use conformal mapping. ◮ Approximation of C i by O i , n : - One approximates C i by a finite union of disks. - One makes slits in this union of disks to get it simply connected. b) Weak compactness. Continuity of the tangency condition First ingredient : Explicit Hodge decomposition. The field u n reads � k � u n ( t , x ) = ∇ ⊥ ψ 0 , n ( t , x ) + � α i , n ( t ) ψ i , n ( x ) i = 1 10 / 19

with ◮ ψ 0 n (rotational part) satisfying ∆ ψ 0 , n = ω n in Ω n , ψ 0 , n | ∂ Ω n = 0 . ◮ ψ i n , i ≥ 1 (harmonic part) satisfying ∆ ψ i , n = 0 in Ω n , ψ i , n | ∂ ˜ ψ i , n | ∂ O j , n = δ ij . Ω n = 0 , Questions: ◮ Bounds on stream functions ψ i , n , i ≥ 0, and coeffts α i , n ? ◮ ∂ τ ψ i , n | ∂ Ω n = 0 ⇒ ∂ τ ψ i | ∂ Ω = 0 ? 11 / 19

Second ingredient : γ -convergence of open sets. Definition: Ω n ⋐ D. We note Ω n γ → Ω if the solution v n of − ∆ v n = 1 in Ω n , v n | ∂ Ω n = 0 converges in H 1 0 ( D ) (after extension by 0 ) to the solution v of ∆ v = 1 in Ω , v | ∂ Ω = 0 . Remark: Equivalent to the Γ -convergence of the associated Dirichlet functionals. Remark: γ -convergence means domain continuity of elliptic equations and Dirichlet conditions. Proposition (Sverak): If the number of connected components of D \ Ω n is uniformly bounded in n, then Ω n H Ω n γ − → Ω ⇒ − → Ω . Allows to handle the asymptotic boundary behavior of ψ i , n . 12 / 19

Question: Coefficients α i , n in the decomposition ? Broadly: they satisfy a linear system with ◮ a source term involving the circulations of u n around the O i n . Ω n ∇ ψ i , n · ∇ ψ j , n ) i , j ◮ a matrix close to ( � Broadly: ◮ The source is controlled thanks to Bernoulli’s theorem. ◮ The matrix is controlled by (A2). � ∇ ψ i · ∇ ψ j ) i , j non-singular ⇒ α i , n → α i cap ( C i ) > 0 ⇒ ( Ω 13 / 19

c) Asymptotics of the momentum equation in Euler Main point: No Aubin-Lions lemma in Ω n . What about Ω ′ ⋐ Ω ? Not really ! Roughly, one has P Ω ′ u n is compact in L q ( 0 , T × Ω ′ ) , for some q > 2 . → harmonic functions p n − h such that u n = u n + ∇ p n is compact in L q ( 0 , T × Ω ′ ) , ˜ for some q > 2 . h Then, one uses an algebraic identity well-known in the theory of Navier-Stokes [Lions et al, 1998], [Woelf, 2004]: div ( u n ⊗ u n ) = div (˜ u n ⊗ ˜ u n ) + weak-strong terms + 1 h | 2 + ∆ p n 2 ∇|∇ p n h ∇ p n h . 14 / 19

4. Navier-Stokes equation in rough domains Physical motivation: Microfluidics. Goal: To make fluids flow through very small devices. Minimizing drag at the walls is crucial. Many theoretical and experimental works. [Tabeling, 2004], [Bocquet, 2007 and 2012], [Vinogradova, 2012]. Some of these works claim that the usual no-slip condition is not always satisfied at the micrometer scale: Some rough surfaces may generate a substantial slip . However, these results have raised controversies . . . ... Maths may help, notably through a homogenization approach. 15 / 19

Ω ε = Ω ∪ Σ ∪ R ε Simple model: 2D rough channel : Ω Σ ε R ◮ Ω : smooth part: R × ( 0 , 1 ) . ◮ R ε : rough part, typical size ε ≪ 1. R ε = ε R , R = { y = ( y 1 , y 2 ) , 0 > y 2 > ω ( y 1 ) } ω with values in ( − 1 , 0 ) , and K-Lipschitz. ◮ Σ : interface: R × { 0 } . 16 / 19

Stationary Navier-Stokes, with given flux: x ∈ Ω ε ,  u · ∇ u − ∆ u + ∇ p = 0 ,    x ∈ Ω ε , div u = 0 ,  (NS ε ) �   u 1 = φ,   σ with φ > 0, σ vertical cross-section. Question: Can we get, for some boundary condition at ∂ Ω ε , an effective (meaning asymptotic ) slip condition at ∂ Ω ? Intuition: yes, at least if we consider some pure slip at ∂ Ω ε : D ( u ) ν ε × ν ε | ∂ Ω ε = 0 . u · ν ε | ∂ Ω ε = 0 , (S) 17 / 19

Answer: No ! As soon as the roughness is "non-degenerate", any weak limit u of a sequence of solutions ( u ε ) in H 1 loc (Ω) will satisfy u | ∂ Ω = 0 ! [Casado-Diaz et al, 03], [Bucur et al, 08] . Here: Refined result, under the non-degeneracy assumption (A) There is C > 0 , such that for all u ∈ C ∞ c ( R ) , u · ν | { y 2 = ω ( y 1 ) } = 0 ⇒ � u � L 2 ( R ) ≤ C �∇ u � L 2 ( R ) Remarks: ◮ Not satisfied for flat boundaries. ◮ Satisfied if there is A > 0 such that � A | ω ′ ( y 1 + t ) | 2 dt > 0 . inf y 1 ∈ R 0 ◮ Satisfied by non-cst periodic and quasiperiodic ω . 18 / 19

There exists φ 0 > 0 such that for all φ < φ 0 , ε ≤ 1 , Theorem: system (NS ε ) - (S) has a unique solution u ε ∈ H 1 uloc (Ω ε ) . Moreover, if (A) holds, uloc (Ω) ≤ C φ √ ε, � u ε − u � H 1 � u ε − u � L 2 uloc (Ω) ≤ C φε, where u is the Poiseuille flow in Ω (that satisfies u | ∂ Ω = 0 ). Remark: The theorem shows that the effective slip can not be more than O ( ε ) . Does not support some physics papers ... Boundary layer analysis: under ergodicity properties of ω , one shows that the effective slip is indeed O ( ε ) . Formal idea: Non-vanishing of the tangential component + high frequency oscillations of the boundary ⇒ blow up of ∇ u ε as ε goes to zero. Incompatible with the control of ∇ u ε in Navier-Stokes. 19 / 19