Weak solutions to the incompressible Euler equations

Weak solutions to the incompressible Euler equations Antoine Choffrut Seminar Talk - 2014

Motivation Time-dependent Euler equations: t v + (v )v + p = 0 div v = 0 Onsager s conjecture: conservation vs. non-conservation of kinetic energy. De Lellis & Székelyhidi (2009, 2012): L -, C 0,α -solutions dissipating energy. Antoine Choffrut University of Edinburgh 2

Non-uniqueness/solutions with compact support in time Scheffer (1993) Shnirelman (1997, 2000) DL-Sz (2009, 2012) Isett (2013) Buckmaster, DL, & Sz (2013) Related: Bardos, Sz., & Wiedemann; Chiodaroli; C.; Córdoba, Faraco, & Gancedo; Daneri; Shvydkoy; Wiedemann, etc. Antoine Choffrut University of Edinburgh 3

h-principle for the stationary case Theorem (C. & Székelyhidi (2013)) Let v 0 :T d R d be a smooth stationary flow: div (v 0 v 0 ) + p 0 = 0, div v 0 = 0. Fix e(x) > v 0 (x) 2 and σ > 0. Then, there exist infinitely many weak solutions v L (T d ;R d ), p L (T d ) satisfying 1. v(x) 2 = e(x) for a.e. x, 2. v v 0 H 1 < σ. Antoine Choffrut University of Edinburgh 4

The Tartar framework Nonlinear: to linear: div (v v) + p = 0, div v = 0 (1) div u + q = 0, div v = 0 (2) via u = v v v 2 d, p = q + v 2 d. Simplifying assumption: e(x) = 1. Antoine Choffrut University of Edinburgh 5

The constraint set K := { } (v, u) R d S0 d d u = v v v 2 d Id, v 2 = 1. (v, p) solves (1) if and only if (v, u, q) solves (2) and (v(x), u(x)) K for a.e. x. Antoine Choffrut University of Edinburgh 6

Convex integration scheme { div un + q n = 0 div v n = 0 K (v n (x),u n (x)) Antoine Choffrut University of Edinburgh 7

Convex integration scheme { div un + q n = 0 div v n = 0 { div un+1 + q n+1 = 0 div v n+1 = 0 Antoine Choffrut University of Edinburgh 8

Convex integration scheme { div un + q n = 0 div v n = 0 Antoine Choffrut University of Edinburgh 9

Convex integration scheme { div un + q n = 0 div v n = 0 Antoine Choffrut University of Edinburgh 10

Convex integration scheme { div un + q n = 0 div v n = 0 Antoine Choffrut University of Edinburgh 11

Convex integration scheme { div un + q n = 0 div v n = 0 Antoine Choffrut University of Edinburgh 12

Convex integration scheme { div un + q n = 0 div v n = 0 Antoine Choffrut University of Edinburgh 13

Convex integration scheme { div un + q n = 0 div v n = 0 Antoine Choffrut University of Edinburgh 14

Implementation x 0 approx. constant (v n (x 0 ),u n (x 0 )) Antoine Choffrut University of Edinburgh 15

Basic phenomenon φ(x) φ (x) Antoine Choffrut University of Edinburgh 16

Basic phenomenon φ(x) A φ (x) B Antoine Choffrut University of Edinburgh 17

Basic phenomenon φ(x) A φ (x) B Antoine Choffrut University of Edinburgh 18

Basic phenomenon φ(x) A φ (x) B Antoine Choffrut University of Edinburgh 19

Basic phenomenon φ(x) A φ (x) B Antoine Choffrut University of Edinburgh 20

Basic phenomenon φ(x) A φ (x) B Antoine Choffrut University of Edinburgh 21

Basic phenomenon A C B Antoine Choffrut University of Edinburgh 22

Basic phenomenon A x C B Antoine Choffrut University of Edinburgh 23

Basic phenomenon A λ 1 λ x C B Antoine Choffrut University of Edinburgh 24

Basic phenomenon A 1 λ 1 λ λ λ x C B Antoine Choffrut University of Edinburgh 25

Basic phenomenon A 1 λ λ x C B Antoine Choffrut University of Edinburgh 26

K K x 0 approx. constant K C K K K Antoine Choffrut University of Edinburgh 27

Plane wave = solution to (2) of the form (v(x), u(x)) = h(x ν)(v 0, u 0 ). Direction of oscillations must be taken in wave cone: Λ := { } (v 0, u 0 ) R d S0 d d plane wave along (v 0, u 0 ). For stationary Euler: Λ = { } (v 0, u 0 ) R d S0 d d q, ν : u 0 ν + qν = 0, ν v = 0. Antoine Choffrut University of Edinburgh 28

A K K K C K A K K Antoine Choffrut University of Edinburgh 29

A K K K C K A K K Antoine Choffrut University of Edinburgh 30

A K K K C K A K K Antoine Choffrut University of Edinburgh 31

A K K K C K A K K Antoine Choffrut University of Edinburgh 32

A K K K C K A K K Antoine Choffrut University of Edinburgh 33

Effect of localization A x 0 approx. constant C A Antoine Choffrut University of Edinburgh 34

Effect of localization A C A Antoine Choffrut University of Edinburgh 35

Effect of localization A C A Antoine Choffrut University of Edinburgh 36

A x 0 approx. constant A C B A Antoine Choffrut University of Edinburgh 37

A A C B A Antoine Choffrut University of Edinburgh 38

A A C B A Antoine Choffrut University of Edinburgh 39

A A C B A Antoine Choffrut University of Edinburgh 40

A A C B A Antoine Choffrut University of Edinburgh 41

A A C B A Antoine Choffrut University of Edinburgh 42

A A C B A Antoine Choffrut University of Edinburgh 43

A A C B A Antoine Choffrut University of Edinburgh 44

Convex integration scheme (revisited) { div u0 + q 0 = 0 div v 0 = 0 U K Antoine Choffrut University of Edinburgh 45

Convex integration scheme (revisited) { div un + q n = 0 div v n = 0 U K Antoine Choffrut University of Edinburgh 46

Convex integration scheme (revisited) { div un+1 + q n+1 = 0 div v n+1 = 0 U K Antoine Choffrut University of Edinburgh 47

1. Can we generate all points in K co with Λ-connections? 2. Convergence? A A C U B A K Antoine Choffrut University of Edinburgh 48

1 Can we generate every point in K co with Λ-connections? A A C B A Antoine Choffrut University of Edinburgh 49

1. Can we generate every point in K co with Λ-connections? A d 3: A C d = 2: U = int K co B A U int K co Antoine Choffrut University of Edinburgh 50

Stationary Euler (d = 2) Identifying v R 2 z C and u S 2 2 0 ζ C one finds Symmetries: { } K = (z, ζ) C C z 2 = 1, ζ = z 2 /2, { } Λ = (z, ζ) C C I(z 2 ζ) = 0. R θ : (z, ζ) (ze iθ, ζe 2iθ ). Antoine Choffrut University of Edinburgh 51

Stationary Euler (d = 2) L := {(z, ζ) C C I(ζ) = 0} (z, ζ) = (a + ib, c) C R: (z, ζ) Λ abc = 0. In the slice K L, U L looks like... Antoine Choffrut University of Edinburgh 52

c r/2 r b a r Antoine Choffrut University of Edinburgh 53

b r/2-c/ r f r (a,b,c) < 1 - r/2-c/ r r/2+c/ r a - r/2+c/ r Antoine Choffrut University of Edinburgh 54

Convergence in L 1 div u n + q n = 0, div v n = 0 (v n, u n ) L1 (v, u) ; dist ((v n (x), u n (x)), K) dx 0. div u + q = 0, div v = 0 (D ) (v(x), u(x)) K a.e. x Antoine Choffrut University of Edinburgh 55

Baire-category method (v, u) K co v 1 (v, u) K (v, u) K co and v = 1 Antoine Choffrut University of Edinburgh 56

Baire-category method (v, u) K co v 1 (v, u) K (v, u) K co and v = 1 For each n: Ω > Ω > Ω Ω Ω v n (x) 2 (x) dx v n+1 (x) 2 dx v n (x) 2 dx + Ω v n+1 (x) v n (x) 2 dx. Antoine Choffrut University of Edinburgh 57

Baire-category method (v, u) K co v 1 (v, u) K (v, u) K co and v = 1 For each n: Ω > Ω > Ω Ω Ω v n (x) 2 (x) dx v n+1 (x) 2 dx v n (x) 2 dx + Ω v n+1 (x) v n (x) 2 dx. Goal: Ω = Ω v(x) 2 dx so that v(x) 2 = 1 a.e. x. Antoine Choffrut University of Edinburgh 58

Perturbation property (P) There exists a continuously strictly increasing function Φ: [0, ) [0, ) with Φ(0) = 0 such that the following holds. Let Q = [0, 1] d. For every w := (v, u) U, there exists w = (v, u) C c (Q;R d S d d 0 ) (and q) such that 1. w solves (2); 2. w + w(x) U for all x; 1 3. Q Q w(x) 2 dx Φ (dist (w, K)). Antoine Choffrut University of Edinburgh 59

Main Theorem: idea of proof Suppose Ω v(x) 2 dx < Ω, so that v(x) < 1 in some Q. This is excluded by the stability/perturbation property. Antoine Choffrut University of Edinburgh 60

Perturbation property (P) ε A C A ε A Var [µ] ε 2 Remark For time-dependent Euler, or stationary Euler with d 3, one branch is enough. Antoine Choffrut University of Edinburgh 61

Smooth approximations to probability measures C = j λ ja j µ = j λ jδ Aj w(x) R 2 2 f(a) dµ(a) Q f(w(x)) dx E[µ] Q w(x) dx Var [µ] Q w(x) 2 dx A C A B A Antoine Choffrut University of Edinburgh 62

Theorem (C. & Székelyhidi, 2013) With d = 2, U = int K co fails the Perturbation Property. Antoine Choffrut University of Edinburgh 63

Antoine Choffrut University of Edinburgh 64

Additional material Antoine Choffrut University of Edinburgh 65

Stationary Euler (d 3) A segment is admissible if σ K co R d S d d 0 σ (a, a a) (b, b b) for some a = b = 1, a ±b. Lemma (DL-Sz, 2009) If d 3, then every admissible segment is a Λ-direction. Antoine Choffrut University of Edinburgh 66

Stationary Euler (d 3) Λ = { } (v, u) R d S0 d d q R, η R d : uη + qη = 0, η v Proof. (of Lemma) Let σ = (a, a a) (b, b b). Choose η 0 such that η a = η b = 0 (d 3). Then, uη = (a a b b)η = 0. Antoine Choffrut University of Edinburgh 67

Digression: time-dependent Euler K := { } (v, u, q) R d S0 d d R u = v v v 2 d Id, v 2 = 1 Λ := {... } Proposition (DL-Sz, 2009) The convex hull Kco can be generated by laminations along Λ. Lemma (DL-Sz, 2009) For every v 0 R d and u 0 S0 d d, there exists q 0 R such that (v 0, u 0, q 0 ) Λ. Antoine Choffrut University of Edinburgh 68

Proof Antoine Choffrut University of Edinburgh 69

K co K Antoine Choffrut University of Edinburgh 70

C K co K Antoine Choffrut University of Edinburgh 71

A 0 C A 2 A 1 Antoine Choffrut University of Edinburgh 72

A 0 C A 2 B A 1 Antoine Choffrut University of Edinburgh 73

A 0 B C A 2 A 1 Antoine Choffrut University of Edinburgh 74

Antoine Choffrut University of Edinburgh 75

B C Ã0 Antoine Choffrut University of Edinburgh 76

à 2 B à 1 C Ã0 Antoine Choffrut University of Edinburgh 77