Weak solutions to the incompressible Euler equations

Transcription

1 Weak solutions to the incompressible Euler equations Antoine Choffrut Seminar Talk

2 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

3 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

4 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

5 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

6 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

7 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

8 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

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

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

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

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

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

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

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

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

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

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

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

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

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

22 Basic phenomenon A C B Antoine Choffrut University of Edinburgh 22

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

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

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

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

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

28 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

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

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

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

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

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

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

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

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

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

38 A A C B A Antoine Choffrut University of Edinburgh 38

39 A A C B A Antoine Choffrut University of Edinburgh 39

40 A A C B A Antoine Choffrut University of Edinburgh 40

41 A A C B A Antoine Choffrut University of Edinburgh 41

42 A A C B A Antoine Choffrut University of Edinburgh 42

43 A A C B A Antoine Choffrut University of Edinburgh 43

44 A A C B A Antoine Choffrut University of Edinburgh 44

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

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

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

48 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

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

50 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

51 Stationary Euler (d = 2) Identifying v R 2 z C and u S 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

52 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

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

54 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

55 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

56 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

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. Antoine Choffrut University of Edinburgh 57

58 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

59 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

60 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

61 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

62 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

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

64 Antoine Choffrut University of Edinburgh 64

65 Additional material Antoine Choffrut University of Edinburgh 65

66 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

67 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

68 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

69 Proof Antoine Choffrut University of Edinburgh 69

70 K co K Antoine Choffrut University of Edinburgh 70

71 C K co K Antoine Choffrut University of Edinburgh 71

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

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

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

75 Antoine Choffrut University of Edinburgh 75

76 B C Ã0 Antoine Choffrut University of Edinburgh 76

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