Weak solutions to the stationary incompressible Euler equations

Transcription

1 Weak solutions to the stationary incompressible Euler equations Antoine Choffrut September 29, 2014 University of Sussex Analysis & PDEs Seminar

2 Euler s equations (1757) Motivation Conservation of energy { t v + (v )v + p = 0 div v = 0 d 1 dt 2 Ω v(x, t) 2 dx = 0 Onsager s conjecture (1949) A solution v C α... α Proof?... must conserve energy. Eyink, Constantin-E-Titi (1994) 1/3 1/5? Isett, Buckmaster-DL-Sz (2013)... may dissipate energy. 1/10 De Lellis-Székelyhidi (2012)

3 Prehistory: Scheffer, Shnirelman. Related: Bardos, Sz., & Wiedemann; Chiodaroli; C; Córdoba, Faraco, & Gancedo; Daneri; Shvydkoy; Wiedemann; etc. Rigidity for stationary Euler: C & Šverák (GAFA, 2012), Local structure of the set of steady-states to the 2D incompressible Euler equations Flexibility for stationary Euler: C & Székelyhidi (2014), Weak solutions to the stationary incompressible Euler equations

4 H-principle for the stationary case Theorem (C & Székelyhidi (2014)) 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.

5 The h-principle (Gromov) solutions sub-solution homotopy

6 The h-principle (Gromov) solutions

7 The h-principle (Müller-Šverák) solutions sub-solution iterates

8 The h-principle (Müller-Šverák) solutions

9 An h-principle... is a statement about the density of solutions. Convex integration... is a technical tool to construct solutions;... can be used to establish an h-principle.

10 Functional Analysis unit sphere in l 2

11 Analogy: continuous, but nowhere differentiable functions. Weierstraß example (1872) a n cos(b n πx) n=0 where 0 < a < 1, ab > π. Baire category argument Most continuous functions are nowhere differentiable. Baire category argument associated with convex integration due to Dacorogna-Marcellini, Kirchheim.

12 The Tartar framework From nonlinear... div (v v) + p = 0, div v = 0 (1)... to linear: 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.

13 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.

14 Convex integration (Nash, Gromov) { div un + q n = 0 div v n = 0 K (v n (x),u n (x))

15 Convex integration (Nash, Gromov) { div un + q n = 0 div v n = 0 { div un+1 + q n+1 = 0 div v n+1 = 0

16 Convex integration (Nash, Gromov) { div un + q n = 0 div v n = 0

17 Convex integration (Nash, Gromov) { div un + q n = 0 div v n = 0

18 Convex integration (Nash, Gromov) { div un + q n = 0 div v n = 0

19 Convex integration (Nash, Gromov) { div un + q n = 0 div v n = 0

20 Convex integration (Nash, Gromov) { div un + q n = 0 div v n = 0

21 Convex integration (Nash, Gromov) { div un + q n = 0 div v n = 0

22 Convex integration (Müller-Šverák) { div u0 + q 0 = 0 div v 0 = 0 U K

23 Convex integration (Müller-Šverák) { div un + q n = 0 div v n = 0 U K

24 Convex integration (Müller-Šverák) { div un+1 + q n+1 = 0 div v n+1 = 0 U K

25 Convex integration (Müller-Šverák) { div un+1 + q n+1 = 0 div v n+1 = 0 U K

26 Implementation? x 0 approx. constant U (v n (x 0 ),u n (x 0 )) K

27 Convex integration: basic phenomenon φ(x) φ (x) x

28 Convex integration: basic phenomenon φ(x) A φ (x) B x

29 Convex integration: basic phenomenon φ(x) A φ (x) B x

30 Convex integration: basic phenomenon φ(x) A φ (x) B x

31 Convex integration: basic phenomenon φ(x) A φ (x) B x

32 Convex integration: basic phenomenon φ(x) A φ (x) B x

33 Convex integration: basic phenomenon A φ(x) φ (x) B x

34 K K x 0 approx. constant K C K K K

35 Plane waves = solutions to (2) of the form (v(x), u(x)) = h(x ν)(v 0, u 0 ). Directions of oscillation must be taken in the 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 0, ν : u 0 ν + q 0 ν = 0, ν v 0 = 0.

36 A K K K C K A K K

37 A K K K C K A K K

38 A K K K C K A K K

39 A K K K C K A K K

40 A K K K C K A K K

41 Effect of localisation A x 0 approx. constant C A

42 Effect of localisation A C A

43 Effect of localisation A C A

44 A x 0 approx. constant A C B A

45 A A C B A

46 A A C B A

47 A A C B A

48 A A C B A

49 A A C B A

50 A A C B A

51 A A C B A Constraint set: K. Lamination convex hull: K l.c.. K l.c. K co

52 The h-principle Let U int K co, X 0 := {smooth sub sol. w w(x) U }, X := weak closure of X 0 in L 2. A problem satisfies an h-principle if {w X w(x) K a.e.} is weakly dense in X. Theorem (Folklore) If K l.c. = K co and has non-empty interior, then the h-principle holds.

53 Smooth approximations to prelaminates C = j λ j A j A 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 B A

54 Smooth approximations to prelaminates C = j λ j A j A 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 B A Remark: min j C A j > ε Var [µ] > ε 2.

55 Proof of Folklore Theorem The Baire category argument Define I(w) := w(x) 2 dx for w X. Fact X is metrizable: (X, d). Fact If I is continuous at w and w j w, then w j w. Fact The points of continuity of I form a residual set in X. To show If I is continuous at w, then w(x) K a.e.

56 Proof of Folklore Theorem The Baire category argument Assume I continuous at w X where dist (w(x), K) dx > ε. Let X 0 w j w with dist (w j (x), K) dx > ε. Let w j X 0 such that w j w j j 0 and w j (x) w j (x) 2 dx > c(ε). Contradiction with w j j w in L 2.

57 Theorem (DL & Sz, 2009) For time-dependent Euler, K l.c. = K co. Corollary (C & Sz, 2014) For stationary Euler in d 3 dimensions, K l.c. = K co. Theorem (C & Sz, 2014) For stationary Euler in d = 2 dimensions, K l.c. K co.

58 Perturbation Property (P) There exists a continuous, 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)).

59 Theorem (C & Sz, 2014) For stationary Euler in d = 2 dimensions, int K co fails (P). Proof: K co

60 Theorem (C & Sz, 2014) For stationary Euler in d = 2 dimensions, int K co fails the perturbation property. Proof: w K co \K K co

61 Theorem (C & Sz, 2014) For stationary Euler in d = 2 dimensions, int K co fails the perturbation property. Proof: w 2 K w 1 K w K co \K K co

62 Theorem (C & Sz, 2014) For stationary Euler in d = 2 dimensions, int K co fails the perturbation property. Proof: w 2 K w 1 K w K co \K unique K co

63 Theorem (C & Sz, 2014) For stationary Euler in d = 2 dimensions, int K co fails the perturbation property. Proof: w 2 K w 1 K w K co \K Λ K co

64 Theorem (C & Sz, 2014) For stationary Euler in d = 2 dimensions, int K co fails the perturbation property. Proof: w 2 K w K co \K w (k) w 1 K K co

65 Theorem (C & Sz, 2014) For stationary Euler in d = 2 dimensions, int K co fails the perturbation property. Proof: w 2 K w K co \K w (k) w 1 K K co

66 Question What is the relaxation set for 2d stationary Euler? H-principle for d = 2 {w X w(x) K a.e.} is weakly dense in X. U int K co, X 0 := {smooth sub sol. w w(x) U }, X := weak closure of X 0 in L 2. Proof: exhibit U int K co satisfying (P).

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

68 Stationary Euler (d = 2) L := {(z, ζ) C C I(ζ) = 0} (z, ζ) = (a + ib, c) C R: (z, ζ) Λ abc = 0. Given construct K L, U L.

69 Constructing U L K L r/2 c r b a r

70 Constructing U L K L Λ r/2 c r b a r

71 Constructing U L K L Λ r/2 c r b a r

72 Constructing U L K L Λ r/2 c r b a r

73 Constructing U L K L Λ r/2 c r b a r

74 c-slice of U L b r/2-c/ r f r (a,b,c) < 1 - r/2-c/ r r/2+c/ r a - r/2+c/ r

75 U L r/2 c r b a r

76 [C & Székelyhidi] (2014) arxiv: Weak solutions to the stationary incompressible Euler equations [Székelyhidi] (2012) Max Planck Lecture Notes 4112 From isometric embeddings to turbulence [De Lellis & Székelyhidi] (2009) Ann. of Math. The Euler equations as a differential inclusion [Müller & Šverák] (2003) Ann. of Math. Cvx integr. for Lipschitz mappings and counterexamples to regularity [Kirchheim] (2003) Univ. Leipzig Habilitation Thesis Rigidity and geometry of microstructures [Müller] (1998) Max Planck Lecture Notes 2 Variational models for microstructure and phase transitions