The h-principle for the Euler equations

Transcription

1 The h-principle for the Euler equations László Székelyhidi University of Leipzig IPAM Workshop on Turbulent Dissipation, Mixing and Predictability 09 January 2017

2 Incompressible Euler t v + v rv + rp =0 div v =0 x 2 T 3 t 2 [0,T] Some facts: To any given suff. smooth initial data there exists, at least for a short time, a unique suff. smooth solution (Lichtenstein 1930s, Kato 1980s). For any suff. smooth solution, the energy is constant in time (classical). There exist non-trivial weak solutions with compact support in time (Scheffer 1993).

3 Notions of solutions of Euler Classical Continuous Weak "Very weak" v 2 C 1, or v 2 H s>5/2 Lichtenstein, Kato v 2 C 0 v 2 L 2 loc v 2 L 1 (M) v 2 C v 2 L 1 De Lellis-Sz Buckmaster Isett Scheffer, Shnirelman, De Lellis-Sz Sz-Wiedemann... measure-valued DiPerna-Majda dissipative P.L.Lions LWP energy conserved blow-up? "wild behaviour", h-principle non-uniqueness, energy not conserved existence (via compactness)

4 Weak solutions and non-uniqueness

5 Non-uniqueness in L 2 Theorem (Scheffer 93, Shnirelman 97, De Lellis - Sz. 2009) There exist nontrivial weak solutions of the Euler equations with compact support in space-time. [Scheffer] in R 2 [Shnirelman] in T 2 [De Lellis-Sz.] works for general domains in any dimension

6 Non-uniqueness in L 2 Theorem (Scheffer 93, Shnirelman 97, De Lellis - Sz. 2009) There exist nontrivial weak solutions of the Euler equations with compact support in space-time. Theorem (De Lellis - Sz. 2010) Given e = e(x, t) > 0 [Scheffer] in R 2, there exist infinitely many weak solutions of the Euler [Shnirelman] in T 2 equations with 1 [De Lellis-Sz.] works for general domains in any dimension 2 v(x, t) 2 = e(x, t)

7 Non-uniqueness in L 2 Theorem (Wiedemann 2011) For any L 2 initial data there exist infinitely many global weak solutions with bounded energy. domain is a torus n 2 first global existence result for weak solutions in dimension n 3

8 Non-uniqueness in L 2 Theorem (Wiedemann 2011) For any L 2 initial data there exist infinitely many global weak solutions with bounded energy. Theorem v 0 v 1 Z Z Given and with v 0 dx = v 1 dx, there exist infinitely T many weak solutions of the Euler d T equations d with domain is a torus n 2 first global v(t existence = 0) = result v 0, for v(t weak = solutions 1) = vin 1 dimension n 3

9 Differential Inclusions

10 Differential Inclusions Toy problem: construct v :[0, 1]! R such that v =1 Baire-category approach (Cellina, Bressan-Flores, Dacorogna-Marcellini, Kirchheim,.) v : v = 1 a.e. v : v apple 1 a.e. residual in L 1 w*

11 Differential Inclusions: Baire category approach Theorem (folklore) Any 1-Lipschitz map u : R n! R n can be uniformly approximated by (weak) Lipschitz isometries, i.e. solutions of Du(x) 2 O(n) a.e. x

12 Differential Inclusions: Baire category approach Du T Du = Id isometric n () curl A =0 A T A = Id

13 Differential Inclusions: Baire category approach Du T Du = Id isometric n () curl A =0 A T A = Id short n curl A =0 A T A apple Id

14 Differential Inclusions: Baire category approach isometric n Du T Du = Id () curl A =0 A T A = Id t v +div(v v)+rp =0 div v =0 n t v +div + rp =0 div v =0 v v = short n curl A =0 A T A apple Id

15 Differential Inclusions: Baire category approach isometric n Du T Du = Id () curl A =0 A T A = Id t v +div(v v)+rp =0 div v =0 n t v +div + rp =0 div v =0 v v = short n curl A =0 A T A apple Id subsolution t v +div + rp =0 div v =0 v v apple

16 Differential Inclusions: Baire category approach Theorem: The typical short map is (weakly) isometric. Theorem: The typical Euler subsolution is a (weak) solution. Euler subsolution t v +div(v v)+rp = div v =0 R 0 div R

17 Differential Inclusions Toy problem: construct v :[0, 1]! R such that v =1 Constructive approach: define inductively v N+1 (x) =v N (x)+ 1 2 (1 v N (x) 2 )s( N x) { amplitude { high-frequency oscillation s(t) = sign sin t v N apple 1 ) v N+1 apple 1

18 Differential Inclusions Toy problem: construct v :[0, 1]! R such that v =1 v N+1 (x) =v N (x)+ 1 2 (1 v N (x) 2 )s( N x) v 2 v 6 v 10

19 Differential Inclusions Toy problem: construct v :[0, 1]! R such that v =1 v N+1 (x) =v N (x)+ 1 2 (1 v N (x) 2 )s( N x) Lemma If N =2 N, there exists > 0 so that Z 1 0 (1 v N 2 ) dx. 2 N

20 Beyond Lipschitz maps Theorem (Nash-Kuiper 1954/55) Any short embedding can be uniformly approximated by isometric embeddings. C 1 M n,! R n+1 an example of Gromov s h-principle method of proof: convex integration C 2 embeddings are rigid Lipschitz version of theorem is trivial

21 Beyond Lipschitz maps Theorem (Nash-Kuiper 1954/55) Any short embedding can be uniformly approximated by isometric embeddings. C 1 M n,! R n+1 Theorem (Borisov , Conti-De Lellis-Sz. 09) The Nash-Kuiper theorem remains valid for isometric embeddings with Du(x) Du(y) apple C x y C 1 1 < 1+2n Note: the embedding S 2,! R 3 is rigid for > 2/3

22 Selection criteria and instabilities

23 Weak-strong uniqueness Admissibility: Z v(x, t) 2 dx apple Z v 0 (x) 2 dx Theorem (P.L.Lions 1996) Given an initial data v 0 v + v T, if there exists a solution to the IVP with then this solution is unique in the class of admissible weak solutions. L The Scheffer-Shnirelman solution clearly not admissible For the solutions of Wiedemann has an instantaneous jump up at E(t) t =0

24 Wild initial data is generic Theorem (De Lellis-Sz / Wiedemann-Sz. 2012) There exists a dense set of initial data exist infinitely many admissible weak solutions. v 0 2 L 2 for which there solutions also satisfy the strong and local energy inequality a posteriori such initial data needs to be irregular

25 Strong instabilities I: Kelvin-Helmholtz T 2 Kelvin-Helmholtz instability v 0 (x) = Theorem (Sz. 2011) There exist infinitely many admissible weak solutions on initial data given by above. v 0 T 2 with c.f. Delort (1991): there exists a weak solution with curl v M + (T 2 )

26 Strong instabilities I: Kelvin-Helmholtz t =0 t>0 turbulent zone 2t selection principle? v = 1 a.e. The solution above is conservative. Strictly dissipative solutions also possible. There exists a maximal dissipation rate Maximally dissipative solution is different from vanishing viscosity limit More realistic limit should include perturbations of the initial condition, i.e. (NS )! (E) (NS," )! (E)

27 Strong instabilities II: Rayleigh-Taylor Incompressible porous medium t + v r =0 div v =0 v + rp = g Darcy s law Theorem (D. Cordoba - D. Faraco - F. Gancedo 2011) There exist nontrivial weak solutions with compact support in time. R. Shvydkoy 2011: Same result holds for general active scalar equations with even (& non-degenerate) kernel

28 Strong instabilities II: Rayleigh-Taylor Incompressible porous medium t + v r =0 div v =0 v + rp = g Darcy s law g = 1 = 2

29 Strong instabilities II: Muskat problem Muskat curve t z(s, t) = P.V. Z 1 1 z 1 (s, t) z 1 (s 0,t) z(s, t) z(s 0,t) 2 (@ sz(s, s z(s 0,t)) ds 0 = 1 = 2 1 < 2 2 < 1 Stable case P.Constantin-F.Gancedo-V.Vicol-R.Shvydkoy 2016 Unstable case A.Castro-D.Cordoba-C.Fefferman-F.Gancedo-M.Lopez-Fernandez 2012

30 Strong instabilities II: Muskat problem regularized Muskat curve evolution: A. Castro - D. Cordoba - D. Faraco t z(s, t) = ct Z ct ct Z 1 1 (@ s z(s, s z(s 0,t)) 1 2ct Z ct ct z 1 (s, t) z 1 (s 0,t) z(s, t) z(s 0,t)+( 0 )g 2 d 0 ds 0 d = 1 mixing zone 2ct = 2 c = 2 1 2

31 Unstable Muskat problem: Selection criteria? Flat initial interface Lagrangian relaxation (F.Otto 1999): gradient flow formulation Eulerian relaxation (Sz. 2012): calculation of lamination hull Maximal mixing (Sz. 2012): c apple ( 2 1 ) = 1 mixing zone 2ct = 2

32 Eulerian relaxation Du T Du = Id short n curl A =0 A T A apple t v +div(v v)+rp =0 div v t + v r =0 div v =0 v + rp = g subsolution subsolution n t v +div + rp =0 div v =0 v v t +div =0 div v =0 curl (v + g) =0 v (1 2 )g apple 1 2 (1 2 )

33 Onsager s conjecture

34 Onsager s Conjecture 1949 For (weak) solutions of Euler with v(x, t) v(y, t) applec x y a) If > 1/3 energy is conserved. b) If < 1/3 dissipation possible. Onsager 1949, Eyink 1994, Constantin-E-Titi 1993, Robert-Duchon 2000, Cheskidov-Constantin-Friedlander- -Shvydkoy 2007, Scheffer 1993, Shnirelman 1999 De Lellis - Sz Buckmaster-De Lellis-Isett-Sz 2013 Buckmaster 2013 Isett 2016

35 Two types of statements for b) Let X be a function space. Theorem A There exists a nontrivial weak solution equations with compact support in time. v 2 X of the Euler Theorem B For any smooth positive function solution of the Euler equations 1 2 Z v 2 X E(t) T 3 v(x, t) 2 dx = E(t) there exists a weak such that for all t

36 Theorem A Theorem A There exists a nontrivial weak solution equations with compact support in time. v 2 X of the Euler P. Isett 2013 X = L 1 (0,T; C 1/5 (T 3 )) T. Buckmaster 2013 v(,t) 2 C 1/3 a.e. t T. Buckmaster-De Lellis-Sz X = L 1 (0,T; C 1/3 (T 3 )) P. Isett 2016 X = L 1 (0,T; C 1/3 (T 3 ))

37 Theorem B Theorem B For any smooth positive function solution of the Euler equations 1 2 Z v 2 X E(t) T 3 v(x, t) 2 dx = E(t) there exists a weak such that for all t De Lellis-Sz A. Choffrut 2013 X = L 1 (0,T; C 1/10 (T 3 )) X = L 1 (0,T; C 1/10 (T 2 )) T. Buckmaster-De Lellis-Isett-Sz X = L 1 (0,T; C 1/5 (T 3 )) T. Buckmaster-De Lellis-Sz.-Vicol work in progress: X = L 1 (0,T; C 1/3 (T 3 ))

38 H-principle and Closure: Subsolutions Reynolds decomposition: v = v + w average fluctuation Euler-Reynolds system: t v +div( v v)+r p = div R div v =0 R = v v v v = w w Closure problem: equation for R? We know: R 0.

39 { H-principle and Closure: Subsolutions Deterministic turbulence via weak convergence (following P.D.Lax) Assume v k * v in L 1 t v k +div(v k v k )+rp k = k div v k =0 v k k! 0 t v +div( v v)+r p = div R div v =0 Energy: Z with R = w lim (v k v) (v k v) 0 k!1 E(t) = 1 2 v 2 +tr Rdx

40 H-principle and Closure Theorem B (S. Daneri - Sz 16) Let ( v, p, R) (v k,p k ) there exist a sequence that be a smooth strict subsolution. For any of weak solutions of Euler such moreover v k * v and v k v k * v v + R in L 1 and 1 2 v k (x, t) v k (y, t) apple C x y Z T 3 v k 2 dx = 1 2 Z T 3 v 2 +tr Rdx < 1/5 8 t Expect same result for all < 1/3.

41 Some ideas of the construction (based on De Lellis-Sz. 2012)

42 } } The Euler-Reynolds equations Euler-Reynolds system: q 2 t v q + r (v q v q )+rp q = r v q =0 r R q v q+1 (x, t) =v q (x, t)+w (x, t, q+1x, q+1t) slow fast fluctuation - analogue of Nash twist

43 } The Euler-Reynolds equations Euler-Reynolds system: q 2 t v q + r (v q v q )+rp q = r v q =0 r R q more precisely: v q+1 (x, t) =v q (x, t)+w v q (x, t),r q (x, t), q+1x, q+1t explicit dependence of previous state

44 Conditions on the "fluctuation" W = W (v, R,, ) (H1) (H2) 7! W (v, R,, ) hw W i = R periodic with average zero: Z hw i = W (v, R,, ) d =0 T 3 prescribed average stress W + v r W + W r W + r P =0 div W =0 in (H4) W. R v W. R R W. R 1/2

45 Estimating new Reynolds stress R q+1 R q+1 = div t v q+1 + v q+1 rv q+1 + rp q+1 i (I) (II) (III) = div t w q+1 + v q rw q+1 i div 1h i r w q+1 w q+1 R q + r(p q+1 p q ) div 1h w q+1 rv q i

46 Estimating new Reynolds stress (III)= div 1h w q+1 rv q i =div 1h X i a k (x, t)e i q+1k x k6=0 stationary phase + (H4) k(iii)k 0. P k ka kk 0 q+1 (H1). kr qk 1/2 0 krv q k 0 q+1

47 Estimating new Reynolds stress slow (II)= div 1h r W W R q +rp i= div 1h X i b k (x, t)e i q+1k x k6=0 (H3) (H2) stationary phase + (H4) k(ii)k 0 = O( 1 q+1 ) k(i)k 0 = O(...and similarly 1 q+1 )

48 Estimating new Reynolds stress Summarizing: v q+1 = v q + W (v q,r q, q+1x, q+1t)+corrector 1) 2) kv q+1 v q k 0. kr q k 1/2 0 kr q+1 k 0. O( 1 q+1 ) leads to convergence in C 0 more careful estimates lead in the ideal case: 2 ) kr q+1 k 0. kr qk 1/2 0 krv q k 0 q+1

49 Conditions on the "fluctuation" W = W (v, R,, ) (H1) (H2) 7! W (v, R,, ) hw W i = R periodic with average zero: Z hw i = W (v, R,, ) d =0 T 3 convection of microstructure W + v r W + W r W + r P =0 div W =0 in (family of) stationary solutions: Beltrami flows (H4) W. R v W. R R W. R 1/2

50 Convection of microstructure Better Ansatz: v q+1 (x, t) =v q (x, t)+w R q (x, t), q+1 q(x, t) P. Isett 2013, PhD Thesis (1/5-Hölder, Theorem A) c.f. also D.W.McLaughlin-G.C.Papanicolaou-O.R.Pironneau 1985 inverse flow of v q (H1) hw i =0 (H3 ) W r W + r P =0 div W =0 (H2) hw W i = R (H4) W. R v W. R R W. R 1/2

51 Convection of microstructure Better Ansatz: v q+1 (x, t) =v q (x, t)+w R q (x, t), q+1 q(x, t) inverse flow of v q Problem: Flow only controlled for very short times v q+1 (x, t) =v q (x, t)+ X i i(t)w i R q (x, t), q+1 q,i(x, t) time cut-off partition of unity inverse flow restarted at time t i

52 Convection of microstructure B. Eckhardt - B. Hof - H.Faisst 2006 turbulent puffs Problem: Flow only controlled for very short times v q+1 (x, t) =v q (x, t)+ X i i(t)w i R q (x, t), q+1 q,i(x, t) P. Isett 2013, PhD Thesis (1/5-Hölder, Theorem B) Buckmaster - Isett - De Lellis - Sz. 2014, (1/5-Hölder, Theorem A) Buckmaster - De Lellis - Sz. 2015, (1/3-Hölder L1 in time, Theorem B) P. Isett 2016,(1/3-Hölder, Theorem B)

53 Convection of microstructure Main problems cell-problem: Beltrami modes not exact any more gluing: orthogonality of Beltrami modes

54 Convection of microstructure Main problems cell-problem: Beltrami modes not exact any more gluing: orthogonality of Beltrami modes Alternative to Beltrami-flows: Mikado flows tensegrity

55 Deterministic approach Navier Stokes Equations formal limit Euler Equations weak solutions weak convergence Turbulence averaging Relaxation dissipative/measure valued solutions

56 Deterministic approach Navier Stokes Equations formal limit Euler Equations weak solutions convex integration Turbulence averaging Relaxation subsolutions Meta-Theorem Any Euler subsolution can be approximated by weak solutions.

57 Thank you for your attention