Norm inflation for incompressible Euler

Transcription

1 Norm inflation for incompressible Euler Dong Li University of British Columbia Nov 21 23, 2014, TexAMP Symposium 1 / 45

2 Outline Intro Folklore Evidence Results Proof 2 / 45

3 Incompressible Euler t u + (u )u + p = 0, (t, x) R R d, u = 0, u t=0 = u 0, where dimension d 2, velocity: u(t, x) = (u 1 (t, x),, u d (t, x)); pressure: p(t, x) : R R d R Well-known: LWP in H s (R d ), s > s c := 1 + d/2 (Old) Folklore problem: s = s c? 3 / 45

4 The appearance of critical index s c = 1 + d/2 Typical energy estimate (+usual regularization/mollification arguments): d ( ) u(t, ) 2 dt Hx s (R d ) C s,d Du(t, ) L x (R d ) u(t, ) 2 Hx s (R d ) To close the H s estimate, need Thus Du L x const u H s x. s > 1 + d/2 =: s c 4 / 45

5 A doomed attempt What about closing estimates in u X = u H s + Du and still hope s s c? Equation for Du (after eliminating pressure), roughly t (Du) + (u )(Du) + (Du )u + R }{{}}{{} ij (Du Du) = 0. }{{} OK OK came from pressure Due to Riesz transform R ij, R ij (Du Du) (Du)(Du) d H 2 +ɛ u d. H 2 +1+ɛ Again need s > s c (= d/2 + 1) 5 / 45

6 Issues with criticality s > s c can also be seen through vorticity formulation Similar questions arise in other function spaces An extensive literature on wellposedness results in non-critical spaces 6 / 45

7 Outline Intro Folklore Evidence Results Proof 7 / 45

8 Classical results: partial list Lichtenstein, Gunther (LWP in C k,α ) Wolibner (GWP of 2D Euler in Hölder), Chemin Ebin-Marsden (LWP of Euler in H d/2+1+ɛ on general compact manifolds, C -boundary allowed) Bouruignon-Brezis W s,p (s > d/p + 1). 8 / 45

9 Wellposedness results: Sobolev Kato 75 : LWP in C 0 t H m x (R d ), integer m > d/ Kato-Ponce 88 : LWP in W s,p (R d ), real s > d/p + 1, 1 < p < Kato-Ponce commutator estimate: J s = (1 ) s/2, s 0, 1 < p < : J s (fg) fj s g p d,s,p Df J s 1 g p + J s f p g, 9 / 45

10 Wellposedness results: Sobolev In Sobolev spaces W s,p (R d ), you need Not surprisingly, it came from s > d/p + 1 Du const u W d/p+1+ɛ,p 10 / 45

11 Wellposedness results: Besov Vishik 98: GWP of 2D Euler in B 2/p+1 p,1 (R 2 ), 1 < p <. Chae 04: LWP in B d/p+1 p,1 (R d ), 1 < p <. Pak-Park 04: LWP in B 1,1 (Rd ). The key idea of Besov refinements: you can push regularity down to critical s = d/p + 1, but you pay summability! Example: But H 1 (R 2 ) = B2,2(R 1 2 ) L (R 2 ) B 1 2,1(R 2 ) L (R 2 ) 11 / 45

12 The Besov l 1 -cheat If you insist on having critical regularity s c = d/p + 1, then you need B sc p,q with q =1!(to accommodate L embedding) NO wellposedness results were known for 1 < q. 12 / 45

13 A common theme Find a Banach space X such that (e.g. f = u, X = B d/p p,1 ) f + R ij f f X some version of Kato-Ponce commutator estimate holds in X. 13 / 45

14 The inevitable Completely breaks down for critical (say) H d 2 +1 (R d ) spaces. Two fantasies: super-good commutator estimates? divergence-free may save the day? NO! Takada 10: divergence-free counterexamples of Kato-Ponce-type commutator estimates in critical Bp,q d/p+1 (R d ) (1 p, 1 < q ) and Fp,q d/p+1 (R d ) (1 < p <, q or p = q = ) space. 14 / 45

15 The slightest clue Consider 2D Euler in vorticity form: ω = x2 u 1 + x1 u 2, Critical space: H 2 (R 2 ) for u. t ω + 1 ω ω = 0. 1 d 2 dt x 1 ω 2 2 = ( 1 1 ω ω) 1 ωdx R } 2 {{} Can be made very large when ω H 1 only Not difficult to show: no C 1 t H 2 wellposedness (for velocity u). But this does not rule out C 0 t H 2, L t H 2, and so on! 15 / 45

16 Folklore problem Conjecture: The Euler equation is illposed for a class of initial data in H d/2+1 (R d ) Rem: analogous versions in W d/p+1,p, Besov, Triebel-Lizorkin... Part of the difficulty: How even to formulate it? Infinitely worse: How to approach it? Need a deep understanding of how critical space typology changes under the Euler dynamics 16 / 45

17 Outline Intro Folklore Evidence Results Proof 17 / 45

18 Explicit solutions Two and a half dimensional shear flow (DiPerna-Majda 87): u(t, x) = (f (x 2 ), 0, g(x 1 tf (x 2 ))), x = (x 1, x 2, x 3 ), f and g are given 1D functions. Solves 3D Euler with pressure p = 0 DiPerna-Lions: 1 p <, T > 0, M > 0, exist u(0) W 1,p (T 3 ) = 1, u(t ) W 1,p (T 3 ) > M Bardos-Titi 10: u(0) C α, but u(t) / C β for any t > 0, 1 > β > α 2 (illposedness in F 1,2 and B1, ). Misio lek and Yoneda: illposedness in LL α, 0 < α 1: f LLα = f + sup 0< x y < 1 2 f (x) f (y) x y log x y α <. 18 / 45

19 Understanding the solution operator Kato 75: the solution operator for the Burgers equation is not Hölder continuous in H s (R), s 2 norm for any prescribed Hölder exponent. Himonas and Misio lek 10: the data-to-solution map of Euler is not uniformly continuous in H s topology Inci 13: nowhere locally uniformly continuous in H s (R d ), s > d/ Cheskidov-Shvydkoy 10: illposedness of Euler in B s r, (T d ), s > 0 if r > 2; s > d(2/r 1) if 1 r 2. Yudovich (Hölder e Ct ), Bahouri-Chemin 94 (not Hölder e t ), Kelliher 10 (no Hölder) No bearing on the critical case! 19 / 45

20 Outline Intro Folklore Evidence Results Proof 20 / 45

21 Strong illposedness of Euler Roughly speaking, the results are as follows: Theorem[Bourgain-L 13]. Let the dimension d = 2, 3. The Euler equation is indeed illposed in the Sobolev space W d/p+1,p for any 1 < p < or the Besov space Bp,q d/p+1 for any 1 < p <, 1 < q. 21 / 45

22 Situation worse/better than we thought Usual scenario (you would think): Initially u(0) X 1 Later u(t ) X 1 Here: u(0) X 1, but Rem: kills even L t X! ess-sup 0<t<T u(t) X = + 22 / 45

23 Generic Illposedness Strongly illposed : any smooth u 0, once can find a nearby v 0, s.t. but v 0 u 0 X < ɛ, ess-sup 0<t<t0 v(t) X = +, t 0 > 0. Illposedness (Norm inflation) is dense in critical X -topology! 23 / 45

24 Vorticity formulation To state more precisely the main results, recall: 2D Euler: ω = u: t ω + ( 1 ω )ω = 0, ω = ω 0. t=0 3D Euler: ω = u, t ω + (u )ω = (ω )u, u = 1 ω, ω = ω 0. t=0 24 / 45

25 Coming up: 2D Euler TWO cases for 2D Euler: a Initial vorticity ω is not compactly-supported b Compactly supported case 3D Euler a lot more involved (comments later) 25 / 45

26 2D Euler non-compact data Theorem 1: For any given ω (g) 0 C c (R 2 ) Ḣ 1 (R 2 ) and any ɛ > 0, there exist C perturbation ω (p) 0 : R 2 R s.t. 1. ω (p) 0 Ḣ 1 (R 2 ) + ω(p) 0 L 1 (R 2 )+ ω (p) 0 L (R 2 )+ ω (p) 0 Ḣ 1 (R 2 ) < ɛ. 2. Let ω 0 = ω (g) 0 + ω (p) 0. The initial velocity u 0 = 1 ω 0 has regularity u 0 H 2 (R 2 ) C (R 2 ) L (R 2 ). 3. There exists a unique classical solution ω = ω(t) to the 2D Euler equation (in vorticity form) satisfying ( ) ω(t, ) L 1 + ω(t, ) L + ω(t, ) Ḣ 1 <. max 0 t 1 Here ω(t) C, u(t) = 1 ω(t) C L 2 L for each 0 t For any 0 < t 0 1, we have ess-sup 0<t t0 ω(t, ) L 2 (R 2 ) = / 45

27 Comments The Ḣ 1 assumption on the vorticity data ω (g) 0 is actually not needed In our construction, although the initial velocity u 0 is C L, u 0 L (R 2 ) = +. Classical C -solutions! (No need to appeal to Yudovich theory) Kato 75 introduced the uniformly local Sobolev spaces L p ul (Rd ), H s ul (Rd ) which contain H s (R d ) and the periodic space H s (T d ). We can refine the result to ess-sup 0<t t0 ω(t, ) L 2 ul (R 2 ) = / 45

28 2D compactly supported case Next result: the compactly supported data for the 2D Euler equation. Carries over (with simple changes) to the periodic case as well For simplicity consider vorticity functions having one-fold symmetry: g is odd in x 1 g( x 1, x 2 ) = g(x 1, x 2 ), x = (x 1, x 2 ) R 2. Preserved by the Euler flow 28 / 45

29 2D Euler compact data Theorem 2: Let ω (g) 0 C c (R 2 ) Ḣ 1 (R 2 ) be any given vorticity function which is odd in x 2. For any any ɛ > 0, we can find a perturbation ω (p) 0 : R 2 R s.t. 1. ω (p) 0 is compactly supported (in a ball of radius 1), continuous and ω (p) 0 Ḣ 1 (R 2 ) + ω(p) 0 L (R 2 ) + ω (p) 0 Ḣ 1 (R 2 ) < ɛ. 2. Let ω 0 = ω (g) 0 + ω (p) 0. Corresponding to ω 0 there exists a unique time-global solution ω = ω(t) to the Euler equation satisfying ω(t) L Ḣ 1. Furthermore ω Ct 0 Cx 0 and u = 1 ω Ct 0 L 2 x Ct 0 Cx α for any 0 < α < ω(t) has additional local regularity in the following sense: there exists x R 2 such that for any x x, there exists a neighborhood N x x, t x > 0 such that ω(t, ) C (N x ) for any 0 t t x. 29 / 45

30 Conti. For any 0 < t 0 1, we have ess-sup 0<t t0 ω(t, ) Ḣ1 = +. More precisely, there exist 0 < tn 1 < tn 2 < 1 n, open precompact sets Ω n, n = 1, 2, 3, such that ω(t) C (Ω n ) for all 0 t tn, 2 and ω(t, ) L 2 (Ω n) > n, t [t 1 n, t 2 n]. 30 / 45

31 Comments: uniqueness Yudovich 63: existence and uniqueness of weak solutions to 2D Euler in bounded domains for L vorticity data. Yudovich 95: improved uniqueness result (for bounded domain in general dimensions d 2) allowing vorticty ω p0 p< L p and ω p Cθ(p) with θ(p) growing relatively slowly in p (such as θ(p) = log p). Vishik 99 uniqueness of weak solutions to Euler in R d, d 2, under the following assumptions: ω L p 0, 1 < p 0 < d, For some a(k) > 0 with the property 1 1 dk = +, a(k) it holds that k P 2 j ω const a(k), k 4. j=2 Uniqueness OK in our construction: uniform in time L control of the vorticity ω 31 / 45

32 3D case Without going into any computation, you realize: One of difficulty in 3D: lifespan of smooth initial data Technical issues: make judicious perturbation in H 5/2 while controlling the lifespan! Vorticity stretching 32 / 45

33 Comments (contin...) Critical norm Ḣ 3 2 for vorticity (H 5 2 for velocity). A technical nuissance: nonlocal fractional differentiation operator 3 2 Theorem 1-Theorem 4 can be sharpened significantly: e.g. u 0 u (g) 0 Bp,q d/p+1 < ɛ, ess-sup 0<t<t0 u(t, ) Ḃd/p+1 p, = + for any t 0 > 0. Similarly: Sobolev W d/p+1,p, Triebel-Lizorkin etc. 33 / 45

34 Outline Intro Folklore Evidence Results Proof 34 / 45

35 Consider 2D Euler in vorticity formulation: t ω + 1 ω ω = 0. Critical space: Ḣ 1 (R 2 ) for ω (H 2 for velocity u) 35 / 45

36 Step 1: Creation of large Lagrangian deformation Flow map φ = φ(t, x) { t φ(t, x) = u(t, φ(t, x)), φ(0, x) = x. For any 0 < T 1, B(x 0, δ) R 2 and δ 1, choose initial (vorticity) data ω a (0) such that and ω (0) a L 1 + ω (0) a L + ω (0) a H 1 1, sup Dφ a (t, ) 1. 0<t T φ a is the flow map associated with the velocity u = u a. Judiciously choose ω a (0) as a chain of bubbles concentrated near origin (respect H 1 assumption!) Deformation matrix Du remains essentially hyperbolic 36 / 45

37 Step 2: Local inflation of critical norm. The solution constructed in Step 1 does not necessarily obey sup 0<t T ω a (t) 2 1. Perturb the initial data ω (0) a and take ω (0) b = ω a (0) + 1 sin(kf (x))g(x), k where k is a very large parameter. g 2 o(1), f captures Dφ a (t, ). A perturbation argument in W 1,4 to fix the change in flow map As a result, in the main order the H 1 norm of the solution corresponding to ω (0) b is inflated through the Lagrangian deformation matrix Dφ a. Rem: Nash twist, Onsager (De Lellis-Szekelyhidi) 37 / 45

38 Step 3: Gluing of patch solutions Repeat the local construction in infinitely many small patches which stay away from each other initially. To glue these solutions: consider two cases. 38 / 45

39 Noncompact case... Case 3a: noncompact data Add each patches sequentially and choose their mutual distance ever larger! REM: this is the analogue of weakly interacting particles in Stat Mech. Key properties exploited here: finite transport speed of the Euler flow; spatial decay of the Riesz kernel. 39 / 45

40 Living on different scales... Case 3b: compact data Patches inevitably get close to each other! Need to take care of fine interactions between patches (Strongly interacting case!) 40 / 45

41 3b (contin...): a very involved analysis For each n 2, define ω n 1 the existing patch and ω n the current (to be added) patch There exists a patch time T n s.t. for 0 t T n, the patch ω n has disjoint support from ω n 1, and obeys the dynamics t ω n + 1 ω n 1 ω n + 1 ω n ω n = 0. By a re-definition of the patch center and change of variable, ω n satisfies t ω n + 1 ω n ω n ( ) y1 + b(t) ω n + r(t, y) ω n = 0, y 2 where b(t) = O(1) and r(t, y) y 2. Re-do a new inflation argument! 41 / 45

42 3b (conti...) Choose initial data for ω n such that within patch time 0 < t T n the critical norm of ω n inflates rapidly. As we take n, the patch time T n 0 and ω n becomes more and more localized the whole solution is actually time-global. During interaction time T n ( Smoothness in limited patch time!) the patch ω n produces the desired norm inflation since it stays well disjoint from all the other patches. 42 / 45

43 Difficulties in 3D: a snapshot First difficulty in 3D: lack of L p conservation of the vorticity. Deeply connected with the vorticity stretching term (ω )u To simplify the analysis, consider the axisymmetric flow without swirl ( ω ) ( ω ) t + (u ) = 0, r = x1 2 r r + x 2 2, x = (x 1, x 2, z). Owing to the denominator r, the solution formula for ω then acquires an additional metric factor (compared with 2D) which represents the vorticity stretching effect in the axisymmetric setting. Difficulty: control the metric factor and still produce large Lagrangian deformation. 43 / 45

44 Difficulties in 3D (conti..) The best local theory still requires ω/r L 3,1 (R 3 ) The dilemma: need infinite ω/r L 3,1 norm to produce inflation. A new perturbation argument: add each new patch ω n with sufficiently small ω n norm (over the whole lifespan) such that the effect of the large ω n /r L 3,1 becomes negligible. The spin-off: local solution with infinite ω/r L 3,1 More technical issues... norm! 44 / 45

45 In summary Our new strategy: Large Lagrangian deformation induces critical norm inflation Exploited both Lagrangian and Eulerian point of view A multi-scale construction! In stark contrast: H 1 -critical NLS in R 3 : i t u + u = u 4 u is wellposed for u 0 Ḣ 1 (R 3 ) Flurry of more recent developments Proof of endpoint Kato-Ponce (conjectured by Grafakos-Maldonado-Naibo) C m case: anisotropic Lagrangian deformation, flow decoupling Misolek-Yoneda, Masmoudi-Elgindi, / 45