Isometric immersions and beyond

Transcription

1 Isometric immersions and beyond László Székelyhidi Jr. Universität Leipzig Fifth Abel Conference Celebrating the Mathematical Impact of John F. Nash Jr. and Louis Nirenberg

2 A problem in geometry Given a smooth Riemannian manifold (M n,g) do there exist isometric immersions in some Euclidean space R N? isometric: preserving the length of curves. A problem in PDE Given a smooth Riemannian manifold (M n,g) do there exist maps u : M n! R N such that, in local i j u = g ij i, j =1...n?

3 PDE aspects / difficulties first order system of s n := 1 2n(n + 1) equations with unknowns does not naturally fit into any of the usual (elliptic/parabolic/hyperbolic) types uniqueness?? local solvability: N Schläfli conjecture (1873): C 1 solvability for N = s n Janet - Cartan Theorem (1927): C! solvability for N = s n General conjecture open. For n=2 partial results by Lin (1986), Nakamura (1987), Han-Hong-Lin (2003), Khuri (2007),.

4 Prominent (geometric) special case: the Weyl problem (S 2,g),! R 3 with positive Gauss curvature apple g > 0 [H. Weyl 1916] i.e. a convex compact surface The Determination of a Closed Convex Surface Having Given Line Elements PhD Thesis of Louis Nirenberg (1949) Complete (PDE) solution of the Weyl problem: The Weyl and Minkowski problems in differential geometry in the large L. Nirenberg CPAM (1953) Independently solved also by Pogorelov by purely geometric methods..

5 The Weyl Problem (S 2,g),! R 3 The continuity method (Weyl): (a) Construct a homotopy {g t } t2[0,1] with positive curvature (b) openness : perturbation problem (IFT) i j u = g ij (c) closedness : a priori estimates for second derivatives using convexity (elliptic Monge-Ampére) Theorem (Nirenberg 1953) g 2 C 4 apple g > 0 S 2 Let be a metric with on. Then there u 2 C 4 (S 2,g),! R 3 exists an isometric immersion of. improved later by E. Heinz in 1962 to C 3

6 The Weyl Problem (S 2,g),! R 3 The continuity method (Weyl): (a) Construct a homotopy {g t } t2[0,1] with positive curvature (b) openness : perturbation problem (IFT) i j u = g ij (c) closedness : a priori estimates for second derivatives using convexity Simpler problem: suppose {u k } k is a sequence of smooth maps with (i) u k! u uniformly i u j u k = g ij Does it follow i j u = g ij? (@ i u j u k g ij )! 0 What if (ii) is replaced by (ii ) uniformly?

7 The Weyl Problem (S 2,g),! R 3 The continuity method (Weyl): (a) Construct a homotopy {g t } t2[0,1] with positive curvature (b) openness : perturbation problem (IFT) i j u = g ij (c) closedness : a priori estimates for second derivatives using convexity i j v j i v = h ij Crucial fact: homogeneous equation has only trivial solutions, i j v j i v =0 8i, j ) v = a u + b infinitesimal rigidity

8 Rigidity for the Weyl Problem (S 2,g),! R 3 i j u = g ij Theorem (Cohn-Vossen 1927, Herglotz 1943) C 2 A isometric immersion of with positive Gauss curvature is unique up to rigid motion. (S 2,g) i j v j i v =0 infinitesimal rigidity Theorem (Blaschke 1916) The linearization around a C 2 isometric immersion of with positive Gauss curvature admits no nontrivial solutions. (S 2,g)

9 Rigidity for the Weyl Problem (S 2,g),! R 3 i j u = g ij Theorem (Cohn-Vossen 1927, Herglotz 1943) C 2 A isometric immersion of with positive Gauss curvature is unique upto rigid motion. (S 2,g) rigidity below C 2 If u 2 C 1 and u(s 2 ) is convex [Pogorelov 1951] If u 2 C 1 and u(s 2 ) has bounded extrinsic curvature (N d finite measure) If u 2 C 1, > 2/3 [Borisov 1950s, Conti-De Lellis-Sz. 2004]

10 Intrinsic versus extrinsic geometry iu ju = g ij u 2 C 1,, g smooth convergence of regularizations: u` := u i j u` (g`) ij = O `2 kuk 2 1, consequently, can define parallel transport if > 1/2 and intrinsic curvature = extrinsic curvature, if > 2/3

11 Nash-Kuiper embedding Theorem (J. Nash N. Kuiper 1955) M n,! R n+1 Any short embedding can be uniformly approximated by C 1 isometric embeddings. short: length of curves gets shortened. an example of Gromov s h-principle method of proof: convex integration C 2 embeddings are rigid Lipschitz version of theorem is easy V. Borrelli - S. Jabrane - F. Lazarus - B. Thibert

12 The case M n,! R n+2 - the Nash spiral Start with a short embedding u : M n,! R n+2 with normals, Spiralling perturbation: ũ(x) =u(x)+ a(x) sin( x) (x) + cos( x) (x) amplitude i j ũ i j u + a 2 i j + O 1

13 The case M n,! R n+2 - the Nash spiral Start with a short embedding u : M n,! R n+2 with normals, Spiralling perturbation: ũ(x) =u(x)+ a(x) sin( x) (x) + cos( x) (x) amplitude i j ũ i j u + a 2 i j + O 1 linear part of perturbation

14 The Nash scheme: an inner iteration strictly short map: g i u@ j u = Xn k=1 a 2 k(x) k i k j Define: u 1 = u, u 1,u 2,...,ũ = u n +1 by u k+1 = u k + a k sin( k k x) k + cos( k k x) k k so i u j u k+1 i u j u k + a 2 k k i k j + O 1 k. frequencies: 1 << 1 << 2 << << n

15 The Nash scheme: inner iteration final i ũ@ j ũ = g ij + O 1 1 kũ uk C 1. Xn a k k=1. 1/2 C 0 kũ uk C

16 The Nash scheme: outer iteration The outer iteration: v = u + 1X n X i=1 k=1 u i k u i k is a spiral at frequency i,k!1

17 PNAS 2012

18 Nash-Kuiper embedding beyond C 1 Theorem (Borisov , Conti-De Lellis-Sz. 09) The Nash-Kuiper theorem remains valid for C 1, embeddings with 1 < 1+2n isometric recall: g i u@ j u = Xn k=1 a 2 k(x) k i k j (for local embeddings n = 1 2n(n + 1) in general) Theorem (De Lellis-Inauen-Sz. 15) For embedding 2-dimensional discs in < 1/5 theorem remains valid for. R 3 the Nash-Kuiper

19 Gromov's h-principle What Nash discovered is not any part of the Riemannian geometry, neither it has much (if anything at all) to do with the classical PDE. M. Gromov 2015

20 h-principle versus local-to-global principle Classical physics The main paradigm of the classical physics is non-existence of non-local interactions..non-surprisingly, we expect observable physical patters, e.g. the positions of moving particles after a given time interval, to be predictable in terms of the local laws and a presence of a particular microscopic law should be manifested by a specific global behaviour. Classical geometry notion of length rigidity of closed surfaces / isometric embeddings.. M. Gromov, 1999

21 h-principle versus local-to-global principle h-principle The infinitesimal structure of a medium does not effect the global geometry but only the topological behaviour of the medium. M. Gromov, 1999 Theorem (J. Nash 1956) u :(M n,g),! R N N 1 2n(3n + 11) Any short embedding, can be uniformly approximated by smooth isometric embeddings. [Nash 1966] Same with real analytic embeddings

22 h-principle versus local-to-global principle h-principle The infinitesimal structure of a medium does not effect the global geometry but only the topological behaviour of the medium. The class of infinitesimal laws subjugated by the h-principle is wide, but it does not include most partial differential equations of physics, with a few exceptions leading to unexpected solutions. M. Gromov, 1999 PDE in classical physics: Hadamard s notion of well-posedness, expect uniqueness PDE in geometry and topology: large invariance group, expect non-uniqueness cf. Müller, Stefan; Šverák, Vladimir: Unexpected solutions of first and second order partial differential equations. Proceedings of the ICM (Berlin, 1998)

23 Nonlinear Elasticity - Non-convex calculus of variations u u( ) Deformation: u : R 3! R 3 Minimize: Z W (Du) dx Expect non-uniqueness! (e.g. buckling, microstructures, ) J.M. Ball - R.D. James

24 Microstructures C. Chu - R.D. James How does the microstructure influence macroscopic elastic properties of a material?

25 Microstructures and differential inclusions Z W (Du) dx! min Replace (wlog W 0, K = {W =0} ) by Du(x) 2 K a.e. x frame indifference: K is SO(3) -invariant, i.e. SO(3)K = K

26 Differential inclusions Du(x) 2 K a.e. x 2 Solid-solid phase transitions and the appearance of microstructure: rigidity liquid-like behaviour K = SO(n) K = SO(2)A [ SO(2)B Müller-Šverák 99 Liouville, Reshetnyak 68 Friesecke-James-Müller 04 K = 3[ SO(3)A i i=1 Dolzmann-Kirchheim 02 crystal structure: cubic tetragonal

27 Differential inclusions Du(x) 2 K a.e. x 2 Solid-solid phase transitions and the appearance of microstructure: rigidity liquid-like behaviour K = SO(n) Liouville, Reshetnyak 68 Friesecke-James-Müller 04 K = SO(2)A [ SO(2)B 3[ K = SO(3)A i i=1 Müller-Šverák 99 Dolzmann-Kirchheim 02 crystal structure: cubic tetragonal typical statement: Du 2 SO(n) a.e. =) u a ne 8F with F Id < &detf =1, 9 u : Du 2 S 3 i=1 SO(3)A i a.e.x 2, u(x) = Fx

28 Differential Inclusions

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

30 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 But note: such maps do not necessarily preserve the length of curves!

31 Non-uniqueness for the Euler equations Theorem (Scheffer 93, Shnirelman 97, De Lellis - Sz. 2009) There exist nontrivial weak solutions of the Euler equations with compact support in t v + v rv + rp =0 r v =0 Classical fact: conservation of energy! 1 2 Z v(x, t) 2 dx

32 Non-uniqueness for the Euler equations 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, there exist infinitely many weak solutions of the Euler equations with 1 2 v(x, t) 2 = e(x, t) Classical fact: conservation of energy! 1 2 Z v(x, t) 2 dx

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

34 Selection criteria and instabilities

35 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, if there exists a solution to the IVP with v + v T L then this solution is unique in the class of admissible weak solutions. The Scheffer-Shnirelman solution clearly not admissible For the solutions of Wiedemann has an instantaneous jump up at E(t) t =0

36 Wild initial data 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

37 Strong instabilities: 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 )

38 Strong instabilities: 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. de Maximal dissipation rate dt = 1 6 Maximally dissipative solution is different from vanishing viscosity limit More realistic limit should include perturbations of the initial condition, i.e. Key point: the admissibility leads to global constraints for the mean velocity. (NS )! (E) (NS," )! (E)

39 Strong instabilities: Kelvin-Helmholtz t =0 t>0 turbulent zone 2t v = 1 a.e. Analogous statements for compressible isentropic Euler (E. Chiodaroli,C. De Lellis, O. Kreml, E. Feireisl) Muskat problem / incompressible porous media (F. Gancedo, D. Faraco, D. Cordoba, Sz.)

40 Regularity and (turbulent) spectra

41 Dissipation anomaly in turbulent t v +div(v v)+rp = div v =0 v Z Energy dissipation rate: = T 3 rv 2 dx Reynolds number: Re = UL Energy dissipation rate in various turbulent flows seems to remain positive as Reynolds number

42 Kolmogorov-Obukhov Spectrum Energy spectrum: U. Frisch: Turbulence

43 Onsager s conjecture 1949 In actual liquids this subdivision of energy is intercepted by the action of viscosity, which destroys the energy more rapidly the greater the wave number. However, various experiments indicate that the viscosity has a negligible effect on the primary process; hence one may inquire about the laws of dissipation in an ideal fluid. In fact it is possible to show that the velocity field in such ideal turbulence cannot obey any LIPSCHITZ condition of the form v(r 0 + r) v(r 0 ) apple (const.)r for any order greater than 1/3.

44 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. Onsager 1949, Eyink 1994, Constantin-E-Titi 1993, Robert-Duchon 2000, Cheskidov-Constantin-Friedlander- -Shvydkoy 2007, b) If < 1/3 dissipation possible. Scheffer 1993, Shnirelman 1999 De Lellis - Sz. 2012, Isett 2012 Buckmaster-De Lellis-Isett-Sz 2014 Buckmaster 2013,.

45 Constantin-E-Titi t v " + v " rv " + rp " =div v " v " (v v) " energy balance Z T v "(T ) 2 dx Z T v "(0) 2 dx = Z T 0 Z T 3 rv " (v " v " (v v) " ) dxdt commutator estimate kv " v " (v v) " k 0. " 2 [v] 2 energy conserved if Z T 0 [v(t)] 3 C dt < 1 for some > 1/3

46 Dissipative Hölder solutions A Theorem A (Buckmaster - De Lellis - Isett - Sz. 14) For any smooth positive function e(t) and any exists a weak solution of the Euler equations such that and v(x, t) v(y, t) applec x y Z v(x, t) 2 dx = e(t) T 3 < 1/5 there C. De Lellis - Sz. 2012, same for continuous + 1/10- S. Daneri - Sz. 2015, non-uniqueness for 1/5-Hölder A. Choffrut 2013, same in 2D

47 Dissipative Hölder solutions B Theorem B (Buckmaster - De Lellis - Sz 15) For any < 1/3 there exists a continuous weak solution of the Euler equations with compact support in time such that Z 1 0 [v(t)] C dt < 1 P. Isett 2012, same with 1/5-Hölder T. Buckmaster 2013, same with 1/3 Hölder a.e. t P. Isett - V. Vicol 2014, same for 1/9 Hölder for general class of active scalar equations [v(t)] =sup x6=y v(x, t) v(y, t) x y

48 Nash iteration for Euler

49 } } 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 spiral

50 } 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

51 Conditions on the "fluctuation" W = W (v, R,, ) (H1) 7! W (v, R,, ) periodic with average zero: Z hw i = W (v, R,, ) d =0 T 3 (H2) hw W i = R 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

52 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

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

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

55 Estimating new Reynolds stress R q+1 Summarizing: v q+1 = v q + W (v q,r q, q+1x, q+1t)+corrector 1) kv q+1 v q k 0. kr q k 1/2 0 2) kr q+1 k 0. O( 1 q+1 ) leads to convergence in C 0

56 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

57 Concluding remarks h-principle for the isometric immersions: abundance of solutions in case of high codimension or low regularity Convex integration: iterative construction, where quadratic is leading order Non-uniqueness for Euler connected to strong instabilities (e.g. Kelvin- Helmholtz) Selection criteria? Vanishing viscosity limit different from maximally dissipating solution. Mean flow corresponds to weak closure

58 h-principle for fluid dynamics The infinitesimal structure of a medium does not effect the global geometry but only the topological behaviour of the medium. The class of infinitesimal laws subjugated by the homotopy principle is wide, but it does not include most partial differential equations of physics, with a few exceptions leading to unexpected solutions. In fact, the presence of the h-principle would invalidate the very idea of a physical law as it yields very limited global information effected by the infinitesimal data. M. Gromov, 1999 BUT: the h-principle seems to apply to certain physical situations where a statistical description of a phenomenon (microstructures, turbulence) seems best suited. In terms of the analysis, one is then lead to consider solutions with low regularity.

59 Thank you for your attention