Paradoxe de Scheffer-Shnirelman revu sous l'angle de l'intégration convexe [d'après C. De Lellis et L. Székelyhidi]

Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1001, 34 p.

MR 2648676 | Zbl 05856440

BibTeX
RIS

@incollection{AST_2010__332__101_0,
     author = {Villani, C\'edric},
     title = {Paradoxe de {Scheffer-Shnirelman} revu sous l'angle de l'int\'egration convexe [d'apr\`es {C.} {De} {Lellis} et {L.} {Sz\'ekelyhidi]}},
     booktitle = {S\'eminaire Bourbaki : volume 2008/2009 expos\'es 997-1011  - Avec table par noms d'auteurs de 1848/49 \`a 2008/09},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:1001},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {332},
     year = {2010},
     zbl = {05856440},
     mrnumber = {2648676},
     language = {fr},
     url = {http://archive.numdam.org/item/AST_2010__332__101_0/}
}

TY  - CHAP
AU  - Villani, Cédric
TI  - Paradoxe de Scheffer-Shnirelman revu sous l'angle de l'intégration convexe [d'après C. De Lellis et L. Székelyhidi]
BT  - Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09
AU  - Collectif
T3  - Astérisque
N1  - talk:1001
PY  - 2010
DA  - 2010///
IS  - 332
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_2010__332__101_0/
UR  - https://zbmath.org/?q=an%3A05856440
UR  - https://www.ams.org/mathscinet-getitem?mr=2648676
LA  - fr
ID  - AST_2010__332__101_0
ER  -

Villani, Cédric. Paradoxe de Scheffer-Shnirelman revu sous l'angle de l'intégration convexe [d'après C. De Lellis et L. Székelyhidi], dans Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1001, 34 p. http://archive.numdam.org/item/AST_2010__332__101_0/

Bibliographie
Cité par

[1] V. I. Arnold & B. A. Khesin - Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer, 1998. | MR 1612569 | Zbl 0902.76001

[2] C. Bardos & E. S. Titi - Euler equations for an ideal incompressible fluid, Uspekhi Mat. Nauk 62 (2007), p. 5-46 ; traduction anglaise : Russian Math. Surveys 62 (2007), p. 409-451. | MR 2355417 | Zbl 1139.76010

[3] Y. F. Borisov - $C^{1, α}$ -isometric immersions of Riemannian spaces, Dokl. Akad. Nauk SSSR 163 (1965), p. 11-13 ; traduction anglaise : Soviet. Math. Dokl. 6 (1965), p. 869-871. | MR 192449 | Zbl 0135.40303

[4] Y. F. Borisov, Irregular surfaces of the class $C^{1, β}$ with an analytic metric, Sibirsk. Mat. Zh. 45 (2004), p. 25-61 ; traduction anglaise : Siberian Math. J. 45 (2004), 19-52. | EuDML 51131 | MR 2047871 | Zbl 1054.53081

[5] A. Bressan & F. Flores - On total differential inclusions, Rend. Sem. Mat. Univ. Padova 92 (1994), p. 9-16. | EuDML 108348 | Numdam | MR 1320474 | Zbl 0821.35158

[6] A. Cellina - On the differential inclusion $x^{'} \in [- 1, + 1]$ , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 69 (1980), p. 1-6. | MR 641583 | Zbl 0922.34009

[7] A. Cheskidov, P. Constantin, S. Friedlander & R. Shvydkoy - Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity 21 (2008), p. 1233-1252. | Article | MR 2422377 | Zbl 1138.76020

[8] P. Constantin, W. E. Titi & E. S. Titi - Onsager's conjecture on the energy conservation for solutions of Euler's equation, Comm. Math. Phys. 165 (1994), p. 207-209. | Article | MR 1298949 | Zbl 0818.35085

[9] P. Constantin, C. Fefferman & A. J. Majda - Geometric constraints on potentially singular solutions for the 3-D Euler equations, Comm. Partial Differential Equations 21 (1996), p. 559-571. | Article | MR 1387460 | Zbl 0853.35091

[10] S. Conti, C. De Lellis & L. Székelyhidi - Travail en cours.

[11] B. Dacorogna & P. Marcellini - General existence theorems for Hamilton- Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1997), p. 1-37. | Article | MR 1448710 | Zbl 0901.49027

[12] B. Dacorogna & P. Marcellini, Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications, vol. 37, Birkhäuser, 1999. | MR 1702252 | Zbl 0938.35002

[13] B. Dacorogna & G. Pisante - A general existence theorem for differential inclusions in the vector valued case, Port. Math. (N.S.) 62 (2005), p. 421-436. | EuDML 53037 | MR 2191629 | Zbl 1107.49008

[14] F. S. De Blasi & G. Pianigiani - A Baire category approach to the existence of solutions of multivalued differential equations in Banach spaces, Funkcial. Ekvac. 25 (1982), p. 153-162. | MR 694909 | Zbl 0535.34009

[15] C. De Lellis & L. Székelyhidi - On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 195 (2010), p. 225-260. | Article | MR 2564474 | Zbl 1192.35138

[16] C. De Lellis & L. Székelyhidi, Communication personnelle.

[17] C. De Lellis & L. Székelyhidi, The Euler equations as a differential inclusion, à paraître dans Ann. of Math., consultable en ligne à arXiv:math/0702079. | MR 2600877 | Zbl 1350.35146

[18] J.- M. Delort - Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991), p. 553-586. | Article | MR 1102579 | Zbl 0780.35073

[19] R. J. Diperna - Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292 (1985), p. 383-420. | Article | MR 808729 | Zbl 0606.35052

[20] S. K. Donaldson - Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), p. 666-705. | Article | MR 1438190 | Zbl 0883.53032

[21] J. Duchon & R. Robert - Dissipation d'énergie pour des solutions faibles des équations d'Euler et Navier-Stokes incompressibles, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), p. 243-248. | Article | MR 1711068 | Zbl 0939.76021

[22] J. Duchon & R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity 13 (2000), p. 249-255. | Article | MR 1734632 | Zbl 1009.35062

[23] Y. Eliashberg & N. Mishachev - Introduction to the $h$ -principle, Graduate Studies in Math., vol. 48, Amer. Math. Soc., 2002. | MR 1909245 | Zbl 1008.58001

[24] G. L. Eyink - Local 4/5-law and energy dissipation anomaly in turbulence, Nonlinearity 16 (2003), p. 137-145. | Article | MR 1950780 | Zbl 1138.76358

[25] A. F. Filippov - Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control 5 (1967), p. 609-621. | Article | MR 220995

[26] U. Frisch - Turbulence : The legacy of A. N. Kolmogorov, Cambridge Univ. Press, 1995. | Article | MR 1428905 | Zbl 0832.76001

[27] P. Gérard - Résultats récents sur les fluides parfaits incompressibles bidi-mensionnels (d'après J.-Y. Chemin et J.-M. Delort), Séminaire Bourbaki, vol. 1991/92, exposé n° 757, Astérisque 206 (1992), p. 411-444. | EuDML 110160 | Numdam | MR 1206075 | Zbl 1154.76321

[28] M. Gromov - Partial differential relations, Ergebnisse Math. Grenzg., vol. 9, Springer, 1986. | MR 864505 | Zbl 0651.53001

[29] B. Kirchheim - Deformations with finitely many gradients and stability of qua-siconvex hulls, C. R. Acad. Sei. Paris Sér. I Math. 332 (2001), p. 289-294. | Article | MR 1817378 | Zbl 0989.49013

[30] B. Kirchheim, Rigidity and geometry of microstructures, thèse d'habilitation, université de Leipzig, 2003.

[31] B. Kirchheim, S. Müller & V. Šverák - Studying nonlinear PDE by geometry in matrix space, in Geometric analysis and nonlinear partial differential equations, Springer, 2003, p. 347-395. | MR 2008346 | Zbl 1290.35097

[32] N. H. Kuiper - On $C^{1}$ -isometric imbeddings. I, Nederl. Akad. Wetensch. Proc. Ser. A. 58. | MR 75640 | Zbl 0067.39601

N. H. Kuiper - On $C^{1}$ -isometric imbeddings. II, Indag. Math. 17 (1955), p. 545-556, 683-689. | MR 75640 | Zbl 0067.39601

[33] N. H. Kuiper, Isometric and short imbeddings, Nederl. Akad. Wetensch. Proc. Ser. A. 62. | MR 100891 | Zbl 0092.14703

N. H. Kuiper, Isometric and short imbeddings, Indag. Math. 21 (1959), p. 11-25. | Article | MR 100891 | Zbl 0092.14703

[34] P.-L. Lions - Mathematical topics in fluid mechanics. Vol. 1 : Incompressible models, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press Oxford Univ. Press, 1996. | MR 1422251 | Zbl 0866.76002

[35] J. Lohkamp - Curvature $h$ -principles, Ann. of Math. 142 (1995), p. 457-498. | Article | MR 1356779 | Zbl 0909.58005

[36] C. Marchioro & M. Pulvirenti - Mathematical theory of incompressible non-viscous fluids, Applied Mathematical Sciences, vol. 96, Springer, 1994. | Article | MR 1245492 | Zbl 0789.76002

[37] S. Müller & V. Šverák - Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. 157 (2003), p. 715-742. | Article | MR 1983780 | Zbl 1083.35032

[38] F. Murat - Compacité par compensation, Ann. Scuola Norm. Sup. Pisa CI. Sci. 5 (1978), p. 489-507, partie II : Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978). | EuDML 83787 | Numdam | MR 506997 | Zbl 0399.46022

[38] F. Murat - Compacité par compensation, Ann. Scuola Norm. Sup. Pisa CI. Sci., Pitagora, Bologna, 1979, p. 245-256. | MR 533170 | Zbl 0427.35008

[39] J. Nash - $C^{1}$ isometric imbeddings, Ann. of Math. 60 (1954), p. 383-396. | Article | MR 65993 | Zbl 0058.37703

[40] Y. Ollivier - Communication personnelle.

[41] L. Onsager - Statistical hydrodynamics, Nuovo Cimento 6, Supplements (1949), p. 279-287. | Article | MR 36116

[42] A. V. Pogorelov - Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs, vol. 35, Amer. Math. Soc., 1973. | MR 346714 | Zbl 0311.53067

[43] V. Scheffer - An inviscid flow with compact support in space-time, J. Geom. Anal. 3 (1993), p. 343-401. | Article | MR 1231007 | Zbl 0836.76017

[44] A. Shnirelman - On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math. 50 (1997), p. 1261-1286. | Article | MR 1476315 | Zbl 0909.35109

[45] A. Shnirelman, Weak solutions with decreasing energy of incompressible Euler equations, Comm. Math. Phys. 210 (2000), p. 541-603. | Article | MR 1777341 | Zbl 1011.35107

[46] S. Smale - A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1958), p. 281-290. | Article | MR 104227 | Zbl 0089.18102

[47] D. Spring - Convex integration theory, solutions to the h-principle in geometry and topology, Monographs in Math., vol. 92, Birkhäuser, 1998. | MR 1488424 | Zbl 0997.57500

[48] M. A. Sychev - A few remarks on differential inclusions, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), p. 649-668. | Article | MR 2227812 | Zbl 1106.35153

[49] L. Székelyhidi - Differential inclusions, ouvrage en cours de rédaction.

[50] L. Tartar - Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics : Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, 1979, p. 136-212. | MR 584398 | Zbl 0437.35004

[51] L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of nonlinear partial differential equations (Oxford, 1982), NATO Adv. Sei. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, 1983, p. 263-285. | Article | MR 725524 | Zbl 0536.35003

[52] W. P. Thurston - Existence of codimension-one foliations, Ann. of Math. 104 (1976), p. 249-268. | Article | MR 425985 | Zbl 0347.57014