These notes are based on a series of lectures given at the meeting Journées EDP in Roscoff in June 2015 on recent developments concerning weak solutions of the Euler equations and in particular recent progress concerning the construction of Hölder continuous weak solutions and Onsager’s conjecture.
@incollection{JEDP_2015____A10_0, author = {L\'aszl\'o Sz\'ekelyhidi Jr}, title = {Weak solutions of the {Euler} equations: non-uniqueness and dissipation}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {10}, pages = {1--34}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2015}, doi = {10.5802/jedp.639}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.639/} }
TY - JOUR AU - László Székelyhidi Jr TI - Weak solutions of the Euler equations: non-uniqueness and dissipation JO - Journées équations aux dérivées partielles PY - 2015 SP - 1 EP - 34 PB - Groupement de recherche 2434 du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.639/ DO - 10.5802/jedp.639 LA - en ID - JEDP_2015____A10_0 ER -
%0 Journal Article %A László Székelyhidi Jr %T Weak solutions of the Euler equations: non-uniqueness and dissipation %J Journées équations aux dérivées partielles %D 2015 %P 1-34 %I Groupement de recherche 2434 du CNRS %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.639/ %R 10.5802/jedp.639 %G en %F JEDP_2015____A10_0
László Székelyhidi Jr. Weak solutions of the Euler equations: non-uniqueness and dissipation. Journées équations aux dérivées partielles (2015), article no. 10, 34 p. doi : 10.5802/jedp.639. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.639/
[1] J. J. Alibert; G. Bouchitté Non-uniform integrability and generalized Young measures, J. Convex Anal., Volume 4 (1997) no. 1, pp. 129-147 | MR | Zbl
[2] J. M. Ball A version of the fundamental theorem for young measures, PDEs and continuum models of phase transitions, Springer-Verlag, Berlin/Heidelberg, 1989, pp. 207-215 | MR | Zbl
[3] C. Bardos; J. M. Ghidaglia; S. Kamvissis Weak Convergence and Deterministic Approach to Turbulent Diffusion (1999) (http://arxiv.org/abs/math/9904119) | Zbl
[4] C. Bardos; L. Székelyhidi Non-uniqueness for the Euler equations: the effect of the boundary, Russ. Math. Surv. (2014) | MR | Zbl
[5] C. Bardos; E. Titi Euler equations for incompressible ideal fluids, Russ. Math. Surv., Volume 62 (2007) no. 3, pp. 409-451 | MR | Zbl
[6] J. T. Beale; T. Kato; A. J. Majda Remarks on the breakdown of smooth solutions for the -D Euler equations, Comm. Math. Phys., Volume 94 (1984) no. 1, pp. 61-66 http://projecteuclid.org/euclid.cmp/1103941230 | MR | Zbl
[7] Ju. F. Borisov The parallel translation on a smooth surface. I, Vestnik Leningrad. Univ., Volume 13 (1958) no. 7, pp. 160-171 | MR | Zbl
[8] Ju. F. Borisov The parallel translation on a smooth surface. II, Vestnik Leningrad. Univ., Volume 13 (1958) no. 19, pp. 45-54 | MR | Zbl
[9] Ju. F. Borisov On the connection bewteen the spatial form of smooth surfaces and their intrinsic geometry, Vestnik Leningrad. Univ., Volume 14 (1959) no. 13, pp. 20-26 | MR | Zbl
[10] Ju. F. Borisov On the question of parallel displacement on a smooth surface and the connection of space forms of smooth surfaces with their intrinsic geometries., Vestnik Leningrad. Univ., Volume 15 (1960) no. 19, pp. 127-129 | MR
[11] Ju. F. Borisov -isometric immersions of Riemannian spaces, Dokl. Akad. Nauk SSSR, Volume 163 (1965), pp. 11-13 | MR | Zbl
[12] Ju. F. Borisov Irregular surfaces of the class with an analytic metric, Sibirsk. Mat. Zh., Volume 45 (2004) no. 1, pp. 25-61 | DOI | EuDML | MR | Zbl
[13] V. Borrelli; S. Jabrane; F. Lazarus; B. Thibert Flat tori in three-dimensional space and convex integration, Proc. Natl. Acad. Sci. USA, Volume 109 (2012) no. 19, pp. 7218-7223 | DOI | MR | Zbl
[14] Y. Brenier Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. PDE, Volume 25 (2000) no. 3, pp. 737-754 | MR | Zbl
[15] Y. Brenier; C. De Lellis; L. Székelyhidi Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys. (2011) | MR | Zbl
[16] Y. Brenier; E. Grenier Limite singulière du système de Vlasov-Poisson dans le régime de quasi neutralité: le cas indépendant du temps, C. R. Acad. Sci. Paris Sér. I Math., Volume 318 (1994) no. 2, pp. 121-124 | MR | Zbl
[17] A. Bressan; F. Flores On total differential inclusions, Rend. Sem. Mat. Univ. Padova, Volume 92 (1994), pp. 9-16 | EuDML | Numdam | MR | Zbl
[18] T. Buckmaster Onsager’s conjecture, University of Leipzig (2014) (Ph. D. Thesis)
[19] T. Buckmaster Onsager’s Conjecture Almost Everywhere in Time, Comm. Math. Phys., Volume 333 (2015) no. 3, pp. 1175-1198 | MR | Zbl
[20] T. Buckmaster; C. De Lellis; P. Isett; L. Székelyhidi Anomalous dissipation for 1/5-Hölder Euler flows, Ann. of Math. (2), Volume 182 (2015), pp. 1-46 | MR | Zbl
[21] T. Buckmaster; C. De Lellis; L. Székelyhidi Dissipative Euler flows with Onsager-critical spatial regularity, Comm. Pure Appl. Math., Volume math.AP (2015), pp. 1-66
[22] A. Cellina On the differential inclusion , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), Volume 69 (1980) no. 1-2, p. 1-6 (1981) | MR | Zbl
[23] A. Cellina A view on differential inclusions, Rend. Semin. Mat. Univ. Politec. Torino, Volume 63 (2005) no. 3 | EuDML | MR | Zbl
[24] A. Cellina; St. Perrotta On a problem of potential wells, J. Convex Anal., Volume 2 (1995) no. 1-2, pp. 103-115 | EuDML | MR | Zbl
[25] A. Cheskidov; P. Constantin; S. Friedlander; R. Shvydkoy Energy conservation and Onsager’s conjecture for the Euler equations, Nonlinearity, Volume 21 (2008) no. 6, pp. 1233-1252 | MR | Zbl
[26] A. Cheskidov; R. Shvydkoy Euler equations and turbulence: analytical approach to intermittency, SIAM J. Math. Anal, Volume 46 (2014) no. 1, pp. 353-374 | MR | Zbl
[27] A. Choffrut h-Principles for the Incompressible Euler Equations, Arch. Rational Mech. Anal., Volume 210 (2013) no. 1, pp. 133-163 | MR | Zbl
[28] A. Choffrut; L. Székelyhidi Weak solutions to the stationary incompressible Euler equations, SIAM J. Math. Anal, Volume 46 (2014) no. 6, pp. 4060-4074 | Zbl
[29] A. J. Chorin Vorticity and turbulence, Applied Mathematical Sciences, 103, Springer-Verlag, New York, 1994, pp. viii+174 | DOI | MR | Zbl
[30] St. Cohn-Vossen Zwei Sätze über die Starrheit der Eisflächen., Nachrichten Göttingen (1927), pp. 125-137 | JFM
[31] P. Constantin The Littlewood–Paley spectrum in two-dimensional turbulence, Theoret. Comp. Fluid Dynamics, Volume 9 (1997) no. 3, pp. 183-189 | Zbl
[32] P. Constantin On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc, Volume 44 (2007) no. 4, pp. 603 | MR | Zbl
[33] P. Constantin; C. Fefferman; A. J. Majda Geometric constraints on potentially singular solutions for the -D Euler equations, Comm. Partial Differential Equations, Volume 21 (1996) no. 3-4, pp. 559-571 | DOI | MR | Zbl
[34] P. Constantin; E. Weinan; E. Titi Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Comm. Math. Phys., Volume 165 (1994) no. 1, pp. 207-209 http://projecteuclid.org/euclid.cmp/1104271041 | MR | Zbl
[35] S. Conti; C. De Lellis; L. Székelyhidi -principle and rigidity for isometric embeddings, Nonlinear partial differential equations (Abel Symp.), Volume 7, Springer, Heidelberg, 2012, pp. 83-116 | DOI | MR | Zbl
[36] B. Dacorogna; P. Marcellini General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math., Volume 178 (1997) no. 1, pp. 1-37 | DOI | MR | Zbl
[37] B. Dacorogna; P. Marcellini; E. Paolini Lipschitz-continuous local isometric immersions: rigid maps and origami, J. Math. Pures Appl. (9), Volume 90 (2008) no. 1, pp. 66-81 | DOI | MR | Zbl
[38] C. M. Dafermos Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2000, pp. xvi+443 | DOI | MR | Zbl
[39] F. S. De Blasi; G. Pianigiani A Baire category approach to the existence of solutions of multivalued differential equations in Banach spaces, Funkcial. Ekvac., Volume 25 (1982) no. 2, pp. 153-162 http://www.math.kobe-u.ac.jp/~fe/xml/mr0694909.xml | MR | Zbl
[40] C. De Lellis; D. Inauen; L. Székelyhidi A Nash-Kuiper theorem for C1,15- immersions of surfaces in 3 dimensions (2015) (http://arxiv.org/abs/1510.01934)
[41] C. De Lellis; L. Székelyhidi The Euler equations as a differential inclusion, Ann. of Math. (2), Volume 170 (2009) no. 3, pp. 1417-1436 | MR | Zbl
[42] C. De Lellis; L. Székelyhidi On Admissibility Criteria for Weak Solutions of the Euler Equations, Arch. Rational Mech. Anal., Volume 195 (2010) no. 1, pp. 225-260 | MR | Zbl
[43] C. De Lellis; L. Székelyhidi The -principle and the equations of fluid dynamics, Bull. Amer. Math. Soc. (N.S.), Volume 49 (2012) no. 3, pp. 347-375 | MR | Zbl
[44] C. De Lellis; L. Székelyhidi Dissipative continuous Euler flows, Invent. Math., Volume 193 (2013) no. 2, pp. 377-407 | MR | Zbl
[45] C. De Lellis; L. Székelyhidi Dissipative Euler flows and Onsager’s conjecture, J. Eur. Math. Soc. (JEMS), Volume 16 (2014) no. 7, pp. 1467-1505 | EuDML | MR | Zbl
[46] R. J. DiPerna Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. (1985), pp. 383-420 | MR | Zbl
[47] R. J. DiPerna; A. J. Majda Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., Volume 108 (1987) no. 4, pp. 667-689 | MR | Zbl
[48] J. Duchon; R. Robert Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, Volume 13 (2000) no. 1, pp. 249-255 | DOI | MR | Zbl
[49] D. G. Ebin; J. Marsden Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2), Volume 92 (1970), pp. 102-163 | MR | Zbl
[50] Y. Eliashberg; N. M. Mishachev Introduction to the -principle, American Mathematical Society, 2002 | MR | Zbl
[51] G. L. Eyink Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Phys. D, Volume 78 (1994) no. 3-4, pp. 222-240 | MR | Zbl
[52] G. L. Eyink; K. R. Sreenivasan Onsager and the theory of hydrodynamic turbulence, Rev. Modern Phys., Volume 78 (2006) no. 1, pp. 87-135 | DOI | MR | Zbl
[53] U. Frisch Turbulence, Cambridge University Press, Cambridge, 1995, pp. xiv+296 (The legacy of A. N. Kolmogorov) | MR | Zbl
[54] M. Gromov Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete, 9, Springer Verlag, Berlin, 1986 | MR | Zbl
[55] M. Gromov Local and global in geometry, IHES preprint (1999), pp. 1-11
[56] D. Hilbert; St. Cohn-Vossen Geometry and the Imagination, American Mathematical Society, 1999
[57] P. Isett Hölder continuous Euler flows with compact support in time, Princeton University (2013) (Ph. D. Thesis) | MR
[58] P. Isett; S.-J. Oh On Nonperiodic Euler Flows with Hölder Regularity (2014) (http://arxiv.org/abs/1402.2305)
[59] P. Isett; V. Vicol Holder Continuous Solutions of Active Scalar Equations (2014) (http://arxiv.org/abs/1405.7656)
[60] T. Kato Nonstationary flows of viscous and ideal fluids in , J. Functional Analysis, Volume 9 (1972), pp. 296-305 | MR | Zbl
[61] B. Kirchheim Rigidity and Geometry of Microstructures, Habilitation Thesis, Univ. Leipzig (2003)
[62] A. Kolmogoroff The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers, C. R. (Doklady) Acad. Sci. URSS (N.S.), Volume 30 (1941), pp. 301-305 | Zbl
[63] N. H. Kuiper On -isometric imbeddings. I, II, Nederl. Akad. Wetensch. Indag. Math., Volume 17 (1955), p. 545-556, 683–689 | MR | Zbl
[64] P. D. Lax Deterministic theories of turbulence, Frontiers in pure and applied mathematics, North-Holland, Amsterdam, 1991, pp. 179-184 | MR | Zbl
[65] L. Lichtenstein Grundlagen der Hydromechanik, Springer Verlag, 1929 | MR | Zbl
[66] P.-L. Lions Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford University Press, 1996 | MR | Zbl
[67] St. Müller Variational models for microstructure and phase transitions, Calculus of Variations and Geometric Evolution Problems, Le ctures given at the 2nd Session of the Centre Internazionale Matematico Estivo, Cetaro (1996) | Zbl
[68] J. Nash isometric imbeddings, Ann. of Math. (2), Volume 60 (1954) no. 3, pp. 383-396 | MR | Zbl
[69] L. Onsager Statistical hydrodynamics, Nuovo Cimento (9), Volume 6 (1949) no. Supplemento, 2(Convegno Internazionale di Meccanica Statistica), pp. 279-287 | MR
[70] V. Scheffer An inviscid flow with compact support in space-time, J. Geom. Anal., Volume 3 (1993) no. 4, pp. 343-401 | DOI | MR | Zbl
[71] A. I. Shnirelman On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math., Volume 50 (1997) no. 12, pp. 1261-1286 | MR | Zbl
[72] A. I. Shnirelman Weak solution of incompressible Euler equations with decreasing energy, C. R. Math. Acad. Sci. Paris, Volume 326 (1998) no. 3, pp. 329-334 | Numdam | MR | Zbl
[73] L. Székelyhidi Weak solutions to the incompressible Euler equations with vortex sheet initial data, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 19-20, pp. 1063-1066 | MR | Zbl
[74] L. Székelyhidi From isometric embeddings to turbulence, HCDTE lecture notes. Part II. Nonlinear hyperbolic PDEs, dispersive and transport equations (AIMS Ser. Appl. Math.), Volume 7, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2013, pp. 63
[75] L. Székelyhidi; E. Wiedemann Young measures generated by ideal incompressible fluid flows, Arch. Rational Mech. Anal. (2012) | MR | Zbl
[76] L. Tartar The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations. Dordrecht (1977), pp. 263–285 | Zbl
[77] R. Temam Navier-Stokes equations, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984, pp. xii+526 | MR | Zbl
[78] E. Wiedemann Existence of weak solutions for the incompressible Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 28 (2011) no. 5, pp. 727-730 | Numdam | MR | Zbl
[79] S. T. Yau Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) (Proc. Sympos. Pure Math.), Volume 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1-28 | MR | Zbl
[80] L. C. Young Lecture on the Calculus of Variations and Optimal Control Theory, American Mathematical Society, 1980
Cited by Sources: