Weak solutions of the Euler equations: non-uniqueness and dissipation
Journées équations aux dérivées partielles (2015), article no. 10, 34 p.

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.

DOI: 10.5802/jedp.639
Székelyhidi Jr, László 1

1 Mathematisches Institut Universität Leipzig Augustusplatz 10 04109 Leipzig Germany
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Székelyhidi Jr, László. 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. http://www.numdam.org/articles/10.5802/jedp.639/

[1] Alibert, J. J.; Bouchitté, G. Non-uniform integrability and generalized Young measures, J. Convex Anal., Volume 4 (1997) no. 1, pp. 129-147 | MR | Zbl

[2] Ball, J. M. 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] Bardos, C.; Ghidaglia, J. M.; Kamvissis, S. Weak Convergence and Deterministic Approach to Turbulent Diffusion (1999) (http://arxiv.org/abs/math/9904119) | Zbl

[4] Bardos, C.; Székelyhidi, L. Jr. Non-uniqueness for the Euler equations: the effect of the boundary, Russ. Math. Surv. (2014) | MR | Zbl

[5] Bardos, C.; Titi, E. Euler equations for incompressible ideal fluids, Russ. Math. Surv., Volume 62 (2007) no. 3, pp. 409-451 | MR | Zbl

[6] Beale, J. T.; Kato, T.; Majda, A. J. Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., Volume 94 (1984) no. 1, pp. 61-66 http://projecteuclid.org/euclid.cmp/1103941230 | MR | Zbl

[7] Borisov, Ju. F. The parallel translation on a smooth surface. I, Vestnik Leningrad. Univ., Volume 13 (1958) no. 7, pp. 160-171 | MR | Zbl

[8] Borisov, Ju. F. The parallel translation on a smooth surface. II, Vestnik Leningrad. Univ., Volume 13 (1958) no. 19, pp. 45-54 | MR | Zbl

[9] Borisov, Ju. F. 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] Borisov, Ju. F. 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] Borisov, Ju. F. C 1,α -isometric immersions of Riemannian spaces, Dokl. Akad. Nauk SSSR, Volume 163 (1965), pp. 11-13 | MR | Zbl

[12] Borisov, Ju. F. Irregular surfaces of the class C 1,β with an analytic metric, Sibirsk. Mat. Zh., Volume 45 (2004) no. 1, pp. 25-61 | DOI | EuDML | MR | Zbl

[13] Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. 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] Brenier, Y. Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. PDE, Volume 25 (2000) no. 3, pp. 737-754 | MR | Zbl

[15] Brenier, Y.; De Lellis, C.; Székelyhidi, L. Jr. Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys. (2011) | MR | Zbl

[16] Brenier, Y.; Grenier, E. 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] Bressan, A.; Flores, F. On total differential inclusions, Rend. Sem. Mat. Univ. Padova, Volume 92 (1994), pp. 9-16 | EuDML | Numdam | MR | Zbl

[18] Buckmaster, T. Onsager’s conjecture, University of Leipzig (2014) (Ph. D. Thesis)

[19] Buckmaster, T. Onsager’s Conjecture Almost Everywhere in Time, Comm. Math. Phys., Volume 333 (2015) no. 3, pp. 1175-1198 | MR | Zbl

[20] Buckmaster, T.; De Lellis, C.; Isett, P.; Székelyhidi, L. Jr. Anomalous dissipation for 1/5-Hölder Euler flows, Ann. of Math. (2), Volume 182 (2015), pp. 1-46 | MR | Zbl

[21] Buckmaster, T.; De Lellis, C.; Székelyhidi, L. Jr. Dissipative Euler flows with Onsager-critical spatial regularity, Comm. Pure Appl. Math., Volume math.AP (2015), pp. 1-66

[22] Cellina, A. On the differential inclusion x ' [-1,+1], Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), Volume 69 (1980) no. 1-2, p. 1-6 (1981) | MR | Zbl

[23] Cellina, A. A view on differential inclusions, Rend. Semin. Mat. Univ. Politec. Torino, Volume 63 (2005) no. 3 | EuDML | MR | Zbl

[24] Cellina, A.; Perrotta, St. On a problem of potential wells, J. Convex Anal., Volume 2 (1995) no. 1-2, pp. 103-115 | EuDML | MR | Zbl

[25] Cheskidov, A.; Constantin, P.; Friedlander, S.; Shvydkoy, R. Energy conservation and Onsager’s conjecture for the Euler equations, Nonlinearity, Volume 21 (2008) no. 6, pp. 1233-1252 | MR | Zbl

[26] Cheskidov, A.; Shvydkoy, R. Euler equations and turbulence: analytical approach to intermittency, SIAM J. Math. Anal, Volume 46 (2014) no. 1, pp. 353-374 | MR | Zbl

[27] Choffrut, A. h-Principles for the Incompressible Euler Equations, Arch. Rational Mech. Anal., Volume 210 (2013) no. 1, pp. 133-163 | MR | Zbl

[28] Choffrut, A.; Székelyhidi, L. Jr. Weak solutions to the stationary incompressible Euler equations, SIAM J. Math. Anal, Volume 46 (2014) no. 6, pp. 4060-4074 | Zbl

[29] Chorin, A. J. Vorticity and turbulence, Applied Mathematical Sciences, 103, Springer-Verlag, New York, 1994, pp. viii+174 | DOI | MR | Zbl

[30] Cohn-Vossen, St. Zwei Sätze über die Starrheit der Eisflächen., Nachrichten Göttingen (1927), pp. 125-137 | JFM

[31] Constantin, P. The Littlewood–Paley spectrum in two-dimensional turbulence, Theoret. Comp. Fluid Dynamics, Volume 9 (1997) no. 3, pp. 183-189 | Zbl

[32] Constantin, P. On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc, Volume 44 (2007) no. 4, pp. 603 | MR | Zbl

[33] Constantin, P.; Fefferman, C.; Majda, A. J. Geometric constraints on potentially singular solutions for the 3-D Euler equations, Comm. Partial Differential Equations, Volume 21 (1996) no. 3-4, pp. 559-571 | DOI | MR | Zbl

[34] Constantin, P.; Weinan, E.; Titi, E. 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] Conti, S.; De Lellis, C.; Székelyhidi, L. Jr. h-principle and rigidity for C 1,α isometric embeddings, Nonlinear partial differential equations (Abel Symp.), Volume 7, Springer, Heidelberg, 2012, pp. 83-116 | DOI | MR | Zbl

[36] Dacorogna, B.; Marcellini, P. 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] Dacorogna, B.; Marcellini, P.; Paolini, E. 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] Dafermos, C. M. 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] De Blasi, F. S.; Pianigiani, G. 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] De Lellis, C.; Inauen, D.; Székelyhidi, L. Jr. A Nash-Kuiper theorem for C1,15-δ immersions of surfaces in 3 dimensions (2015) (http://arxiv.org/abs/1510.01934)

[41] De Lellis, C.; Székelyhidi, L. Jr. The Euler equations as a differential inclusion, Ann. of Math. (2), Volume 170 (2009) no. 3, pp. 1417-1436 | MR | Zbl

[42] De Lellis, C.; Székelyhidi, L. Jr. 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] De Lellis, C.; Székelyhidi, L. Jr. The h-principle and the equations of fluid dynamics, Bull. Amer. Math. Soc. (N.S.), Volume 49 (2012) no. 3, pp. 347-375 | MR | Zbl

[44] De Lellis, C.; Székelyhidi, L. Jr. Dissipative continuous Euler flows, Invent. Math., Volume 193 (2013) no. 2, pp. 377-407 | MR | Zbl

[45] De Lellis, C.; Székelyhidi, L. Jr. Dissipative Euler flows and Onsager’s conjecture, J. Eur. Math. Soc. (JEMS), Volume 16 (2014) no. 7, pp. 1467-1505 | EuDML | MR | Zbl

[46] DiPerna, R. J. Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. (1985), pp. 383-420 | MR | Zbl

[47] DiPerna, R. J.; Majda, A. J. 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] Duchon, J.; Robert, R. 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] Ebin, D. G.; Marsden, J. Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2), Volume 92 (1970), pp. 102-163 | MR | Zbl

[50] Eliashberg, Y.; Mishachev, N. M. Introduction to the h-principle, American Mathematical Society, 2002 | MR | Zbl

[51] Eyink, G. L. 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] Eyink, G. L.; Sreenivasan, K. R. Onsager and the theory of hydrodynamic turbulence, Rev. Modern Phys., Volume 78 (2006) no. 1, pp. 87-135 | DOI | MR | Zbl

[53] Frisch, U. Turbulence, Cambridge University Press, Cambridge, 1995, pp. xiv+296 (The legacy of A. N. Kolmogorov) | MR | Zbl

[54] Gromov, M. Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete, 9, Springer Verlag, Berlin, 1986 | MR | Zbl

[55] Gromov, M. Local and global in geometry, IHES preprint (1999), pp. 1-11

[56] Hilbert, D.; Cohn-Vossen, St. Geometry and the Imagination, American Mathematical Society, 1999

[57] Isett, P. Hölder continuous Euler flows with compact support in time, Princeton University (2013) (Ph. D. Thesis) | MR

[58] Isett, P.; Oh, S.-J. On Nonperiodic Euler Flows with Hölder Regularity (2014) (http://arxiv.org/abs/1402.2305)

[59] Isett, P.; Vicol, V. Holder Continuous Solutions of Active Scalar Equations (2014) (http://arxiv.org/abs/1405.7656)

[60] Kato, T. Nonstationary flows of viscous and ideal fluids in R 3 , J. Functional Analysis, Volume 9 (1972), pp. 296-305 | MR | Zbl

[61] Kirchheim, B. Rigidity and Geometry of Microstructures, Habilitation Thesis, Univ. Leipzig (2003)

[62] Kolmogoroff, A. 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] Kuiper, N. H. On C 1 -isometric imbeddings. I, II, Nederl. Akad. Wetensch. Indag. Math., Volume 17 (1955), p. 545-556, 683–689 | MR | Zbl

[64] Lax, P. D. Deterministic theories of turbulence, Frontiers in pure and applied mathematics, North-Holland, Amsterdam, 1991, pp. 179-184 | MR | Zbl

[65] Lichtenstein, L. Grundlagen der Hydromechanik, Springer Verlag, 1929 | MR | Zbl

[66] Lions, P.-L. Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford University Press, 1996 | MR | Zbl

[67] Müller, St. 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] Nash, J. C 1 isometric imbeddings, Ann. of Math. (2), Volume 60 (1954) no. 3, pp. 383-396 | MR | Zbl

[69] Onsager, L. Statistical hydrodynamics, Nuovo Cimento (9), Volume 6 (1949) no. Supplemento, 2(Convegno Internazionale di Meccanica Statistica), pp. 279-287 | MR

[70] Scheffer, V. An inviscid flow with compact support in space-time, J. Geom. Anal., Volume 3 (1993) no. 4, pp. 343-401 | DOI | MR | Zbl

[71] Shnirelman, A. I. 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] Shnirelman, A. I. 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] Székelyhidi, L. Jr. 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] Székelyhidi, L. Jr. 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] Székelyhidi, L. Jr.; Wiedemann, E. Young measures generated by ideal incompressible fluid flows, Arch. Rational Mech. Anal. (2012) | MR | Zbl

[76] Tartar, L. The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations. Dordrecht (1977), pp. 263–285 | Zbl

[77] Temam, R. Navier-Stokes equations, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984, pp. xii+526 | MR | Zbl

[78] Wiedemann, E. 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] Yau, S. T. 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] Young, L. C. Lecture on the Calculus of Variations and Optimal Control Theory, American Mathematical Society, 1980

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