Weak solutions of the Euler equations: non-uniqueness and dissipation

Székelyhidi Jr, László

Journées équations aux dérivées partielles, (2015), p. 1-34 / Harvested from Numdam

Résumé

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.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/jedp.639

Bibliographie

[1] Alibert, J. J.; Bouchitté, G. Non-uniform integrability and generalized Young measures, J. Convex Anal., Tome 4 (1997) no. 1, pp. 129-147 | MR 1459885 | Zbl 0981.49012

[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 1036070 | Zbl 0991.49500

[3] Bardos, C.; Ghidaglia, J. M.; Kamvissis, S. Weak Convergence and Deterministic Approach to Turbulent Diffusion (1999) (http://arxiv.org/abs/math/9904119) | Zbl 1301.35097

[4] Bardos, C.; Székelyhidi, L. Jr. Non-uniqueness for the Euler equations: the effect of the boundary, Russ. Math. Surv. (2014) | MR 1777632 | Zbl 0967.35112

[5] Bardos, C.; Titi, E. Euler equations for incompressible ideal fluids, Russ. Math. Surv., Tome 62 (2007) no. 3, pp. 409-451 | MR 2355417 | Zbl 1139.76010

[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., Tome 94 (1984) no. 1, pp. 61-66 http://projecteuclid.org/euclid.cmp/1103941230 | MR 763762 | Zbl 0573.76029

[7] Borisov, Ju. F. The parallel translation on a smooth surface. I, Vestnik Leningrad. Univ., Tome 13 (1958) no. 7, pp. 160-171 | MR 104277 | Zbl 0080.15105

[8] Borisov, Ju. F. The parallel translation on a smooth surface. II, Vestnik Leningrad. Univ., Tome 13 (1958) no. 19, pp. 45-54 | MR 104278 | Zbl 0121.17101

[9] Borisov, Ju. F. On the connection bewteen the spatial form of smooth surfaces and their intrinsic geometry, Vestnik Leningrad. Univ., Tome 14 (1959) no. 13, pp. 20-26 | MR 116295 | Zbl 0126.37302

[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., Tome 15 (1960) no. 19, pp. 127-129 | MR 131225

[11] Borisov, Ju. F. $C^{1,\,\alpha }$-isometric immersions of Riemannian spaces, Dokl. Akad. Nauk SSSR, Tome 163 (1965), pp. 11-13 | MR 192449 | Zbl 0135.40303

[12] Borisov, Ju. F. Irregular surfaces of the class $C^{1,\beta }$ with an analytic metric, Sibirsk. Mat. Zh., Tome 45 (2004) no. 1, pp. 25-61 | Article | MR 2047871 | Zbl 1054.53081

[13] Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. Flat tori in three-dimensional space and convex integration, Proc. Natl. Acad. Sci. USA, Tome 109 (2012) no. 19, pp. 7218-7223 | Article | MR 2935570 | Zbl 1267.53004

[14] Brenier, Y. Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. PDE, Tome 25 (2000) no. 3, pp. 737-754 | MR 1748352 | Zbl 0970.35110

[15] Brenier, Y.; De Lellis, C.; Székelyhidi, L. Jr. Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys. (2011) | MR 1798434 | Zbl 1219.35182

[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., Tome 318 (1994) no. 2, pp. 121-124 | MR 1260322 | Zbl 0803.35110

[17] Bressan, A.; Flores, F. On total differential inclusions, Rend. Sem. Mat. Univ. Padova, Tome 92 (1994), pp. 9-16 | Numdam | MR 1320474 | Zbl 0821.35158

[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., Tome 333 (2015) no. 3, pp. 1175-1198 | MR 3302631 | Zbl 1308.35184

[20] Buckmaster, T.; De Lellis, C.; Isett, P.; Székelyhidi, L. Jr. Anomalous dissipation for 1/5-Hölder Euler flows, Ann. of Math. (2), Tome 182 (2015), pp. 1-46 | MR 3374958 | Zbl 1330.35303

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

[22] Cellina, A. On the differential inclusion $x^\prime \in [-1,\,+1]$, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), Tome 69 (1980) no. 1-2, p. 1-6 (1981) | MR 641583 | Zbl 0922.34009

[23] Cellina, A. A view on differential inclusions, Rend. Semin. Mat. Univ. Politec. Torino, Tome 63 (2005) no. 3 | MR 2201565 | Zbl 1182.34013

[24] Cellina, A.; Perrotta, St. On a problem of potential wells, J. Convex Anal., Tome 2 (1995) no. 1-2, pp. 103-115 | MR 1363363 | Zbl 0880.49005

[25] Cheskidov, A.; Constantin, P.; Friedlander, S.; Shvydkoy, R. Energy conservation and Onsager’s conjecture for the Euler equations, Nonlinearity, Tome 21 (2008) no. 6, pp. 1233-1252 | MR 2422377 | Zbl 1138.76020

[26] Cheskidov, A.; Shvydkoy, R. Euler equations and turbulence: analytical approach to intermittency, SIAM J. Math. Anal, Tome 46 (2014) no. 1, pp. 353-374 | MR 3152734 | Zbl 1295.76010

[27] Choffrut, A. h-Principles for the Incompressible Euler Equations, Arch. Rational Mech. Anal., Tome 210 (2013) no. 1, pp. 133-163 | MR 3073150 | Zbl 1291.35200

[28] Choffrut, A.; Székelyhidi, L. Jr. Weak solutions to the stationary incompressible Euler equations, SIAM J. Math. Anal, Tome 46 (2014) no. 6, pp. 4060-4074 | Zbl 1311.35194

[29] Chorin, A. J. Vorticity and turbulence, Springer-Verlag, New York, Applied Mathematical Sciences, Tome 103 (1994), pp. viii+174 | Article | MR 1281384 | Zbl 0795.76002

[30] Cohn-Vossen, St. Zwei Sätze über die Starrheit der Eisflächen., Nachrichten Göttingen (1927), pp. 125-137 | JFM 53.0712.01

[31] Constantin, P. The Littlewood–Paley spectrum in two-dimensional turbulence, Theoret. Comp. Fluid Dynamics, Tome 9 (1997) no. 3, pp. 183-189 | Zbl 0907.76042

[32] Constantin, P. On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc, Tome 44 (2007) no. 4, pp. 603 | MR 2338368 | Zbl 1132.76009

[33] Constantin, P.; Fefferman, C.; Majda, A. J. Geometric constraints on potentially singular solutions for the $3$-D Euler equations, Comm. Partial Differential Equations, Tome 21 (1996) no. 3-4, pp. 559-571 | Article | MR 1298949 | Zbl 0853.35091 | Zbl 0818.35085

[34] Constantin, P.; Weinan, E.; Titi, E. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Comm. Math. Phys., Tome 165 (1994) no. 1, pp. 207-209 http://projecteuclid.org/euclid.cmp/1104271041 | MR 1387460 | Zbl 0818.35085 | Zbl 0853.35091

[35] Conti, S.; De Lellis, C.; Székelyhidi, L. Jr. $h$-principle and rigidity for $C^{1,\alpha }$ isometric embeddings, Nonlinear partial differential equations, Springer, Heidelberg (Abel Symp.) Tome 7 (2012), pp. 83-116 | Article | MR 3289360 | Zbl 1255.53038

[36] Dacorogna, B.; Marcellini, P. General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math., Tome 178 (1997) no. 1, pp. 1-37 | Article | MR 1448710 | Zbl 0901.49027

[37] Dacorogna, B.; Marcellini, P.; Paolini, E. Lipschitz-continuous local isometric immersions: rigid maps and origami, J. Math. Pures Appl. (9), Tome 90 (2008) no. 1, pp. 66-81 | Article | MR 2435215 | Zbl 1147.53042

[38] Dafermos, C. M. Hyperbolic conservation laws in continuum physics, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 325 (2000), pp. xvi+443 | Article | MR 1763936 | Zbl 1078.35001

[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., Tome 25 (1982) no. 2, pp. 153-162 http://www.math.kobe-u.ac.jp/~fe/xml/mr0694909.xml | MR 694909 | Zbl 0535.34009

[40] De Lellis, C.; Inauen, D.; Székelyhidi, L. Jr. A Nash-Kuiper theorem for C1,15-$\delta $ 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), Tome 170 (2009) no. 3, pp. 1417-1436 | MR 2600877 | Zbl 1350.35146

[42] De Lellis, C.; Székelyhidi, L. Jr. On Admissibility Criteria for Weak Solutions of the Euler Equations, Arch. Rational Mech. Anal., Tome 195 (2010) no. 1, pp. 225-260 | MR 2564474 | Zbl 1192.35138

[43] De Lellis, C.; Székelyhidi, L. Jr. The $h$-principle and the equations of fluid dynamics, Bull. Amer. Math. Soc. (N.S.), Tome 49 (2012) no. 3, pp. 347-375 | MR 2917063 | Zbl 1254.35180

[44] De Lellis, C.; Székelyhidi, L. Jr. Dissipative continuous Euler flows, Invent. Math., Tome 193 (2013) no. 2, pp. 377-407 | MR 3090182 | Zbl 1280.35103

[45] De Lellis, C.; Székelyhidi, L. Jr. Dissipative Euler flows and Onsager’s conjecture, J. Eur. Math. Soc. (JEMS), Tome 16 (2014) no. 7, pp. 1467-1505 | MR 3254331 | Zbl 1307.35205

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

[47] Diperna, R. J.; Majda, A. J. Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., Tome 108 (1987) no. 4, pp. 667-689 | MR 877643 | Zbl 0626.35059

[48] Duchon, J.; Robert, R. Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, Tome 13 (2000) no. 1, pp. 249-255 | Article | MR 1734632 | Zbl 1009.35062

[49] Ebin, D. G.; Marsden, J. Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2), Tome 92 (1970), pp. 102-163 | MR 271984 | Zbl 0211.57401

[50] Eliashberg, Y.; Mishachev, N. M. Introduction to the $h$-principle, American Mathematical Society (2002) | MR 1909245 | Zbl 1008.58001

[51] Eyink, G. L. Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Phys. D, Tome 78 (1994) no. 3-4, pp. 222-240 | MR 1302409 | Zbl 0817.76011

[52] Eyink, G. L.; Sreenivasan, K. R. Onsager and the theory of hydrodynamic turbulence, Rev. Modern Phys., Tome 78 (2006) no. 1, pp. 87-135 | Article | MR 2214822 | Zbl 1205.01032

[53] Frisch, U. Turbulence, Cambridge University Press, Cambridge (1995), pp. xiv+296 (The legacy of A. N. Kolmogorov) | MR 1428905 | Zbl 0832.76001

[54] Gromov, M. Partial differential relations, Springer Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete, Tome 9 (1986) | MR 864505 | Zbl 0651.53001

[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 3153420

[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 ${\bf R}^{3}$, J. Functional Analysis, Tome 9 (1972), pp. 296-305 | MR 481652 | Zbl 0229.76018

[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.), Tome 30 (1941), pp. 301-305 | Zbl 0025.37602

[63] Kuiper, N. H. On $C^1$-isometric imbeddings. I, II, Nederl. Akad. Wetensch. Indag. Math., Tome 17 (1955), p. 545-556, 683–689 | MR 75640 | Zbl 0067.39601

[64] Lax, P. D. Deterministic theories of turbulence, Frontiers in pure and applied mathematics, North-Holland, Amsterdam (1991), pp. 179-184 | MR 1110599 | Zbl 0727.76063

[65] Lichtenstein, L. Grundlagen der Hydromechanik, Springer Verlag (1929) | MR 228225 | Zbl 0157.56701

[66] Lions, P.-L. Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford University Press (1996) | MR 1422251 | Zbl 1264.76002

[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 0968.74050

[68] Nash, J. $C^1$ isometric imbeddings, Ann. of Math. (2), Tome 60 (1954) no. 3, pp. 383-396 | MR 65993 | Zbl 0058.37703

[69] Onsager, L. Statistical hydrodynamics, Nuovo Cimento (9), Tome 6 (1949) no. Supplemento, 2(Convegno Internazionale di Meccanica Statistica), pp. 279-287 | MR 36116

[70] Scheffer, V. An inviscid flow with compact support in space-time, J. Geom. Anal., Tome 3 (1993) no. 4, pp. 343-401 | Article | MR 1231007 | Zbl 0836.76017

[71] Shnirelman, A. I. On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math., Tome 50 (1997) no. 12, pp. 1261-1286 | MR 1476315 | Zbl 0909.35109

[72] Shnirelman, A. I. Weak solution of incompressible Euler equations with decreasing energy, C. R. Math. Acad. Sci. Paris, Tome 326 (1998) no. 3, pp. 329-334 | Numdam | MR 1648461 | Zbl 0913.35110

[73] Székelyhidi, L. Jr. Weak solutions to the incompressible Euler equations with vortex sheet initial data, C. R. Math. Acad. Sci. Paris, Tome 349 (2011) no. 19-20, pp. 1063-1066 | MR 2842999 | Zbl 1230.35093

[74] Székelyhidi, L. Jr. From isometric embeddings to turbulence, HCDTE lecture notes. Part II. Nonlinear hyperbolic PDEs, dispersive and transport equations, Am. Inst. Math. Sci. (AIMS), Springfield, MO (AIMS Ser. Appl. Math.) Tome 7 (2013), pp. 63

[75] Székelyhidi, L. Jr.; Wiedemann, E. Young measures generated by ideal incompressible fluid flows, Arch. Rational Mech. Anal. (2012) | MR 2968597 | Zbl 1256.35072

[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 0536.35003

[77] Temam, R. Navier-Stokes equations, North-Holland Publishing Co., Amsterdam, Studies in Mathematics and its Applications, Tome 2 (1984), pp. xii+526 | MR 769654 | Zbl 0568.35002

[78] Wiedemann, E. Existence of weak solutions for the incompressible Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 28 (2011) no. 5, pp. 727-730 | Numdam | MR 2838398 | Zbl 1228.35172

[79] Yau, S. T. Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 54 (1993), pp. 1-28 | MR 1216572 | Zbl 0801.53001

[80] Young, L. C. Lecture on the Calculus of Variations and Optimal Control Theory, American Mathematical Society (1980)