A note on weak solutions of conservation laws and energy/entropy conservation. (English) Zbl 1398.35168

What is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality? The authors answered the question by extending the approach in [E. Feireisl et al., Arch. Ration. Mech. Anal. 223, No. 3, 1375–1395 (2017; Zbl 1365.35113)] for a general class of conservation laws. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied. This general scenario might be of independent interest for some experts.


35Q35 PDEs in connection with fluid mechanics
35L65 Hyperbolic conservation laws
35Q31 Euler equations
76W05 Magnetohydrodynamics and electrohydrodynamics
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35D30 Weak solutions to PDEs


Zbl 1365.35113
Full Text: DOI arXiv


[1] Ball, J.M.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics, pages 3-59. Springer, New York, 2002. · Zbl 1054.74008
[2] Buckmaster, T., Onsager’s conjecture almost everywhere in time, Commun. Math. Phys.,, 333, 1175-1198, (2015) · Zbl 1308.35184
[3] Buckmaster, T.; De Lellis, C.; Isett, P.; Székelyhidi, L., Anomalous dissipation for 1/5-Hölder Euler flows, Ann. Math., (2), 182, 127-172, (2015) · Zbl 1330.35303
[4] Buckmaster, T.; De Lellis, C.; Székelyhidi, L., Dissipative Euler flows with Onsager-critical spatial regularity, Commun. Pure Appl. Math.,, 69, 1613-1670, (2016) · Zbl 1351.35109
[5] Buckmaster, T., De Lellis, C., Székelyhidi, L. Jr., Vicol V.: Onsager’s conjecture for admissible weak solutions. arXiv:1701.08678, 2017.
[6] Caflisch, R.E.; Klapper, I.; Steele, G., Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Commun. Math. Phys.,, 184, 443-455, (1997) · Zbl 0874.76092
[7] Cheskidov, A.; Constantin, P.; Friedlander, S.; Shvydkoy, R., Energy conservation and onsager’s conjecture for the Euler equations, Nonlinearity,, 21, 1233-1252, (2008) · Zbl 1138.76020
[8] Constantin, P., E,W., Titi, E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys., 165(1), 207-209 (1994) · Zbl 0818.35085
[9] Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, fourth edition, 2016. · Zbl 1364.35003
[10] Dafermos, C.M.; Hrusa, W.J., Energy methods for quasilinear hyperbolic initial-boundary value problems. applications to elastodynamics, Arch. Ration. Mech. Anal.,, 87, 267-292, (1985) · Zbl 0586.35065
[11] De Lellis, C.; Székelyhidi, L., The Euler equations as a differential inclusion, Ann. Math. (2),, 170, 1417-1436, (2009) · Zbl 1350.35146
[12] De Lellis, C.; Székelyhidi, L., Dissipative continuous Euler flows, Invent. Math.,, 193, 377-407, (2013) · Zbl 1280.35103
[13] De Lellis, C.; Székelyhidi, L., Dissipative Euler flows and onsager’s conjecture, J. Eur. Math. Soc. (JEMS),, 16, 1467-1505, (2014) · Zbl 1307.35205
[14] Demoulini, S.; Stuart, D.M.A.; Tzavaras, A.E., A variational approximation scheme for three-dimensional elastodynamics with polyconvex energy, Arch. Ration. Mech. Anal.,, 157, 325-344, (2001) · Zbl 0985.74024
[15] Drivas, T.D., Eyink, G.L.: An Onsager singularity theorem for turbulent solutions of compressible Euler equations. Commu. Math. Phys. 2017. https://doi.org/10.1007/s00220-017-3078-4. · Zbl 1397.35191
[16] Duchon, J.; Robert, R., Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity,, 13, 249-255, (2000) · Zbl 1009.35062
[17] Eyink, G.L., Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Physica D,, 78, 222-240, (1994) · Zbl 0817.76011
[18] Feireisl, E.; Gwiazda, P.; Świerczewska-Gwiazda, A.; Wiedemann, E., Regularity and energy conservation for the compressible Euler equations, Arch. Ration. Mech. Anal.,, 223, 1-21, (2017) · Zbl 1365.35113
[19] Godunov, S.K., An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR,, 139, 521-523, (1961)
[20] Isett, P.: A Proof of Onsager’s Conjecture. arXiv:1608.08301, 2016. · Zbl 1335.58018
[21] Isett, P.: Hölder continuous Euler flows in three dimensions with compact support in time, volume 196 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2017. · Zbl 1367.35001
[22] Kang, E.; Lee, J., Remarks on the magnetic helicity and energy conservation for ideal magnetohydrodynamics, Nonlinearity,, 20, 2681-2689, (2007) · Zbl 1142.76062
[23] Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media. Volume VIII of Course of Theoretical Physics. Pergamon Press, 1961. · Zbl 1330.35303
[24] Leslie, T.M.; Shvydkoy, R., The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations, J. Differ. Equ.,, 261, 3719-3733, (2016) · Zbl 1383.76080
[25] Shvydkoy, R., On the energy of inviscid singular flows, J. Math. Anal. Appl.,, 349, 583-595, (2009) · Zbl 1184.35256
[26] Shvydkoy, R., Lectures on the Onsager conjecture, Discrete Contin. Dyn. Syst. Ser. S,, 3, 473-496, (2010) · Zbl 1210.76086
[27] Yu, C., Energy conservation for the weak solutions of the compressible Navier-Stokes equations, Arch. Ration. Mech. Anal.,, 225, 1073-1087, (2017) · Zbl 1375.35336
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