On uniqueness of dissipative solutions to the isentropic Euler system. (English) Zbl 1428.35325

Summary: The dissipative solutions can be seen as a convenient generalization of the concept of weak solution to the isentropic Euler system. They can be seen as expectations of the Young measures associated to a suitable measure-valued solution of the problem. We show that dissipative solutions coincide with weak solutions starting from the same initial data on condition that: (i) the weak solution enjoys certain Besov regularity; (ii) the symmetric velocity gradient of the weak solution satisfies a one-sided Lipschitz bound.


35Q31 Euler equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35D30 Weak solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI arXiv


[1] Benzoni-Gavage, S.; Serre, D., Multidimensional Hyperbolic Partial Differential Equations, First Order Systems and Applications.. Oxford Mathematical Monographs, (2007), Oxford: The Clarendon Press Oxford University Press, Oxford · Zbl 1113.35001
[2] Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Vol. 53, (1984), New York: Springer-Verlag, New York · Zbl 0537.76001
[3] Smoller, J., Shock Waves and Reaction-Diffusion Equations, (1967), New York: Springer, New York
[4] DiPerna, R. J., Convergence of the viscosity method for isentropic gas dynamics, Communmath. Phys., 91, 1, 1-30, (1983) · Zbl 0533.76071
[5] Lions, P.-L.; Perthame, B.; Souganidis, E., Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates, Commun. Pure Appl. Math., 49, 6, 599-638, (1998) · Zbl 0853.76077
[6] Chiodaroli, E., A counterexample to well-posedness of entropy solutions to the compressible euler system, J. Hyperbolic Differ. Eq., 11, 3, 493-519, (2014) · Zbl 1304.35515
[7] Feireisl, E., Mathematical Fluid Dynamics, Present and Future, Vol. 183: Springer Proceedings in Mathematics and Statistics, Weak solutions to problems involving inviscid fluids, 377-399, (2016), New York: Springer, New York · Zbl 1371.35204
[8] Donatelli, D.; Feireisl, E.; Marcati, P., Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Commun. Partial Differen. Eq., 40, 7, 1314-1335, (2015) · Zbl 1326.35253
[9] De Lellis, C.; Székelyhidi, L. Jr., On admissibility criteria for weak solutions of the euler equations, Arch. Rational Mech. Anal, 195, 1, 225-260, (2010) · Zbl 1192.35138
[10] Chiodaroli, E.; De Lellis, C.; Kreml, O., Global ill-posedness of the isentropic system of gas dynamics, Commun. Pure Appl. Math., 68, 7, 1157-1190, (2015) · Zbl 1323.35137
[11] Chiodaroli, E.; Kreml, O., On the energy dissipation rate of solutions to the compressible isentropic euler system, Arch. Rational Mech. Anal, 214, 3, 1019-1049, (2014) · Zbl 1304.35516
[12] Chiodaroli, E.; Kreml, O.; Mácha, V.; Schwarzacher, S., Nonuniqueness of admissible weak solutions to the compressible euler equations with smooth initial data, Arxiv Preprint Series,, (2019)
[13] Dafermos, C. M., The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70, 2, 167-179, (1979) · Zbl 0448.73004
[14] Gwiazda, P.; Świerczewska-Gwiazda, A.; Wiedemann, E., Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity, 28, 11, 3873-3890, (2015) · Zbl 1336.35291
[15] Feireisl, E.; Kreml, O., Uniqueness of rarefaction waves in multidimensional compressible euler system, J. Hyperbolic Differen. Eq., 12, 3, 489-499, (2015) · Zbl 1327.35230
[16] Chen, G.-Q.; Chen, J., Stability of rarefaction waves and vacuum states for the multidimensional euler equations, J. Hyperbolic Differen. Eq., 4, 1, 105-122, (2007) · Zbl 1168.35312
[17] Bouchut, F.; James, F., One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32, 7, 891-933, (1998) · Zbl 0989.35130
[18] Bouchut, F.; James, F., Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Commun. Partial Differen. Eq., 24, 11-12, 2173-2189, (1999) · Zbl 0937.35098
[19] Chen, G. Q.; Glimm, J., Kolmogorov-type theory of compressible turbulence and inviscid limit of the Navier-Stokes equations in, Arxiv Preprint Series, (2018)
[20] Constantin, P.; Weinan, E.; 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
[21] Feireisl, E.; Gwiazda, P.; Świerczewska-Gwiazda, A.; Wiedemann, E., Regularity and energy conservation for the compressible euler equations, Arch. Rational Mech. Anal, 223, 3, 1375-1395, (2017) · Zbl 1365.35113
[22] Dafermos, C. M., The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differ. Eq., 14, 2, 202-212, (1973) · Zbl 0262.35038
[23] Breit, D.; Feireisl, E.; Hofmanová, M., Solution semiflow to the isentropic Euler system, Arxiv Preprint Series,, (2019)
[24] Lions, P.-L, Mathematical Topics in Fluid Dynamics, Vol. 1: Incompressible Models, (1996), Oxford: Oxford Science Publication, Oxford
[25] Li, T. T.; Yu, W. C., Boundary Value Problems for Quasilinear Hyperbolic Systems, (1985), Durham, NC: V. Duke University, Mathematics Department, Durham, NC
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.