×

Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-\(BV\) perturbations. (English) Zbl 1261.35090

The authors consider a wide class of systems of conservation laws endowed with a convex entropy. Based on the notion of relative entropy, they develop the theory to establish the uniqueness and global \(L^2\)-stability of extremal entropy Rankine-Hugoniot shock and contact discontinuities in the class of weak entropy solutions with the trace property. The assumptions are quite general. In particular, neither a smallness condition nor global strict hyperbolicity is required. Applications are given to systems of fluid mechanics.

MSC:

35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35B35 Stability in context of PDEs
76L05 Shock waves and blast waves in fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Software:

HE-E1GODF
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bardos C., Golse F., Levermore C.D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Stat. Phys. 63(1–2), 323–344 (1991) · doi:10.1007/BF01026608
[2] Bardos C., Golse F., Levermore C.D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46(5), 667–753 (1993) · Zbl 0817.76002 · doi:10.1002/cpa.3160460503
[3] Berthelin F., Tzavaras A.E., Vasseur A.: From discrete velocity Boltzmann equations to gas dynamics before shocks. J. Stat. Phys. 135(1), 153–173 (2009) · Zbl 1168.82322 · doi:10.1007/s10955-009-9709-1
[4] Berthelin F., Vasseur A.: From kinetic equations to multidimensional isentropic gas dynamics before shocks. SIAM J. Math. Anal. 36(6), 1807–1835 (2005) · Zbl 1130.35090 · doi:10.1137/S0036141003431554
[5] Bressan A.: Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford University Press, Oxford (2000) · Zbl 0997.35002
[6] Bressan A., Colombo R.: Unique solutions of 2 {\(\times\)} 2 conservation laws with large data. Indiana Univ. Math. J. 44(3), 677–725 (1995) · Zbl 0852.35092 · doi:10.1512/iumj.1995.44.2004
[7] Bressan A., Crasta G., Piccoli B.: Well-posedness of the Cauchy problem for n {\(\times\)} n systems of conservation laws. Mem. Am. Math. Soc. 146(694), viii+134 (2000) · Zbl 0958.35001
[8] Bressan A., Liu T.-P., Yang T.: L 1 stability estimates for n {\(\times\)} n conservation laws. Arch. Rational Mech. Anal. 149(1), 1–22 (1999) · Zbl 0938.35093 · doi:10.1007/s002050050165
[9] Chen G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics. III. Acta Math. Sci. 6(1), 75–120 (1986) · Zbl 0643.76086
[10] Chen G.-Q., Frid H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Rational Mech. Anal. 147(2), 89–118 (1999) · Zbl 0942.35111 · doi:10.1007/s002050050146
[11] Chen G.-Q., Frid H.: Large-time behavior of entropy solutions of conservation laws. J. Differ. Equ. 152(2), 308–357 (1999) · Zbl 0926.35085 · doi:10.1006/jdeq.1998.3527
[12] Chen G.-Q., Frid H., Li Y.: Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics. Commun. Math. Phys. 228(2), 201–217 (2002) · Zbl 1029.76045 · doi:10.1007/s002200200615
[13] Chen G.-Q., Li Y.: Stability of Riemann solutions with large oscillation for the relativistic Euler equations. J. Differ. Equ. 202(2), 332–353 (2004) · Zbl 1068.35173 · doi:10.1016/j.jde.2004.02.009
[14] Chen G.-Q., Rascle M.: Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Rational Mech. Anal. 153(3), 205–220 (2000) · Zbl 0962.35122 · doi:10.1007/s002050000081
[15] Dafermos, C.: Entropy and the stability of classical solutions of hyperbolic systems of conservation laws. In: Recent Mathematical Methods in Nonlinear Wave Propagation (Montecatini Terme, 1994). Lecture Notes in Mathematics, vol. 1640, pp. 48–69. Springer, Berlin, 1996 · Zbl 0878.35072
[16] Dafermos C.M.: The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 70(2), 167–179 (1979) · Zbl 0448.73004 · doi:10.1007/BF00250353
[17] De Lellis C., Otto F., Westdickenberg M.: Structure of entropy solutions for multi-dimensional scalar conservation laws. Arch. Rational Mech. Anal. 170(2), 137–184 (2003) · Zbl 1036.35127 · doi:10.1007/s00205-003-0270-9
[18] DiPerna R.J.: Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28(1), 137–188 (1979) · Zbl 0409.35057 · doi:10.1512/iumj.1979.28.28011
[19] Glimm J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965) · Zbl 0141.28902 · doi:10.1002/cpa.3160180408
[20] Golse F., Saint-Raymond L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155(1), 81–161 (2004) · Zbl 1060.76101 · doi:10.1007/s00222-003-0316-5
[21] Hájek, O.: Discontinuous differential equations. I, II. J. Differ. Equ. 32(2), 149–170, 171–185 (1979)
[22] Kwon Y.-S.: Strong traces for degenerate parabolic-hyperbolic equations. Discrete Contin. Dyn. Syst. 25(4), 1275–1286 (2009) · Zbl 1180.35309 · doi:10.3934/dcds.2009.25.1275
[23] Kwon Y.-S., Vasseur A.: Strong traces for solutions to scalar conservation laws with general flux. Arch. Rational Mech. Anal. 185(3), 495–513 (2007) · Zbl 1121.35078 · doi:10.1007/s00205-007-0055-7
[24] Lax, P.: Shock waves and entropy. In: Contributions to Nonlinear Functional Analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), pp. 603–634. Academic Press, New York, 1971 · Zbl 0268.35014
[25] Leger, N.: L 2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method. Preprint (2009) · Zbl 1241.35134
[26] Lewicka M., Trivisa K.: On the L 1 well posedness of systems of conservation laws near solutions containing two large shocks. J. Differ. Equ. 179(1), 133–177 (2002) · Zbl 0994.35086 · doi:10.1006/jdeq.2000.4000
[27] Lions, P.-L., Masmoudi, N.: From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Rational Mech. Anal. 158(3), 173–193, 195–211 (2001) · Zbl 0987.76088
[28] Lions P.-L., Perthame B., Tadmor E.: Kinetic formulation of the isentropic gas dynamics and p-systems. Commun. Math. Phys. 163(2), 415–431 (1994) · Zbl 0799.35151 · doi:10.1007/BF02102014
[29] Liu T.-P.: The Riemann problem for general systems of conservation laws. J. Differ. Equ. 18, 218–234 (1975) · Zbl 0297.76057 · doi:10.1016/0022-0396(75)90091-1
[30] Liu T.-P., Ruggeri T.: Entropy production and admissibility of shocks. Acta Math. Appl. Sin. Engl. Ser. 19(1), 1–12 (2003) · Zbl 1029.35172 · doi:10.1007/s10255-003-0074-6
[31] Liu T.-P., Yang T.: Well-posedness theory for hyperbolic conservation laws. Commun. Pure Appl. Math. 52(12), 1553–1586 (1999) · Zbl 1034.35073 · doi:10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S
[32] Masmoudi N., Saint-Raymond L.: From the Boltzmann equation to the Stokes- Fourier system in a bounded domain. Commun. Pure Appl. Math. 56(9), 1263–1293 (2003) · Zbl 1024.35031 · doi:10.1002/cpa.10095
[33] Mellet A., Vasseur A.: Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations. Commun. Math. Phys. 281(3), 573–596 (2008) · Zbl 1155.35415 · doi:10.1007/s00220-008-0523-4
[34] Panov E.Y.: Existence of strong traces for generalized solutions of multidimensional scalar conservation laws. J. Hyperbolic Differ. Equ. 2(4), 885–908 (2005) · Zbl 1145.35429 · doi:10.1142/S0219891605000658
[35] Panov E.Y.: Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4(4), 729–770 (2007) · Zbl 1144.35037 · doi:10.1142/S0219891607001343
[36] Saint-Raymond L.: Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. Arch. Rational Mech. Anal. 166(1), 47–80 (2003) · Zbl 1016.76071 · doi:10.1007/s00205-002-0228-3
[37] Saint-Raymond L.: From the BGK model to the Navier-Stokes equations. Ann. Sci. École Norm. Sup. (4) 36(2), 271–317 (2003) · Zbl 1067.76078
[38] Smoller J.A., Johnson J.L.: Global solutions for an extended class of hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 32, 169–189 (1969) · Zbl 0167.10204 · doi:10.1007/BF00247508
[39] Toro E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 2nd edn. Springer, Berlin (1999) · Zbl 0923.76004
[40] Tzavaras A.E.: Relative entropy in hyperbolic relaxation. Commun. Math. Sci. 3(2), 119–132 (2005) · Zbl 1098.35104 · doi:10.4310/CMS.2005.v3.n2.a2
[41] Vasseur A.: Time regularity for the system of isentropic gas dynamics with {\(\gamma\)} = 3. Commun. Partial Differ. Equ. 24(11–12), 1987–1997 (1999) · Zbl 0940.35169 · doi:10.1080/03605309908821491
[42] Vasseur A.: Existence and properties of semidiscrete shock profiles for the isentropic gas dynamic system with {\(\gamma\)} = 3. SIAM J. Numer. Anal. 38(6), 1886–1901 (2001) · Zbl 1002.76062 · doi:10.1137/S0036142999355714
[43] Vasseur A.: Strong traces for solutions of multidimensional scalar conservation laws. Arch. Rational Mech. Anal. 160(3), 181–193 (2001) · Zbl 0999.35018 · doi:10.1007/s002050100157
[44] Vasseur, A.: Recent results on hydrodynamic limits. In: Handbook of Differential Equations: Evolutionary Equations, vol. IV, pp. 323–376. Elsevier/North-Holland, Amsterdam, 2008 · Zbl 1185.35003
[45] Yau H.-T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22(1), 63–80 (1991) · Zbl 0725.60120 · doi:10.1007/BF00400379
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.