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Stability of Riemann solutions with large oscillation for the relativistic Euler equations. (English) Zbl 1068.35173

Entropy solutions of the \(2\times 2\) relativistic Euler equations are considered for perfect fluids in special relativity. The existence of Riemann solutions is established for a class of entropy solutions with arbitrarily large oscillations. The uniqueness of the Riemann solutions then yields their inviscid large-time stability with respect to arbitrarily large perturbations of the initial Riemann data. Finally, an extension is made to show the stability and uniqueness of vacuum Riemann solutions in a class of entropy solutions with arbitrarily large oscillations.

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
35B35 Stability in context of PDEs
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
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[1] S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Preprint, 2001.; S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Preprint, 2001. · Zbl 1082.35095
[2] Bressan, A., Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem (2000), Oxford University Press: Oxford University Press Oxford · Zbl 0997.35002
[3] Chang, T.; Hsiao, L., The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 41 (1989), Longman Scientific & Technical: Longman Scientific & Technical Essex, England · Zbl 0698.76078
[4] Chen, G.-Q, Vacuum states and global stability of rarefaction waves for compressible flow, Methods Appl. Anal., 7, 75-120 (2000)
[5] Chen, G.-Q; Frid, H., Large-time behavior of entropy solutions of conservation laws, J. Differential Equations, 152, 308-357 (1999) · Zbl 0926.35085
[6] Chen, G.-Q; Frid, H., Divergence-measure fields and hyperbolic conservation laws, Arch. Rational Mech. Anal., 147, 89-118 (1999) · Zbl 0942.35111
[7] Chen, G. Q.; Frid, H., Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations, Trans. Amer. Math. Soc., 353, 1103-1117 (2000) · Zbl 0958.35094
[8] Chen, G.-Q; Frid, H., Extended divergence-measure fields and the Euler equations for gas dynamics, Comm. Math. Phys., 236, 251-280 (2003) · Zbl 1036.35125
[9] Chen, G.-Q; Frid, H.; Li, Y. C., Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics, Comm. Math. Phys., 228, 201-217 (2002) · Zbl 1029.76045
[10] G.-Q. Chen, P. LeFloch, Existence theory for the relativistic Euler equations, in preparation, 2004.; G.-Q. Chen, P. LeFloch, Existence theory for the relativistic Euler equations, in preparation, 2004.
[11] Chen, G.-Q; Rascle, M., Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws, Arch. Rational Mech. Anal., 153, 205-220 (2000) · Zbl 0962.35122
[12] G.-Q. Chen, D. Wang, The Cauchy Problem for the Euler Equations for Compressible Fluids, Handbook of Mathematical Fluid Dynamics, Vol. 1, Elsevier Science B.V, Amsterdam, The Netherlands, 2002, pp. 421-543.; G.-Q. Chen, D. Wang, The Cauchy Problem for the Euler Equations for Compressible Fluids, Handbook of Mathematical Fluid Dynamics, Vol. 1, Elsevier Science B.V, Amsterdam, The Netherlands, 2002, pp. 421-543. · Zbl 1230.35096
[13] Chen, J., Conservation laws for the relativistic \(p\)-system, Comm. Partial Differential Equations, 20, 1605-1646 (1995) · Zbl 0877.35096
[14] Dafermos, C. M., Generalized characteristics in hyperbolic systems of conservation laws, Arch. Rational Mech. Anal., 107, 127-155 (1989) · Zbl 0714.35046
[15] Dafermos, C. M., Entropy and the stability of classical solutions of hyperbolic systems of conservation laws, (Ruggeri, Y., Recent Mathematical Methods in Nonlinear Wave Propagation (Montecatini Terme, 1994), Lecture Notes in Math, 1640 (1996), Springer: Springer Berlin), 48-69 · Zbl 0878.35072
[16] Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics (1999), Springer: Springer New York · Zbl 0433.73005
[17] DiPerna, R., Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., 28, 137-188 (1979) · Zbl 0409.35057
[18] Filippov, A. F., Differential equations with discontinuous right-hand side, Mat. Sb. (N.S.), 51, 99-128 (1960), English transl.: Amer. Math. Soc. Transl. Ser. 2 42 (1960) 199-231 · Zbl 0138.32204
[19] Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18, 95-105 (1965) · Zbl 0141.28902
[20] P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS, Vol. 11, SIAM, Philadelphia, 1973.; P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS, Vol. 11, SIAM, Philadelphia, 1973. · Zbl 0268.35062
[21] LeFloch, P., Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves (2002), ETH Lecture Notes: ETH Lecture Notes Birkhäuser, Basel · Zbl 1019.35001
[22] Lewick, M.; Trivisa, K., On the \(L^1\) well posedness of systems of conservation laws near solutions containing two large shocks, J. Differential Equations, 179, 133-177 (2002) · Zbl 0994.35086
[23] Li, Y. C., Global stability of solutions with discontinuous initial data containing vacuum states for the isentropic Euler equations, Z. Angew. Math. Phys., 54, 1-15 (2003)
[24] Liu, T.-P; Yang, T., Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math., 52, 1553-1586 (1999) · Zbl 1034.35073
[25] Serre, D., Systems of Conservation Laws I: Hyperbolicity, Entropies, Shock Waves; II: Geometric Structures, Oscillations, and Mixed Problems (2000), Cambridge University Press: Cambridge University Press Cambridge
[26] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1983), Springer: Springer New York · Zbl 0508.35002
[27] Smoller, J.; Temple, B., Global solutions of the relativistic Euler equations, Comm. Math. Phys., 156, 67-99 (1993) · Zbl 0780.76085
[28] Taub, A. H., Approximate solutions of the Einstein equations for isentropic motions of plane symmetric distributions of perfect fluids, Phys. Rev., 107, 884-900 (1957) · Zbl 0080.18306
[29] Thompson, K., The special relativistic shock tube, J. Fluid Mech., 171, 365-375 (1986) · Zbl 0609.76133
[30] Thorne, K. S., Relativistic shocksthe Taub adiabatic, Astrophys. J., 179, 897-907 (1973)
[31] Volpert, A. I., The space BV and quasilinear equations, Mat. Sb. (N.S.), 73, 255-302 (1967), Math. USSR Sbornik 2 (1967) 225-267 (in English)
[32] Weinberg, S., Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (1972), Wiley: Wiley New York
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