On stability of shock waves in relativistic magnetohydrodynamics
Author:
Yu. L. Trakhinin
Journal:
Quart. Appl. Math. 59 (2001), 25-45
MSC:
Primary 76W05; Secondary 35L50, 35Q35, 76L05
DOI:
https://doi.org/10.1090/qam/1811093
MathSciNet review:
MR1811093
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Abstract: The structural stability of relativistic magnetohydrodynamic shock waves is studied. Stability results are obtained for the special case of fast parallel shock waves. It is proved that the instability and linear stability domains coincide with those of shock waves in relativistic gas dynamics. The domain of structural (nonlinear) stability, where the uniform Lopatinski condition is fulfilled for the stability problem, is found. It is shown that the structural stability domain is smaller than that of relativistic gas dynamic shock waves.
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A. M. Anile and G. Russo, Linear stability for plane relativistic shock waves, Phys. Fluids 30, 1045–1051 (1987)
A. M. Blokhin, The estimate of energy integral of mixed problem for equations of gas dynamics with boundary conditions on a shock wave, Sibirsk. Mat. Zh. (4) 22, 23–51 (1981); English transl. in Siberian Math. J. 22 (1981)
A. M. Blokhin, Uniqueness of classical solution of mixed problem for equations of gas dynamics with boundary conditions on a shock wave, Sibirsk. Mat. Zh. (5) 23, 17–30 (1982); English transl. in Siberian Math. J. 23 (1982)
A. M. Blokhin, Energy integrals and their applications in problems of gas dynamics, Nauka, Novosibirsk, 1986 (in Russian)
A. M. Blokhin, Strong Discontinuities in Magnetohydrodynamics, Nova Science Publishers, Inc., Commack, NY, 1994
A. M. Blokhin and E. V. Mishchenko, Investigation on shock waves stability in relativistic gas dynamics, Matematiche (Catania) 48, 53–75 (1993)
A. M. Blokhin and Yu. L. Trakhinin, Investigation of the well-posedness of the mixed problem on the stability of a fast shock wave in magnetohydrodynamics, Matematiche (Catania) 49, 123–141 (1994)
A. M. Blokhin, Symmetrization of continuum mechanics equations, Siberian J. Differential Equations 2, 3–47 (1995)
A. M. Blokhin, V. Romano, and Yu. L. Trakhinin, Some mathematical properties of radiating gas model obtained with a variable Eddington factor, Z. Angew. Math. Phys. (ZAMP) 47, 639–658 (1996)
G. Boillat, Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systèmes hyperboliques, Comptes Rendues de l’Academie des Sciences Paris Sér. A 278, 909–912 (1974)
K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proceedings of the National Academy of Sciences, U.S.A. 68, 1686–1688 (1971)
K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 31, 123–131 (1974)
C. S. Gardner and M. D. Kruskal, Stability of plane magnetohydrodynamic shocks, Phys. Fluids 7, 700–706 (1964)
S. K. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR 39, 521–523 (1961)
S. K. Godunov, Symmetrization of magnetohydrodynamics equations, Chislennye Metody Mekhaniki Sploshnoi Sredy 3, 26 34 (1972) (in Russian)
A. Jeffrey, Quasilinear Hyperbolic Systems and Waves, Pitman, New York, 1976
T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal. 58, 181–205 (1975)
H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure and Appl. Math. 23, 277–296 (1970)
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, New York and Oxford, 1997
P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM Reg. Conf. Lecture, No. 11, Philadelphia, 1973
A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics, Benjamin, New York, 1967
A. Lichnerowicz, Relativistic Fluid Dynamics, Cremonese, Rome, 1971
A. Lichnerowicz, Shock waves in relativistic magnetohydrodynamics under general assumptions, J. Math. Phys. 17, 2135–2141 (1975)
A. Majda, The stability of multi-dimensional shock fronts—a new problem for linear hyperbolic equations, Mem. Amer. Math. Soc. 41, No. 275, Providence, RI, 1983
A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 43, No. 281, Providence, RI, 1983
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984
T. Ruggeri and A. Strumia, Convex covariant entropy density, symmetric conservative form, and shock waves in relativistic magnetohydrodynamics, J. Math. Phys. 22, 1824–1827 (1981)
T. Ruggeri and A. Strumia, Main field and convex covariant density for quasilinear hyperbolic systems, Ann. Inst. Henri Poincaré Sect. A (N.S.) 34, 65–84 (1981)
G. Russo and A. M. Anile, Stability properties of relativistic shock waves: Basic results, Phys. Fluids 30, 2406–2413 (1987)
S. L. Sobolev, Some applications of functional analysis to the problems of mathematical physics, Publishing House of Leningrad State University, Leningrad, 1950 (in Russian)
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© Copyright 2001
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