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Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace

Author: Guy Métivier
Journal: Trans. Amer. Math. Soc. 296 (1986), 431-479
MSC: Primary 35L65; Secondary 76L05
MathSciNet review: 846593
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Abstract: The existence of shock front solutions to a system of conservation laws in several space variables has been proved by A. Majda, solving a Cauchy problem, with a suitable discontinuous Cauchy data. But, in general, the solution to such a Cauchy problem will present $ N$ singularities, $ N$ being the number of laws. In this paper we solve (locally) this Cauchy problem, with a Cauchy data which is piecewise smooth, in the case where all the singularities are expected to be shock waves. Actually the construction is written for a system of two laws, with two space variables and similarly, for such a system, the same method enables us to study the interaction of two shock waves. The key point, in the construction below, is the study of a nonlinear, free boundary Goursat problem.

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  • [1] S. Alinhac, Le problème de Goursat hyperbolique en dimension deux, Comm. Partial Differential Equations 1 (1976), no. 3, 231–282 (French). MR 0415082
  • [2] Jean-Michel Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209–246 (French). MR 631751
  • [3] Jacques Chazarain and Alain Piriou, Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier-Villars, Paris, 1981 (French). MR 598467
  • [4] Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 0437941
  • [5] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653
  • [6] Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603–634. MR 0393870
  • [7] Andrew Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), no. 275, iv+95. MR 683422, 10.1090/memo/0275
  • [8] -, The existence of multi dimensional shock fronts, Amer. Math. Soc. No. 281 (1983).
  • [9] Yves Meyer, Remarques sur un théorème de J.-M. Bony, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 1–20 (French). MR 639462
  • [10] -, Nouvelles conditions pour les solutions d'équations aux dérivées partielles non linéaires, Séminaire Goulaouic-Schwartz, École Polytechnique, année 1981-1982.

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Article copyright: © Copyright 1986 American Mathematical Society