Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d’espace
HTML articles powered by AMS MathViewer
- by Guy Métivier
- Trans. Amer. Math. Soc. 296 (1986), 431-479
- DOI: https://doi.org/10.1090/S0002-9947-1986-0846593-9
- PDF | Request permission
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.References
- S. Alinhac, Le problème de Goursat hyperbolique en dimension deux, Comm. Partial Differential Equations 1 (1976), no. 3, 231–282 (French). MR 415082, DOI 10.1080/03605307608820011
- 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, DOI 10.24033/asens.1404
- Jacques Chazarain and Alain Piriou, Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier-Villars, Paris, 1981 (French). MR 598467
- Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 437941, DOI 10.1002/cpa.3160230304
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Publ. Math. Res. Center Univ. Wisconsin, No. 27, Academic Press, New York, 1971, pp. 603–634. MR 0393870
- Andrew Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), no. 275, iv+95. MR 683422, DOI 10.1090/memo/0275 —, The existence of multi dimensional shock fronts, Amer. Math. Soc. No. 281 (1983).
- 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 —, Nouvelles conditions pour les solutions d’équations aux dérivées partielles non linéaires, Séminaire Goulaouic-Schwartz, École Polytechnique, année 1981-1982.
Bibliographic Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 296 (1986), 431-479
- MSC: Primary 35L65; Secondary 76L05
- DOI: https://doi.org/10.1090/S0002-9947-1986-0846593-9
- MathSciNet review: 846593