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Multivalued solutions to some nonlinear and non-strictly hyperbolic systems. (English) Zbl 0768.35058
Summary: We introduce and study a general notion of multivalued solution for nonlinear hyperbolic systems, which need not to be strictly hyperbolic and in conservative form. Then we focus our attention on the system of conservation laws of fluid mechanics with constant pressure which is used in plasma physics. This system is in conservative form but not strictly hyperbolic, and is not solvable in the setting of measurable and bounded single valued functions. However we prove that multivalued solutions in the above sense can be found for this system. Moreover, these solutions are the physically meaningful solutions of the problem.

MSC:
35L45 Initial value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
35D05 Existence of generalized solutions of PDE (MSC2000)
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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[1] [AK] J. Albritton and P. Koch, Cold plasma wave breaking: production of energetic electrons. Phys. Fluids,18 (1975), 1136–1139. · doi:10.1063/1.861300
[2] [Au] J. Audounet, Solutions discontinues paramétriques des systèmes de lois de conservation et des problèmes aux limites associés. Séminaire d’Analyse Numérique, Toulouse, France, 1984–1985.
[3] [Be] L. Ben-Aim, Applications des méthodes particulaires en mécanique des fluides et en physique des plasmas. Thèse de Doctorat, Univ. Paris VI, France, 1988.
[4] [Br] Y. Brenier, Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal.,21 (1984), 1013–1037. · Zbl 0565.65054 · doi:10.1137/0721063
[5] [DLM] G. Dal Maso, Ph. Le Floch and F. Murat, Definition and weak stability of a nonconservative product. Internal Report, CMAP, Ecole Polytechnique, Palaiseau, France (to be submitted).
[6] [Da] J.M. Dawson, Nonlinear electron oscillations in a cold plasma. Phys. Rev., Second Series,113 (1959), 383–388. · Zbl 0098.22904
[7] [DL] R. Dipernar and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. To appear.
[8] [Du] F. Dubois, Boundary conditions and the Osher scheme for the Euler equations of gas dynamics. Internal Report no170, Ecole Polytechnique, CMAP, Palaiseau, France, 1988.
[9] [Gl] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of conservation laws. Comm. Pure Appl. Math.,18 (1965), 677–697. · Zbl 0141.28902 · doi:10.1002/cpa.3160180408
[10] [HL] T. Hou and P. Le Floch, Error estimate for difference schemes in nonconservative form. Submitted to Math. Comp.
[11] [IMP] E. Isaacson, D. Marchesin and B. Plohr, Transitional waves for conservation laws. Preprint, Center for the Mathematical Sciences, Univ. Wisconsin, USA, 1988. · Zbl 0707.35088
[12] [Ka] T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems. Arch. Rational Mech. Anal.,58 (1975), 181–205. · Zbl 0343.35056 · doi:10.1007/BF00280740
[13] [KK 1] B. Keyfitz and H.C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Ratinal Mech. Anal.,72 (1980), 219–241. · Zbl 0434.73019 · doi:10.1007/BF00281590
[14] [KK 2] B.L. Keyfitz and H.C. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution. Nonlinear Hyperbolic Problems, Bordeaux, France, June 13–17, 1988, Lecture Notes in Math. 1402, 1990, 185–197.
[15] [Ko] D.J. Korchinski, Solution of a Riemann problem for a 2\(\times\)2 system of conservation laws possessing no classical weak solution. Ph. D. Thesis, Adelphi Univ., 1977.
[16] [Kr] N. Kruskov, First order quasilinear equations in several independent variables. Math. USSR-Sb.,10 (1970), 127–243. · Zbl 0215.07901 · doi:10.1070/SM1970v010n01ABEH001590
[17] [La] P. Lax, Hyperbolic systems of Conservation Laws and the Mathematical Theory of Shock Waves. CMBS Monographs no 11, SIAM, 1973. · Zbl 0268.35062
[18] [Le 1] Ph. Le Floch, Entropy weak solutions to nonlinear hyperbolic system in nonconservative form. Comm. Partial Differential Equations,13 (1988), 669–727. · Zbl 0683.35049 · doi:10.1080/03605308808820557
[19] [Le 2] Ph. Le Floch, Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form. Proc. of the Second Intern. Conference on Nonlinear Hyperbolic Problems, Aachen FRG, 1988, pp. 362–373.
[20] [Le 3] Ph. Le Floch, Shock waves of nonlinear hyperbolic systems in nonconservative form. IMA Preprint Series #513, Univ. Minnesota, Minneapolis.
[21] [Le 4] Ph. Le Floch, An existence and uniqueness result for two nonstrictly hyperbolic systems. Proc. of the IMA Workshop on Nonlinear Evolution Equations that Change Type, Univ. Minnesota, USA, 1989.
[22] [LL] P. Le Floch and T.-P. Liu, Existence theory for nonlinear hyperbolic systems in non conservative form. To appear in Forum Mathematica, 1992.
[23] [LX] Ph. Le Floch and Z. Xin, Uniqueness via the adjoint problem for a class of systems of conservation laws. Submitted to Comm. Pure Appl. Math.
[24] [Li] T.P. Liu, Admissible Solutions of Hyperbolic Conservation Laws. Mem. Amer. Math. Soc., vol. 30, no 240, 1981. · Zbl 0446.76058
[25] [Os] S. Osher, Riemann solvers, the entropy condition and difference approximations. SIAM J. Numer. Anal.,21 (1984), 217–235. · Zbl 0592.65069 · doi:10.1137/0721016
[26] [OS] S. Osher and F. Solomon, Upwind difference schemes for hyperbolic systems of conservation laws. Math. Comp.,38 no 158 (1982), 339–374. · Zbl 0483.65055 · doi:10.1090/S0025-5718-1982-0645656-0
[27] [SS] Ch. Sack and H. Schamel, Nonlinear dynamics in expanding plasmas. Phys. Lett.,110A (1985), 206–212.
[28] [Se] D. Serre, Les ondes planes en électromagnétisme non linéaire. Phys. D,31 (1988), 227–251. · Zbl 0736.35133 · doi:10.1016/0167-2789(88)90078-4
[29] [Sm 1] K.T. Smith, Primer of Modern Analysis, Undergraduate Course in Mathematics. Springer Verlag, New York, 1983.
[30] [Sm 2] J. Smoller, Shock Waves and Reaction-Diffusion Equations. Springer Verlag, New York, 1983. · Zbl 0508.35002
[31] [Vo] A.I. Volpert, The spaceBV and quasilinear equations. Mat. Sb.,73(115) (1967), 225–267. · Zbl 0168.07402 · doi:10.1070/SM1967v002n02ABEH002340
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