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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.

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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