Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form

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Abstract
For nonlinear hyperbolic systems under nonconservative form (∗) we consider weak solutions in the class of bounded functions of bounded variation in the sense of Volpert. A generalized global entropy inequality for selecting the physically relevant solutions of (∗) is proposed and studied. It is shown to be equivalent for weak shocks to the Lax admissibility criterion which, as we note here, make sense for (∗). In this framework, we solve the Riemann problem with small initial data. Hence, we get a generalization of a classical result due to P. Lax for systems of conservation laws. Using the random—choice method introduced by J. Glimm for conservation laws, we construct approximate solutions of the Cauchy problem for (∗), which are uniformly bounded in norms L∞ and BV. Then, we prove the consistancy of the method with both the nonconservative system (∗) and our generalized entropy inequality. Our theory is applied to nonconservative systems issued from Elasticity and Hydrodynamics.

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