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For a class of nonconservative hyperbolic systems of partial differentialequations endowed with a strictly convex mathematical entropy, we formulate theinitial value problem by supplementing the equations with a kinetic relationprescribing the rate of entropy dissipation across shock waves. Our conditioncan be regarded as a generalization to nonconservative systems of a similarconcept introduced by Abeyaratne, Knowles, and Truskinovsky for subsonic phasetransitions and by LeFloch for nonclassical undercompressive shocks tononlinear hyperbolic systems. The proposed kinetic relation for nonconservativesystems turns out to be equivalent, for the class of systems underconsideration at least, to Dal Maso, LeFloch, and Murat's definition based on aprescribed family of Lipschitz continuous paths. In agreement with previoustheories, the kinetic relation should be derived from a phase plane analysis oftraveling solutions associated with an augmented version of the nonconservativesystem. We illustrate with several examples that nonconservative systemsarising in the applications fit in our framework, and for a typical model ofturbulent fluid dynamics, we provide a detailed analysis of the existence andproperties of traveling waves which yields the corresponding kinetic function.

Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems