  For a class of nonconservative hyperbolic systems of partial differential
equations endowed with a strictly convex mathematical entropy, we formulate the
initial value problem by supplementing the equations with a kinetic relation
prescribing the rate of entropy dissipation across shock waves. Our condition
can be regarded as a generalization to nonconservative systems of a similar
concept introduced by Abeyaratne, Knowles, and Truskinovsky for subsonic phase
transitions and by LeFloch for nonclassical undercompressive shocks to
nonlinear hyperbolic systems. The proposed kinetic relation for nonconservative
systems turns out to be equivalent, for the class of systems under
consideration at least, to Dal Maso, LeFloch, and Murat's definition based on a
prescribed family of Lipschitz continuous paths. In agreement with previous
theories, the kinetic relation should be derived from a phase plane analysis of
traveling solutions associated with an augmented version of the nonconservative
system. We illustrate with several examples that nonconservative systems
arising in the applications fit in our framework, and for a typical model of
turbulent fluid dynamics, we provide a detailed analysis of the existence and
properties of traveling waves which yields the corresponding kinetic function.
