Relative Entropy for Hyperbolic-Parabolic Systems and Application to the Constitutive Theory of Thermoviscoelasticity

We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful for providing stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an L (p) framework. It also provides a weak-strong uniqueness theorem for measure valued solutions of the hyperbolic problem. In the second part, the relative entropy identity is developed for the systems of gas dynamics for viscous and heat conducting gases and for the system of thermoviscoelasticity both including viscosity and heat-conduction effects. The dissipation mechanisms and the concentration measures play different roles when applying the method to the general class of hyperbolic-parabolic systems and to the specific examples, and their ramifications are highlighted.