Relative entropy for hyperbolic-parabolic systems and application to the constitutive theory of thermoviscoelasticity

TL;DR

This paper extends the relative entropy method to hyperbolic-parabolic systems, enabling stability analysis, convergence in the zero-viscosity limit, and uniqueness results, with applications to thermoviscoelasticity and gas dynamics.

Contribution

It develops a generalized relative entropy identity for symmetrizable hyperbolic-parabolic systems, including thermoviscoelasticity, and demonstrates its use in stability, convergence, and uniqueness theorems.

Findings

Established stability of viscous solutions.

Proved convergence to smooth solutions in the zero-viscosity limit.

Demonstrated measure-valued versus strong uniqueness for hyperbolic problems.

Abstract

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 to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an $L^p$ framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosity and heat-conduction. Existing differences in applying the relative entropy method between the general hyperbolic-parabolic theory and the examples are underlined.