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

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.

54 pages, no figures

References in corpus (2)

• Weak-strong uniqueness for measure-valued solutions of some compressible fluid models
• Relative entropy in hyperbolic relaxation for balance laws

Cited by in corpus (5)

• Stability of Multidimensional Thermoelastic Contact Discontinuities
• Hyperbolic-parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion
• Convergence of a finite volume scheme and dissipative measure-valued-strong stability for a hyperbolic-parabolic cross-diffusion system
• The Relative Entropy Method for Inhomogeneous Systems of Balance Laws
• Mv-strong uniqueness for density dependent, non-Newtonian, incompressible fluids