A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness

TL;DR

This paper extends polyconvex thermoelasticity to a symmetrizable hyperbolic system, proving convergence in zero-viscosity limits and establishing weak-strong uniqueness for entropy solutions.

Contribution

It introduces a symmetrizable extension of polyconvex thermoelasticity and applies relative entropy methods to analyze zero-viscosity limits and weak-strong uniqueness.

Findings

Proves convergence from thermoviscoelasticity to thermoelasticity as viscosity and heat conduction vanish.

Establishes weak-strong uniqueness for entropy weak solutions of adiabatic thermoelasticity.

Develops a relative entropy framework for the extended system.

Abstract

We embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hyperbolic system and derive a relative entropy identity in the extended variables. Following the relative entropy formulation, we prove the convergence from thermoviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth solutions of the system of adiabatic thermoelasticity as both parameters tend to zero. Also, convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. Finally, we establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions.