Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity

TL;DR

This paper introduces a framework for measure-valued solutions to polyconvex adiabatic thermoelasticity equations, establishing weak-strong uniqueness via relative entropy without relying on symmetric variables.

Contribution

It defines dissipative measure-valued solutions for the system and proves their uniqueness compared to classical solutions using an averaged relative entropy inequality.

Findings

Established measure-valued weak-strong uniqueness

Embedded the system into a symmetrizable hyperbolic form

Derived relative entropy in original variables

Abstract

For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. However, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure- valued weak versus strong uniqueness using the averaged relative entropy inequality.