Global weak solutions in nonlinear 3D thermoelasticity

TL;DR

This paper proves the first global existence of weak solutions for large initial data in a thermodynamically consistent 3D nonlinear thermoelasticity model, ensuring positive temperature and entropy conditions.

Contribution

It introduces a novel weak solution concept with defect measure for a thermodynamically consistent 3D thermoelasticity system, establishing global existence and uniqueness results.

Findings

Global weak solutions exist for large initial data.

Solutions satisfy entropy inequality and maintain positive temperature.

Results hold for both constant and non-constant heat capacities and conductivities.

Abstract

Here we study a nonlinear thermoelasticity hyperbolic-parabolic system describing the balance of momentum and internal energy of a heat-conducting elastic body, preserving the positivity of temperature. So far, no global existence results in such a natural case were available. Our result is obtained by using thermodynamically justified variables which allow us to obtain an equivalent system in which the internal energy balance is replaced with entropy balance. For this system, a concept of weak solution with defect measure is introduced, which satisfies entropy inequality instead of balance and has a positive temperature almost everywhere. Then, the global existence, consistency and weak-strong uniqueness are shown in the cases where heat capacity and heat conductivity are both either constant or non-constant. Let us point out that this is the first result concerning global existence for large initial data in nonlinear thermoelasticity where the model is in full accordance with the laws of thermodynamics.