Existence and weak-strong uniqueness for damage systems in viscoelasticity

TL;DR

This paper establishes existence and weak-strong uniqueness of solutions for a damage model in viscoelasticity, using different solution concepts and energy inequalities to analyze the PDE system.

Contribution

It introduces a framework for proving global existence of weak solutions and local existence of strong solutions, along with a weak-strong uniqueness result in viscoelastic damage models.

Findings

Global-in-time existence of weak solutions

Local-in-time existence of strong solutions

Weak-strong uniqueness via relative energy inequality

Abstract

In this paper we investigate the existence of solutions and their weak-strong uniqueness property for a PDE system modelling damage in viscoelastic materials. In fact, we address two solution concepts, weak and strong solutions. For the former, we obtain a global-in-time existence result, but the highly nonlinear character of the system prevents us from proving their uniqueness. For the latter, we prove local-in-time existence. Then, we show that the strong solution, as long as it exists, is unique in the class of weak solutions. This weak-strong uniqueness statement is proved by means of a suitable relative energy inequality.