Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

TL;DR

This paper proves the existence of weak solutions for a complex PDE system modeling phase separation and damage in stressed alloys, incorporating inertial effects, using advanced mathematical techniques.

Contribution

It establishes the first existence results for a hyperbolic-parabolic PDE system modeling phase separation and damage with inertial effects.

Findings

Existence of weak solutions proven for the PDE system.

Application of regularization and time-discretization methods.

Extension of previous models to include inertial effects.

Abstract

In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.