A degenerating Cahn-Hilliard system coupled with complete damage processes

TL;DR

This paper develops a novel analytical framework to prove the existence of weak solutions for a complex, degenerating coupled PDE system modeling phase separation and damage processes on evolving domains.

Contribution

It introduces a new approach for analyzing highly nonlinear, degenerating PDE systems with nonsmooth domains and coupled damage and phase separation processes.

Findings

Established a suitable notion of weak solutions for the system

Proved existence of weak solutions using new analytical techniques

Handled highly nonsmooth regions with complete damage and shrinking domains

Abstract

In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a nonsmooth time-dependent domain with mixed boundary conditions. The evolution of the system is described by a \textit{degenerating} Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a \textit{degenerating} quasi-static balance equation for the displacement field. All these equations are highly nonlinearly coupled. Because of the doubly degenerating character of the system, the doubly nonlinear differential inclusion and the nonsmooth domain, the structure of the model is very complex from an analytical point of view. A novel approach is introduced for proving existence of weak solutions for such degenerating coupled system. To this end, we first establish a suitable notion of weak solutions, which consists of weak formulations of the diffusion and the momentum balance equation, a variational inequality for the damage process and a total energy inequality. To show existence of weak solutions, several new ideas come into play. Various results on shrinking sets and its corresponding local Sobolev spaces are used. It turns out that, for instance, on open sets which shrink in time a quite satisfying analysis in Sobolev spaces is possible. The presented analysis can handle highly nonsmooth regions where complete damage takes place. To mention only one difficulty, infinitely many completely damaged regions which are not connected with the Dirichlet boundary may occur in arbitrary small time intervals.