Existence results for diffuse interface models describing phase separation and damage

Abstract

In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫ Ω ½Γ∇ c : ∇ c + ½|∇ z | 2 + W ch ( c )+ W el ( e,c,z )dx with a polynomial or logarithmic chemical energy density W ch , an inhomogeneous elastic energy density W el and a quadratic structure of the gradient of damage variable z . For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e .