Global solutions to a generalized Cahn-Hilliard equation with viscosity

Abstract

We address a viscous Cahn-Hilliard equation describing the phase separation process in a binary alloy. Such a material is subject to the influence of internal and external mechanical stresses, whose contribution is assumed to be known. The physical modelling refers to a recent work by Dreyer and Müller. The main features and difficulties of this model are given by a highly nonlinear fourth order elliptic term, a strong constraint imposed by the presence of a double obstacle energy potential, and the dependence of the mobility matrix on the concentration variable (i.e., the unknown of the problem). We are able to prove global existence of solutions of the corresponding initial boundary value problem by using approximation and compactness tools. Hence, we perform some asymptotic analysis of the problem and obtain, at the limit, two models of independent physical interest.