An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity

Abstract

This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter rho and the chemical potential mu; each equation includes a viscosity term -- respectively, varepsilon,partialtmu and delta,partialtrho -- with varepsilon and delta two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its (varepsilon,delta)−solutions. Here we discuss the asymptotic limit of the system as eps tends to 0. We prove convergence of (varepsilon,delta)−solutions to the corresponding solutions for the case eps =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.