WIAS Preprint No. 1652, (2011)

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

Authors

• Colli, Pierluigi
• Gilardi, Gianni
• Podio-Guidugli, Paolo
• Sprekels, Jürgen

2010 Mathematics Subject Classification

• 74A15 35K55 35A05 35B40

Keywords

• viscous Cahn-Hilliard system, phase field model, asymptotic limit, existence of solutions

DOI

10.20347/WIAS.PREPRINT.1652

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, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- 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.

Appeared in

• Discrete Contin. Dyn. Syst. Ser. S, 6 (2013) pp. 353--368.