The Cahn–Hilliard equation as limit of a conserved phase-field system

Abstract

Recently, in Bonfoh and Enyi [Commun. Pure Appl. Anal. 15 2016, 1077–1105], we considered the conserved phase-field system $$\begin{array}{c}\left\{\begin{array}{c}\mathit{\tau }{\mathit{\varphi }}_{\mathit{t}}-\mathrm{\Delta }\mathrm{(}\mathit{\delta }{\mathit{\varphi }}_{\mathit{t}}-\mathrm{\Delta }\mathit{\varphi }+\mathit{g}\mathrm{(}\mathit{\varphi }\mathrm{)}-\mathit{u}\mathrm{)}=0\mathrm{,}\hfill \\ \mathit{\epsilon }{\mathit{u}}_{\mathit{t}}+{\mathit{\varphi }}_{\mathit{t}}-\mathrm{\Delta }\mathit{u}=0\mathrm{,}\hfill \end{array}\right.\end{array}$$ in a bounded domain of ${\mathbb{R}}^{\mathit{d}}$ , $\mathit{d}=1\mathrm{,}2\mathrm{,}3$ , where $\mathit{\tau }\mathrm{>}0$ is a relaxation time, $\mathit{\delta }\mathrm{>}0$ is the viscosity parameter, $\mathit{\epsilon }\in \mathrm{(}0\mathrm{,}1]$ is the heat capacity, ϕ is the order parameter, u is the absolute temperature and $\mathit{g}:\mathbb{R}\to \mathbb{R}$ is a nonlinear function. The system is subject to the boundary conditions of either periodic or Neumann type. We proved a well-posedness result, the existence and continuity of the global and exponential attractors at $\mathit{\epsilon }=0$ . Then, we proved the existence of inertial manifolds in one space dimension, and in the case of two space dimensions in rectangular domains. Stability properties of the intersection of inertial manifolds with a bounded absorbing set were also proven. In the present paper, we show the above-mentioned existence and continuity properties at $\mathrm{(}\mathit{\epsilon }\mathrm{,}\mathit{\delta }\mathrm{)}=\mathrm{(}0\mathrm{,}0\mathrm{)}$ . To establish the existence of inertial manifolds of dimension independent of the two parameters δ and ε, we require ε to be dominated from above by δ. This work shows the convergence of the dynamics of the above mentioned problem to the one of the Cahn–Hilliard equation, improving and extending some previous results.