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

Recently, in Bonfoh and Enyi [Commun. Pure Appl. Anal. 15 2016, 1077-1105], we considered the conserved phasefield system 'Equation presented' in a bounded domain of R^d, d = 1, 2, 3, where τ < 0 is a relaxation time, δ < 0 is the viscosity parameter, ϵ Ε (0, 1] is the heat capacity, φ is the order parameter, u is the absolute temperature and g : R → 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 ϵ = 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 (ϵ, δ) = (0, 0). 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.