Dynamics of a conserved phase-field system

Abstract

Recently, in Bonfoh [Ann. Mat. Pura Appl. 2011;190:105–144], we investigated the dynamics of a nonconserved phase-field system whose singular limit is the viscous Cahn–Hilliard equation. More precisely, we proved the existence of the global attractor, exponential attractors, and inertial manifolds and we showed their continuity with respect to a singular perturbation parameter. In the present paper, we extend most of these results to a conserved phase-field system whose singular limit is the nonviscous Cahn–Hilliard equation. These equations describe phase transition processes. Here, we give a direct proof of the existence of inertial manifolds that differs from our previous method that was based on introducing a change of variables and an auxiliary problem.

Keywords:

• phase-field equations
• singular perturbation
• global attractors
• exponential attractors
• inertial manifolds

AMS Subject Classifications:

• 35B25
• 35B45
• 37L25
• 82C26

Acknowledgements

The author is grateful to the referees for their helpful and careful reading of previous versions of this manuscript.

Notes

The author would like to acknowledge the support provided by the DSR at KFUPM for funding this work through project No. FT121008.