Upper-semicontinuity of the global attractors for a class of nonlocal Cahn-Hilliard equations

TL;DR

This paper studies how the global attractors of a class of nonlocal Cahn-Hilliard equations behave under small perturbations, proving upper-semicontinuity and explicit estimates for the convergence of solutions.

Contribution

It establishes the upper-semicontinuity of global attractors for nonlocal Cahn-Hilliard equations and provides explicit bounds on the convergence of trajectories as parameters vanish.

Findings

Global attractors are upper-semicontinuous with respect to perturbation parameters.

Explicit estimates relate the difference between relaxation and limit solutions.

Results hold under suitable assumptions on the equations.

Abstract

The aim of this work is to examine the upper-semicontinuity properties of the family of global attractors admitted by a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. We prove that the family of global attractors is upper-semicontinuous as the perturbation parameters vanish. Additionally, under suitable assumptions, we prove that the family of global attractors satisfies a further upper-semicontinuity type estimate whereby the difference between trajectories of the relaxation problem and the limit isothermal non-viscous problem is explicitly controlled, in the topology of the relaxation problem, in terms of the relaxation parameters.