Longtime behavior of nonlocal Cahn-Hilliard equations

TL;DR

This paper studies the long-term behavior of nonlocal Cahn-Hilliard equations, proving the existence of exponential attractors and analyzing convergence to stationary states under various potential conditions.

Contribution

It establishes the existence of exponential attractors for nonlocal Cahn-Hilliard equations with regular and singular potentials, extending previous results on their long-term dynamics.

Findings

Existence of exponential attractors for regular potentials.

Uniform boundedness of the order parameter via Alikakos-Moser argument.

Convergence of solutions to stationary states under certain conditions.

Abstract

Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well.