On the separation property and the global attractor for the nonlocal Cahn-Hilliard equation in three dimensions

TL;DR

This paper proves that solutions to the 3D nonlocal Cahn-Hilliard equation with singular potential become uniformly separated from pure phases over time, and uses this to enhance the understanding of the system's long-term behavior.

Contribution

It establishes the instantaneous and uniform separation property for solutions, leading to improved regularity results for the global attractor of the system.

Findings

Solutions are confined within a strict interval after any positive time.

The separation property holds uniformly for all solutions with finite initial energy.

The regularity of the global attractor is improved based on the separation results.

Abstract

In this note, we consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. Given any global solution (whose existence and uniqueness are already known), we prove the so-called {\it instantaneous} and {\it uniform} separation property: any global solution with initial finite energy is globally confined (in the $L^\infty$ metric) in the interval $[-1+\delta,1-\delta]$ on the time interval $[\tau,\infty)$ for any $\tau>0$, where $\delta$ only depends on the norms of the initial datum, $\tau$ and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.