Existence and energy estimates of weak solutions for nonlocal Cahn--Hilliard equations on unbounded domains

TL;DR

This paper proves the existence and energy bounds of weak solutions for a nonlocal Cahn--Hilliard equation on unbounded domains, extending previous results from bounded to unbounded settings where compactness issues arise.

Contribution

It introduces new methods to establish existence and energy estimates for weak solutions of the nonlocal Cahn--Hilliard equation on unbounded domains, overcoming the lack of compactness.

Findings

Existence of weak solutions on unbounded domains.

Energy estimates for the solutions.

Extension of previous bounded domain results.

Abstract

This paper considers the initial-boundary value problem for the nonlocal Cahn--Hilliard equation $$ \partial_t\varphi + (-\Delta+1)(a(\cdot)\varphi -J\ast\varphi + G'(\varphi)) = 0 \quad \mbox{in}\ \Omega\times(0, T) $$ in an unbounded domain $\Omega \subset \mathbb{R}^N$ with smooth bounded boundary, where $N\in\mathbb{N}$, $T>0$, and $a(\cdot), J, G$ are given functions. In the case that $\Omega$ is a bounded domain and $-\Delta+1$ is replaced with $-\Delta$, this problem has been studied by using a Faedo--Galerkin approximation scheme considering the compactness of the Neumann operator $-\Delta+1$ (cf. Colli--Frigeri--Grasselli (2012), Gal--Grasselli (2014)). However, the compactness of the Neumann operator $-\Delta+1$ breaks down when $\Omega$ is an unbounded domain. The present work establishes existence and energy estimates of weak solutions for the above problem on an unbounded domain.