Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations

TL;DR

This paper investigates the convergence of solutions from a nonlocal anisotropic Cahn-Hilliard equation to its local counterpart, establishing existence and convergence results for singular and anisotropic kernels.

Contribution

It proves the convergence of weak solutions from nonlocal to local anisotropic Cahn-Hilliard equations and establishes existence results under singular kernel conditions.

Findings

Weak solutions of nonlocal equations converge to local solutions

Existence of weak solutions is proven for singular and anisotropic kernels

Results apply to a broad class of kernels including localized and singular types

Abstract

We study the nonlocal-to-local convergence for a nonlocal Cahn-Hilliard equation with anisotropic and singular kernels. In particular, we show convergence of weak solutions of the nonlocal Cahn-Hilliard equation to weak solutions of a corresponding anisotropic Cahn-Hilliard equation for suitable subsequences. Moreover, we show existence of weak solutions for the nonlocal equation under a condition, which guarantees existence of weak solutions for suitably localized or singular kernels.