Strong Nonlocal-to-Local Convergence of the Cahn-Hilliard Equation and its Operator

TL;DR

This paper proves that solutions of the nonlocal Cahn-Hilliard equation converge to the local version in smooth domains, establishing strong operator convergence with a rate, using the relative entropy method.

Contribution

It demonstrates the strong convergence of nonlocal to local Cahn-Hilliard solutions and operators, providing a rigorous mathematical foundation for the nonlocal-to-local transition.

Findings

Weak solutions of nonlocal Cahn-Hilliard converge to local solutions

Nonlocal operator converges strongly to the Laplacian with a quantifiable rate

Results are valid in smooth bounded domains with Neumann boundary conditions

Abstract

We prove convergence of a sequence of weak solutions of the nonlocal Cahn-Hilliard equation to the strong solution of the corresponding local Cahn-Hilliard equation. The analysis is done in the case of sufficiently smooth bounded domains with Neumann boundary condition and a $W^{1,1}$-kernel. The proof is based on the relative entropy method. Additionally, we prove the strong $L^2$-convergence of the nonlocal operator to the negative Laplacian together with a rate of convergence.