On the limit problem arising in the kinetic derivation of the Cahn-Hilliard equation

TL;DR

This paper investigates the local limit of a non-local degenerate Cahn-Hilliard equation derived from the Vlasov equation, introducing a new kernel condition to handle degeneracy and establish compactness.

Contribution

It introduces a novel kernel condition that enables the analysis of the local limit in degenerate non-local Cahn-Hilliard equations, applicable to various systems.

Findings

Established a new condition on the nonlocal kernel.

Proved the local limit as the delocalization parameter tends to zero.

Applicable to a broad class of kernels in related models.

Abstract

The non-local degenerate Cahn-Hilliard equation is derived from the Vlasov equation with long-range attraction. We study the local limit as the delocalization parameter converges to 0. The difficulty arises from the degeneracy which requires compactness estimates, but all necessary a priori estimates can be obtained only on the nonlocal quantities yielding almost no information on the limiting solution itself. We introduce a novel condition on the nonlocal kernel which allows us to exploit the available nonlocal a priori estimates. The condition, satisfied by most of the kernels appearing in the applications, can be of independent interest. Our approach is flexible and systems can be treated as well.