Thin-film limit of the Cahn-Hilliard equation in a curved thin domain

TL;DR

This paper rigorously derives the surface Cahn-Hilliard equation as the thin-film limit of the three-dimensional Cahn-Hilliard equation in curved thin domains, providing new insights into phase separation on curved surfaces.

Contribution

It establishes the first rigorous thin-film limit of the nonlinear fourth-order Cahn-Hilliard equation in curved thin domains, including detailed analysis and convergence results.

Findings

Weighted average of solutions converges to a limit on the surface.

Derived the surface Cahn-Hilliard equation with weighted Laplacian.

Provided explicit estimates depending on the domain's thickness.

Abstract

We consider the Cahn-Hilliard equation with Neumann boundary conditions in a three-dimensional curved thin domain around a given closed surface. When the thickness of the curved thin domain tends to zero, we show that the weighted average in the thin direction of a weak solution to the thin-domain problem converges on the limit surface in an appropriate sense. Moreover, we rigorously derive a limit problem, which is the surface Cahn-Hilliard equation with weighted Laplacian, by characterizing the limit function as a unique weak solution to the limit problem. The proof is based on a detailed analysis of the weighted average and the use of Sobolev inequalities and elliptic regularity estimates on the curved thin domain with constants explicitly depending on the thickness. This is the first result on a rigorous thin-film limit of nonlinear fourth order equations in general curved thin domains.