A Hilbert expansions method for the rigorous sharp interface limit of the generalized Cahn-Hilliard Equation

TL;DR

This paper develops a Hilbert expansion method to rigorously analyze the sharp interface limit of generalized Cahn-Hilliard equations with external forcing, extending previous techniques to more complex scenarios.

Contribution

It introduces a Hilbert expansion approach for generalized Cahn-Hilliard equations, enabling rigorous derivation of sharp interface limits even with external forcing.

Findings

Constructed approximate solutions using Hilbert expansions.

Proved convergence of solutions to a moving boundary problem.

Provided spectral estimates for solution differences.

Abstract

We consider Cahn-Hilliard equations with external forcing terms. Energy decreasing and mass conservation might not hold. We show that level surfaces of the solutions of such generalized Cahn-Hilliard equations tend to the solutions of a moving boundary problem under the assumption that classical solutions of the latter exist. Our strategy is to construct approximate solutions of the generalized Cahn-Hilliard equation by the Hilbert expansion method used in kinetic theory and proposed for the standard Cahn-Hilliard equation, by Carlen, Carvalho and Orlandi, \cite {CCO}. The constructed approximate solutions allow to derive rigorously the sharp interface limit of the generalized Cahn-Hilliard equations. We then estimate the difference between the true solutions and the approximate solutions by spectral analysis, as in \cite {A-B-C}