The Sharp interface limit for the stochastic Cahn-Hilliard equation

TL;DR

This paper investigates the behavior of the stochastic Cahn-Hilliard equation as the interface width parameter approaches zero, showing convergence to a stochastic or deterministic Hele-Shaw problem depending on noise strength.

Contribution

It provides a formal asymptotic analysis and rigorous proof of the convergence of stochastic Cahn-Hilliard solutions to Hele-Shaw problems in the sharp interface limit, including noise scaling effects.

Findings

Solutions converge to Hele-Shaw problem as epsilon approaches zero

Small noise leads to deterministic Hele-Shaw limit

Analysis of estimates needed for larger noise regimes

Abstract

We study the two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit, where the positive parameter $\epsilon$ tends to zero, which measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling we indicate that the solutions of stochastic Cahn-Hilliard converge to a solution of a Hele-Shaw problem with stochastic forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.