The stochastic Cahn-Hilliard equation with degenerate mobility and logarithmic potential

TL;DR

This paper establishes the existence of solutions for a stochastic Cahn-Hilliard equation with complex features like degenerate mobility and logarithmic potential, advancing mathematical understanding of stochastic phase-field models.

Contribution

It extends deterministic regularization techniques to the stochastic setting, enabling analysis of solutions with physically relevant constraints under complex potentials.

Findings

Existence of martingale solutions proven.

Uniform energy and magnitude estimates established.

Applicability to stochastic phase-field models demonstrated.

Abstract

We prove existence of martingale solutions for the stochastic Cahn-Hilliard equation with degenerate mobility and multiplicative Wiener noise. The potential is allowed to be of logarithmic or double-obstacle type. By extending to the stochastic framework a regularization procedure introduced by C. M. Elliott and H. Garcke in the deterministic setting, we show that a compatibility condition between the degeneracy of the mobility and the blow-up of the potential allows to confine some approximate solutions in the physically relevant domain. By using a suitable Lipschitz-continuity property of the noise, uniform energy and magnitude estimates are proved. The passage to the limit is then carried out by stochastic compactness arguments in a variational framework. Applications to stochastic phase-field modelling are also discussed.