On the stochastic Cahn-Hilliard equation with a singular double-well potential

TL;DR

This paper establishes well-posedness and regularity for the stochastic Cahn-Hilliard equation with a highly general double-well potential, accommodating both additive and multiplicative noise without growth or smoothness restrictions.

Contribution

It extends the analysis of the stochastic Cahn-Hilliard equation to include very general double-well potentials with multivalued maximal monotone graphs, broadening the scope of existing results.

Findings

Proved well-posedness of the stochastic Cahn-Hilliard equation.

Established regularity results for solutions.

Provided a variational formulation accommodating general potentials.

Abstract

We prove well-posedness and regularity for the stochastic pure Cahn-Hilliard equation under homogeneous Neumann boundary conditions, with both additive and multiplicative Wiener noise. In contrast with great part of the literature, the double-well potential is treated as generally as possible, its convex part being associated to a multivalued maximal monotone graph everywhere defined on the real line on which no growth nor smoothness assumptions are assumed. The regularity result allows to give appropriate sense to the chemical potential and to write a natural variational formulation of the problem. The proofs are based on suitable monotonicity and compactness arguments in a generalized variational framework.