Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection

TL;DR

This paper studies a stochastic Cahn-Hilliard equation with singular nonlinearities and reflection, establishing existence, uniqueness, and invariant measures using advanced infinite-dimensional techniques.

Contribution

It introduces a novel approach to handle the lack of maximum principle in bi-Laplacian driven SPDEs with reflection and conservation constraints.

Findings

Proved existence and uniqueness of solutions for nonnegative initial data.

Analyzed invariant measures and reflection measures.

Developed a new approximation method inspired by Debussche and Zambotti.

Abstract

We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche and Zambotti, we use a method based on infinite dimensional equations, approximation by regular equations and convergence of the approximated semi-group. We obtain existence and uniqueness of solution for nonnegative intial conditions, results on the invariant measures, and on the reflection measures.