Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections

TL;DR

This paper studies a complex stochastic PDE with double singular nonlinearities and reflections, establishing existence, uniqueness, ergodicity, and exponential mixing of solutions under challenging conditions.

Contribution

It introduces a novel approach to handle the bi-Laplacian's lack of maximum principle, proving well-posedness and ergodic properties for the stochastic Cahn-Hilliard equation with singular nonlinearities.

Findings

Existence and uniqueness of solutions for initial data in (-1,1)

Identification of the unique invariant measure as ergodic

Proof of exponential mixing for the system

Abstract

We consider a stochastic partial differential equation with two logarithmic nonlinearities, with two reflections at 1 and -1 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, Gouden\`ege and Zambotti, we obtain existence and uniqueness of solution for initial conditions in the interval $(-1,1)$. Finally, we prove that the unique invariant measure is ergodic, and we give a result of exponential mixing.