Stochastic Cahn–Hilliard Equation with Double Singular Nonlinearities and Two Reflections

Abstract

We consider a stochastic partial differential equation with two logarithmic nonlinearities, two reflections at 1 and $-1$, and 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 a 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, Goudenège, and Zambotti, we obtain existence and uniqueness of a 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.