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

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, Goudenège 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. Introduction and main results The Cahn-Hilliard-Cook equation is a model to describe phase separation in a binary alloy (see [6], [7] and [8]) in the presence of thermal fluctuations (see [11] and [27]). It takes the form:  ∂tu = − 1 2 ∆ (∆u− ψ(u)) + ξ̇, on Ω ⊂ R, ∇u · ν = 0 = ∇(∆u− ψ(u)) · ν, on ∂Ω, (0.1) where t denotes the time variable and ∆ is the Laplace operator. Also u ∈ [−1, 1] represents the ratio between the two species and the noise term ξ̇ accounts for the thermal fluctuations. The nonlinear term ψ has the double-logarithmic form: ψ : u 7→ θ 2 ln ( 1 + u 1− u ) − θcu, (0.2) AMS 2000 subject classifications. 60H15, 60H07, 37L40.