Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection

Ludovic Goudenège (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Cahn-Hilliard; stochastic partial differential equations; integration by parts formulae; reflection measures; invariant measures; singular non linearity

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Ludovic Goudenège. Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection. Stochastic Processes and their Applications, 2009, 119 (10), pp.3516-3548. ⟨10.1016/j.spa.2009.06.008⟩. ⟨hal-00336601⟩