Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection

Ludovic Goudenège

Stochastic Processes and their Applications, 2009, vol. 119, issue 10, 3516-3548

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 semigroup. We obtain existence and uniqueness of a solution for nonnegative initial conditions, results on the invariant measures, and on the reflection measures.

Keywords: Cahn-Hilliard; Stochastic; partial; differential; equations; Integration; by; parts; formulae; Reflection; measures; Invariant; measures; Singular; nonlinearity (search for similar items in EconPapers)
Date: 2009
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