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

Arnaud Debussche (1, 2) , Ludovic Goudenège (2)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Cahn-Hilliard; stochastic partial differential equations; integration by parts formulae; reflection measure; invariant measure; singular nonlinearity; double; two

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

Arnaud Debussche, Ludovic Goudenège. Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections. SIAM Journal on Mathematical Analysis, 2011, 43 (3), pp.1473-1494. ⟨10.1137/090769636⟩. ⟨hal-00411599⟩