Existence and ergodicity for the two-dimensional stochastic Allen–Cahn–Navier–Stokes equations

Abstract

We study in this article a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional bounded domain. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter, both endowed with suitable boundary conditions. We prove the existence of solutions via a semigroup approach. We also obtain the existence and uniqueness of an invariant measure via coupling methods.