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

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. [ABSTRACT FROM AUTHOR]