On the weak solutions to a stochastic two-phase flow model

Abstract

ABSTRACT We study, in this article, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two- or three-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. These equations are motivated by the dynamic of binary fluids under the influence of stochastic external forces. We prove the existence of a probabilistic weak solutions. The proof relies on a Galerkin approximation as well as some compactness results. In the two-dimensional case, we prove the uniqueness of the weak solutions.