On the weak solutions to a 3D stochastic Cahn–Hilliard–Navier–Stokes model

Abstract

We study in this article a stochastic version of a coupled Cahn–Hilliard–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 Cahn–Hilliard model for the order (phase) parameter. We prove the existence of a probabilistic weak solution. The proof relies on a Galerkin approximation as well as some compactness results. In the two-dimensional case, we also derive the uniqueness of the weak solutions.