Exponential stability and stabilization of a stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise

Abstract

In this article, we study the asymptotic stability of the unique strong solution to a stochastic version of a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. The model consists of the 2D Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence, uniqueness and regularity results for the stationary solution of the stochastic 2D nonlocal Cahn-Hilliard Navier-Stokes equations. Furthermore, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. Finally, we also prove a result related to the stabilization of the stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations.