The exponential behavior of a stochastic globally modified Cahn-Hilliard-Navier-Stokes model with multiplicative noise

In this article, we study the stability of weak solutions to a stochastic version of a globally modified coupled Cahn-Hilliard-Navier-Stokes model with multiplicative noise. The model consists of the globally modified Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard model for the order (phase) parameter. We prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.