Existence of Optimal and ε-Optimal Controls for the Stochastic Cahn-Hilliard Navier-Stokes System

We study in this article a stochastic version of a coupled of the Cahn-Hilliard Navier-Stokes model in two dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity, coupled with a Cahn-Hilliard model for the order (phase) parameter. We prove the existence of optimal and ε-optimal controls for the stochastic Cahn-Hilliard Navier-Stokes model.