Global Existence of Martingale Solutions and Large Time Behavior for a 3D Stochastic Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Shear Dependent Viscosity

Abstract

In this paper, we consider a stochastic version of a nonlinear system which consists of the incompressible Navier–Stokes equations with shear dependent viscosity controlled by a power $$p>2$$ , coupled with a convective nonlocal Cahn–Hilliard-equations. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluid having the same density. We prove the existence of a weak martingale solutions when $$p\in [11/5,12/5)$$ , and their exponential decay when the time goes to infinity.