Strong solutions for the stochastic Cahn-Hilliard-Navier-Stokes system

Abstract

A well-known diffuse interface model consists of the Navier-Stokes equations for the average velocity, nonlinearly coupled with a convective Cahn-Hilliard type equation for the order (phase) parameter. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a stochastic version of this model forced by a multiplicative white noise on a bounded domain of Rd, d=2,3. We prove the existence and uniqueness of a local maximal strong solution when the initial data (u0,ϕ0) takes values in H1×H2. Moreover in the two-dimensional case, we prove that our solution is global.