Regularity Results for a Cahn-Hilliard-Navier-Stokes System with Shear Dependent Viscosity

Abstract

We analyze a diffuse interface model describing the behavior of a mixture of two incompressible fluids. More precisely, we consider Navier-Stokes type equations with power-law like shear dependent viscosity. Such equations are nonlinearly coupled with a convective Cahn-Hilliard equation for the order parameter. The resulting system is endowed with no-slip and no-flux boundary conditions. We prove some regularity properties of weak solutions under rather general conditions. This is a generalization of previous results already proven for single fluids. Some consequences of such results are also addressed.