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.

Keywords

• Incompressible viscous fluids
• shear dependent viscosity
• binary mixtures
• diffuse interfaces
• Cahn-Hilliard equations
• regularity of solutions