Well-posedness for a diffuse interface model of non-Newtonian two-phase flows

Abstract: The evolution of two partially miscible, nonhomogeneous, incompressible viscous fluids of non-Newtonian type, can be governed by the Navier-Stokes-Cahn-Hilliard system. In the present work, we prove the global existence of weak solutions for the case of initial density containing zero and the concentration depending viscosity with free energy potential equal to the Landau potential in a bounded domain of three dimensions. Furthermore, we show that a strong solutions exist locally in time in the case of three dimensions periodic domain ${\mathbb T}^3.$ The proof relies on a suitable semi-Galerkin scheme and the monotonicity method.