Weak and strong solutions to the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system

We study the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system in a bounded smooth domain in R^d, d=2,3. This model arises from the Diffuse Interface theory of binary mixtures accounting for density variation, capillarity effects at the interface and partial mixing. We consider the case of initial density away from zero and concentration-depending viscosity with free energy potential equal to either the Landau potential or the Flory-Huggins logarithmic potential. In this setting, we prove the existence of global weak solutions in two and three dimensions, and the existence of strong solutions with bounded and strictly positive density. The strong solutions are local in time in three dimensions and global in time in two dimensions.