We study the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system
in a bounded smooth domain in Rd, 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.