We prove existence of weak solutions for a diffuse interface model for the
ow of two
viscous incompressible Newtonian
fluids in a bounded domain in two and three space dimensions.
In contrast to previous works, we study a new model recently developed by Abels,
Garcke, and Grün for
fluids with different densities, which leads to a solenoidal velocity field.
The model is given by a non-homogeneous Navier-Stokes system with a modified convective
term coupled to a Cahn-Hilliard system. The density of the mixture depends on an order
parameter.