Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase   Flows of Incompressible Fluids with Different Densities
  We prove existence of weak solutions for a diffuse interface model for the flow 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 Gr\"un 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. 
Helmut Abels, Daniel Depner, Harald Garcke
math.AP physics.flu-dyn