Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities and Nonlocal Free Energies

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions. In contrast to previous works, we study a model with a singular non-local free energy, which controls the $H^{\alpha/2}$-norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.