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

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