  We analyze a diffuse interface model for multi-phase flows of $N$
incompressible, viscous Newtonian fluids with different densities. In the case
of a bounded and sufficiently smooth domain existence of weak solutions in two
and three space dimensions and a singular free energy density is shown.
Moreover, in two space dimensions global existence for sufficiently regular
initial data is proven. In three space dimension, existence of strong solutions
locally in time is shown as well as regularization for large times in the
absence of exterior forces. Moreover, in both two and three dimensions,
convergence to stationary solutions as time tends to infinity is proved.
