  We study a diffuse interface model for the flow of two viscous incompressible
Newtonian fluids in a bounded domain. The fluids are assumed to be
macroscopically immiscible, but a partial mixing in a small interfacial region
is assumed in the model. Moreover, diffusion of both components is taken into
account. In contrast to previous works, we study a model for the general case
that the fluids have different densities due to Lowengrub and Truskinovski.
This leads to an inhomogeneous Navier-Stokes system coupled to a Cahn-Hilliard
system, where the density of the mixture depends on the concentration, the
velocity field is no longer divergence free, and the pressure enters the
equation for the chemical potential. We prove existence of unique strong
solutions for the non-stationary system for sufficiently small times.
