  We show well-posedness of a diffuse interface model for a two-phase flow of
two viscous incompressible fluids with different densities locally in time. The
model leads to an inhomogeneous Navier-Stokes/Cahn-Hilliard system with a
solenoidal velocity field for the mixture, but a variable density of the fluid
mixture in the Navier-Stokes type equation. We prove existence of strong
solutions locally in time with the aid of a suitable linearization and a
contraction mapping argument. To this end we show maximal $L^2$-regularity for
the Stokes part of the linearized system and use maximal $L^p$-regularity for
the linearized Cahn-Hilliard system.
