On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities

We study a diffuse interface model for the flow of two viscous incompress-ible Newtonian fluids of the same density in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small inter-facial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier-Stokes/Cahn-Hilliard system, which is capable to describe the evolution of droplet formation and collision during the flow. We prove existence of weak solutions of the non-stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities, which ensures – in contrast to the usual case of a smooth free energy density – that the concentration stays in the physical reasonable interval. Furthermore, we present some results on reg-ularity and uniqueness of weak solutions. In particular, we obtain that unique “strong ” solutions exist in two dimensions globally in time and in three di-mensions locally in time. Moreover, we show that for any weak solution th