On sharp interface limits for diffuse interface models for two-phase flows

We discuss the sharp interface limit of a diffuse interface model for a two-phase flow of two partly miscible viscous Newtonian fluids of different densities, when a certain parameter epsilon > 0 related to the interface thickness tends to zero. In the case that the mobility stays positive or tends to zero slower than linearly in epsilon we will prove that weak solutions tend to varifold solutions of a corresponding sharp interface model. But, if the mobility tends to zero faster than epsilon(3) we will show that certain radially symmetric solutions tend to functions, which will not satisfy the Young-Laplace law at the interface in the limit.