Sharp interface limits for diffuse interface models for two-phase ﬂows of viscous incompressible ﬂuids

We seek a rigorous understanding of sharp interface limits of so-called diffuse interface models for the ﬂow of two viscous ﬂuids like oil and water. In diffuse interface models a partial mixing of the macroscopically immiscible ﬂuids on a small length scale ε > 0 and diffusion of the mass particles are taken into account. These models are capable of describing such two-phase ﬂows beyond the occurrence of topological singularities of the interface due to collision or droplet formation. Both for theoretical and numerical purposes a deeper understanding of the limit ε → 0 in dependence of the scaling of the mobility mε is of interest. Here the mobility is the inverse of the Peclet number and controls the strength of the diffusion. In particular, we want to understand the inﬂuence of the scaling of the mobility coefficient mε as ε → 0 on the limit system and on the convergence rates rigorously. This way we want to clarify numerical observations and formal derivations in the literature and obtain new insights.