CANCELED – For personal reasons, Helmut Abels has unfortunately been forced to cancel his seminar. – CANCELED

Interfaces separating two or more species or components of a material are omnipresent in applications in the sciences. Nowadays there are two main classes of models to describe interfaces, both from a theoretical and practical point of view: In classical so-called « sharp interface models » the species or components under consideration fill disjoint domains that are separated by lower dimensional surfaces of a certain regularity. On the other hand in « diffuse interface models » a partial mixing of the species or components on a small length scale is taken into account, which leads to an interfacial layer of small but positive thickness. This has the advantage that the interfaces do not need to be resolved explicitly and singularities in the interfaces can be described consistently. 

In the first part of the talk we will give an overview of some basic diffuse interface models with applications to material sciences. Moreover, we will discuss some analytic results on the relation between diffuse and sharp interface models, when the interfacial thickness of the diffuse interface tends to zero. The analysis is based on formally matched asymptotic expansions (in a rigorous manner) together with uniform estimates for the linearized operator. In this part we will focus on the Allen-Cahn equation as basic model.

In the second part we will discuss a Navier-Stokes/Allen-Cahn system, which describes the flow of two macroscopically immiscible viscous incompressible fluid, separated by a diffuse interface. We will present some recent results on novel higher-order estimates for the sharp interface limit of this system. With the aid of a suitable weight taking the distance to the interface and the interfacial thickness into account we obtain optimal regularity estimates, which are uniformy in the interfacial thickness. This enables to improve previous convergence results in two space dimensions significantly and extend results to three space dimensions. This is a joint-work with Mingwen Fei, Yadong Liu, and Maximilian Moser.