Diffuse Interface Models and their Sharp Interface Limit

In many applications system with two or more phases occur, which are separated by certain "interfaces", e.g. two immiscible fluids or a material in different states. Macroscopically the interface is usually modeled as a lower/two-dimensional surface. These models are called "sharp intereface models". But on a small length scale the interface usually is "diffuse" e.g. since the immiscible fluids mix partly on a small length scale. This effect is taken into account in so-called diffuse interface models. They have many practical and theoretical advantages since the interfaces between the phases does not need to be described explicitly and singularities in the interfaces can be treated easily. In this talk we will discuss several diffuse interface interface models, starting with the so-called Allen-Cahn equation, and their relation to sharp interface models, when a parameter, which is proportional to the "thickness" of the diffuse interface, tends to zero. In particular, we will present results and techniques how this limit can be treated rigorously.