# Partial differential equations

## Partial differential equations

Partial differential equations (PDEs) model numerous phenomena from physics, biology, and our daily life.

The distribution of milk in our morning coffee is described by convection-diffusion-equations. The shock-like dynamics of traffic jams can be modeled by the so-called Burgers equation. Even the behavior of sheep under the influence of shepherd dogs can be explained by a PDE. In addition, partial differential equations can also be applied as a tool to address problems which — at first sight — do not follow any physical laws, such as sharpening of images and training of neural nets.

The mathematical challenges which are posed by the study of PDEs are numerous. Firstly, the obligatory questions asking for existence and uniqueness of solutions have to be addressed. Another important branch of research is the analysis of qualitative and quantitative properties of solutions, like for instance their regularity or long-time behavior. More applied research deals with numerical methods for solving PDEs approximatively, exploring new areas of application, and deriving PDEs to model a concrete problem.

Our research area “Partial Differential Equations” studies PDEs from biology and social science which model populations dynamics of several concurring species or the behavior of cell membranes. Furthermore, we work on PDEs for image processing and pattern recognition. Our focus lies both on modeling of complex processes and on the mathematical analysis of the resulting equations.