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AG 6: Partial Differential Equations and Applications

Scientific Objectives

Unit 6 is dedicated to the theory of partial differential equations and variational methods. Particular interest is given to problems which are directly related to problems in the physical and engineering sciences. The main emphasis of these investigations is the study of nonlinear problems.

Following are some examples of problems which have been studied or are currently being studied by members of this group:

• Geometrical optics: Do high frequency sound waves and light waves propagate according to the laws of geometrical optics? This question leads to the study of the propagation of singularities of solutions to hyperbolic partial differential equations and microlocal analysis.
• Scattering theory and inverse problems: How are waves scattered when they encounter an obstacle? Is it possible to determine the shape of such an obstacle through measurements of the scattered waves?
• When do shocks develop in the solutions of nonlinear partial differential equations?
• Avalanches: Analysis of conservation laws with source term which model the flow of granular materials such as wet snow avalanches; existence, long time behaviour and shocks.
• Fluid mechanics: Construction of weak and strong solutions of the Stokes or Navier-Stokes equations in bounded and unbounded domains; qualitative analysis of solutions (stability, asymptotic behaviour, singularities in space or time, bifurcation).
• Viscoelasticity: Is there a solution to the equations which describe the plastic deformation of steel?

Open seminar: In this seminar current research problems are discussed. Members of the group often present their own research results, as well as invited guests.

Weg zum Diplom: This page gives an overview about possible lectures and seminars, which can lead to a thesis in our research group.