DFG-Forschungszentrum "Mathematik für Schlüsseltechnologien - MATHEON": Local existence, uniqueness and smooth dependence for quasiliniar parabolic problems with non-smooth data" (Teilprojekt D 19)

The project concerns initial boundary value problems (IBVPs) for quasilinear second order parabolic PDEs with non-smooth data and for weakly coupled systems of such PDEs. Typical applications are transport processes of charged particles in semiconductor heterostructures, phase separation processes of nonlocally interacting particles, or chemotactic aggregation in heterogeneous enviroments. The goal is to apply the Implicit Function Theorem to get local in time existence, uniqueness and smooth dependence for such IBVPs. This is the first, but main step in order to apply the powerful theory of smooth dynamical systems to those IBVPs. The main working tools are new maximal parabolic regularity results in Sobolev-Morrey spaces.