Nonlinear partial differential equations are investigated with methods of evolution equations and harmonic analysis. Of particular interest are the fundamental equations of fluid dynamics, geophysical equations, nematic liquid crystals, complex fluids, fluid-structure interaction, and free boundary value problems.

Equations of fluid dynamics are also studied by means of convex integration, a method which is central for non-uniqueness results of turbulent flows.

Important is also the interaction with related methods of geometric analysis, interpolation theory, periodic solutions and stochastic PDEs for the investigation of, for example, stochastic or also non-local boundary value problems.

M. Hieber

DFG Forschungsgroup

The DFG Research Group: Mathematical Investigations of Geophysical Fluid Models: Analysis and Numerics was approved in 2023.

We focus on the mathematical modelling and computational analysis of complex flow problems, especially two-phase flows with transport processes at interfaces and dynamic wetting phenomena.

This research is based on thermodynamically consistent modelling with further development of sharp interface and diffuse interface models with additional physico-chemical properties of the interfaces. For a fundamental understanding of the transport and transfer processes, direct numerical simulations are performed, employing and extending complementary numerical methods such as volume-of-fluid, interface tracking, combined level set / front tracking as well as phase field methods.

D. Bothe ,H. Marschall

We develop new tools and techniques in harmonic analysis, operator theory and geometric measure theory to study partial differential equations under minimal regularity assumptions.

Of particular interest are elliptic and parabolic boundary value problems, including the functional calculus of their associated Cauchy-Riemann operators, fine properties of elliptic and parabolic measures, and regularity estimates in the realm of the Kato square root problem. The latter is also key to treating quasilinear equations with mixed boundary conditions and data in distributions spaces via maximal regularity methods (see also Applied Analysis)

M. Egert , R. Haller-Dintelmann

Chemotaxis is important in many biological processes involving cell movement. We study the qualitative behavior of solutions to different types of chemotaxis systems. In particular, we investigate global existence, blow-up, large time behavior and periodic solutions.

C. Stinner