Research areas

Main research areas are theory and application of partial differential equations as well as mathematical modeling.

Research areas

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, S. Modena

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

Convex integration is a technique for the construction of ``strange'' solutions to certain nonlinear systems of partial differential equations. The technique originates in the work of John Nash 1954 on continuously differentiable isometric embeddings and has been developed into a powerful general method for certain problems in differential geometry and in the calculus of variations.

In the last years new versions of this technique have been developed primarily for applications in fluid mechanics, becoming a crucial tool to investigate fluid problems in low regularity and low integrability regimes.

S. Modena

Discretization methods for integral equations yield sequences of approximation operators which can be viewed of as elements of suitable C*-algebras. The goals are to employ C*-algebraic techniques to study numerical properties such as the stability of the method, and to describe the structure of the resulting algebras.

S. Roch

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

Period Project Spokesperson Funding
2009–2018 International Research Training Group Mathematical Fluid Dynamics Matthias Hieber DFG/Japan Society for the Promotion of Science (JSPS)
2010–2019 Transport processes at fluidic interfaces Dieter Bothe DFG