Research

The analytical treatment of models related to

- free boundary problems
- multiphase flows
- phase transition problems
- boundary layer problems
- rotating fluids
- electrokinetic flows
- contact line dynamics
- heterogeneous catalysis
- living fluids

The further development of functional analytic tools for the treatment
of the models such as

- R-bounded operator families, maximal regularity
- Fourier multipliers
- Stokes operators
- H^\infty-functional calculus
- Newton-polygon method
- self-similarity
- interpolation and multiplication of anisotropic spaces
- Fourier transformed Radon measures

Publication list of Prof. Dr. Jürgen Saal

For a deeper understanding of complex processes in natural sciences a rigorous mathematical analysis of the corresponding mathematical models is indispensable. By the intricate structure of the systems of nonlinear PDEs such a treatment, however, requires the use of deep and subtle analytical tools.
On the other hand, well-posedness of the models represents the fundamental prerequisite not only for further analytical investigations but also for numerical computations and simulations.

The development of those analytical tools, its application to models describing the processes, as for instance models arising in fluid dynamics, as well as the modeling itself are the main purposes of our research. This includes, for example, the treatment of free boundary/interface flows, multiphase flows, phase transition problems, boundary layer problems, heterogeneous catalysis, as well as living fluids. Further it is examined, how additional effects such as rotation, non-Newtonian behavior, or electrokinetics can influence flow properties in bulks and on the interface.

Besides existence and uniqueness of solutions, stability issues are of preferential interest. This includes examinations on convergence to equilibria or rigorous proofs of linear and nonlinear (in-)stability.
Further main objectives are asymptotics and precise dependence on related parameters of solutions. For example, it is of great interest, how solutions of rotating fluid models depend on viscosity, boundary layer thickness, and angular velocity of rotation.

Such a knowledge is significant for both pure mathematics and applications such as geostrophic flows and spin-coating processes. A precise knowledge of asymptotics and dependence of parameters can also give an important feedback to numerical simulations and it can help to reduce computational costs.