Kinetic models for multiscale problems

This research group will work on kinetic modelling of partial differential equations for multiscale problems. The group consists of researcher whose work is mainly orientated in analytical techniques and of researchers specialized in modeling and numerical techniques. The important objective of this focused research group is to combine modeling, analytical and numerical insights in order to push the frontier of understanding kinetic systems. The interaction of these three areas will be very tight since modern pde modeling/analysis and numerics/simulation cannot survive without one another in the near future. The following topics will be dealt with:

1) Kinetic models in population dynamics and mathematical biology: the dynamic interaction of genotype/phenotype and spatial (geography) effects will be studied by means of kinetic modelling (the "genotype/phenotype" variables play the role of the momentum variable, dual to the spatial variable). Chemotaxis will be studied on the kinetic level, with particular emphasis ondiffusion limits and microscopic modelling of the motion of bacteria. It is conceivable that finite-time blow-up can be avoided on the kinetic level by careful modelling of reorientation processes. 2) Macroscopic limits of kinetic models: Applications include charge transport in semiconductors, ionisation of gases, and kinetic models for wave-particle scattering. An understanding of fluid limits of the latter might lead to a new approach for understanding macroscopic models of gas dynamics. 3) Modelling and Simulation of Bose-Einstein condensation: Bose-Einstein condensation is one of the extremely hot topics in low-temperature atomic physics (cf. the Nobel Prize in physics 2001). The typical model after condensation is the Gross-Pitaevskii equation (cubically non-linear Schrödinger equation with harmonic confinement), while the bosonic Boltzmann equation is used to describe the condensation process. We expect to continue the analysis of the condensation process (transition regime of the bosonic Boltzmann equation to the Gross-Pitaevskii equation). 4) Kinetic modelling of granular gases: Granular media (as sand, powders, seeds and sements) which consists of macroscopically-sized solid particles, exhibit rich and interesting behavior which defies conventional fluid/gas/solid classification. They also exhibit interesting collective phenomena such size segregation, pattern formation, shock waves, and density inhomogeneities. The reseach will be focused on new modeling and computational techniques.

Confirmed Participants