Austrian-French Cooperation, Amadée Project No. 19/2003

Nonlinear Kinetic Modelling in Stellar Physics, Charged Particle Physics, and Mathematical Biology


Participants (France): Participants (Austria): Funding period: 2003/04
Funding agency: ÖAD



Scientific aims

The gravitational response of stellar objects, the transport of electrons and ions in ionized gases and semiconductors, and the chemotactic response of certain bacteria and microorganisms lead to mathematical models with strong similarities. In all cases, movement is driven by a field generated by the particles (stars, electrons, microorganisms, ...) themselves. Nonlinear effects like blowup of solutions in finite time for example, occur both in gravitational models as well as in models for chemotaxis. Also these fields share the need for multiscale modelling. The usually very complex phenomenology is based on mechanisms with strongly different length and time scales. Mathematically, this fact is reflected by hierarchies of models ranging from microscopic and mesoscopic (kinetic) descriptions to macroscopic (fluid type) models.
The proponents have been involved in the analysis of the qualitative behaviour of these models and in the connection between different members of model hierarchies. These subjects have been dealt with in several longstanding cooperations between the French and Austrian proponents. In the proposed project, these efforts should be continued concentrating on the following issues:
1) Vlasov type models: These kinetic models include a driving force generated by gravitational or Coulomb interaction. An issue to be adressed is the transition from kinetic to fluid models in scaling limits, like the nonrelativtistic-quasi neutral limit from the Vlasov-Maxwell system towards incompressible Euler fluid equations or magnetohydrodynamic equations [BrMPu1]. Also, the derivation of nonlinear 1 particle Vlasov equations from linear N particle equations in weak coupling limits will be investigated in a collaboration between Nice, Paris and Vienna, along the lines of the very successful collaboration on the derivation of nonlinear Schroedinger equations [BGM1]. Also, numerical simulations of nonlinear Vlasov equations will tackled in joint work Paris-Vienna-Nice.
2) Chemotaxis: Several issues will be addressed. A challenging modelling problem is the derivation of kinetic models from microscopic biological information. Also the derivation of macroscopic models by scaling limits will be considered. Blow up of solutions has been analyzed extensively for macroscopic models. In this project, we intend to attack the question of blow up for kinetic models. A recent joint effort Paris-Vienna will be continued.
3) Ionized gases and semiconductors: An analysis of formally derived SHE-models (diffusion models for an energy dependent distribution function) for gases subject to ionization and strong electric fields wil be carried out. This is the continuation of a long cooperation Nice-Toulouse-Vienna. A second focus will be on nonlinear quantum mechanical models and their singular limits such as semiclassical or nonrelativistic limits.
4) Long time behaviour: The entropy-entropy dissipation approach will be carried further to include nonhomogeneous kinetic models for semiconductors and wave-particle interaction in cometary flows.