research partners    
Ph.D. students    
consortium members    

Piotr Gwiazda
Institute of Applied Mathematics and Mechanics
University of Warsaw
Banacha 2,
02-097 Warsaw
phone: +48 22 5544551
fax: +48 22 5544700

Analytic and numerical study of interacting particle systems modelling aggregation: from discrete to continuous models
Grzegorz Karch and Benoit Perthame in collaboration with Angela Stevens
The project is carried out by Rafał Celiński at University of Wrocław
The analysis of the long time behaviour of a collection of self-interacting individuals via pairwise potentials arises in the modelling of animal collective behaviour. The simplest models based on systems of ordinary differential equations led to continuum descriptions for the evolution of densities of individuals. The Patlak-Keller-Segel model of chemotaxis and the aggregation equation (sometimes supplemented with a diffusion term) are perhaps the most popular continuous models. These equations belong to the same family of nonlinear friction equations that appear in the modelling of granular media.

In real systems featuring aggregation, the diffusion is always present and may be either nonlinear or fractal (described by the generators of L\'evy processes). In such cases, the questions of global in time existence of solutions versus finite time blow up are much more delicate and need a new approach compared to the inviscid model. There are some preliminary studies in that direction but still many questions are left open.

Ph.D. student will systematically study properties of solutions of such models, including the asymptotic behaviour of solutions as well as the criteria for regularity of solutions and finite time blow up for complicated real models featuring general aggregation nonlinearities and anomalous diffusion terms.

This Ph.D. project will be performed in Wroclaw (5 semesters) as well as in Heidelberg and Paris-VI (3 semesters).

[1] A.L. Bertozzi, J.A. Carrillo and Th. Laurent, (2009) Nonlinearity, 22: 683-710.
[2] P. Biler and G. Karch, (2008) arXiv:~0812.4982} preprint.
[3] B. Perthame, (2007) Transport equations in biology, Frontiers in Mathematics. Birkhauser Verlag, Basel.
[4] A. Stevens, (2000) SIAM J. Appl. Math., 61: 183--212.
[5] E.E. Espejo Arenas, A. Stevens and J.J.L. Velazquez, (2009) preprint.