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S. Hittmeir:
"Chemotaxis Models";
Vortrag: Summer School Fluid2Bio 2012, L'Aquila (eingeladen); 03.06.2012 - 06.06.2012.

In the first part of the course we introduce the famous chemotaxis model by Keller and Segel that has been introduced to describe the aggregation process of cellular slime molds, who secrete a chemical which again attracts the cells. The interesting feature of the system is that solutions may under certain conditions on the initial data blow up in finite time. In two space dimensions the situation is very well understood, since there a critical threshold for the mass exists, which decides if solutions remain globally bounded or grow above all bounds. We present the main ideas for the existence and blow up proofs in higher space dimension as well as in the particular case 2d.
Furthermore we introduce modifications of the Keller Segel system, which act as regularisations of the system and therefore allow for global solutions for any initial data. We explain several mechanisms and in particular focus on the model whith additional cross-diffusion, since there the entropy structure changes completely and the existence result is obtained using a very different approach.
We also study the main result of Keller and Segel on travelling waves for a chemotaxis model for bacteria in a tube that are now consuming the substrate. We explain the necessary conditions on the chemotactic sensitivity of singular form.
In the last part of the course we introduce a kinetic model, or a so called velocity jump process, for chemotactic movements in zick-zack form that occurs e.g. in the case of E. Coli bacteria. We present a global existence result,which implies that the finer kinetic description can be also understood as a regularisation of the Keller Segel model. And we also explain how in the diffusion limit we can raise the Keller Segel model. Clearly this limit only holds for short time intervals, since the kinetic model allows for global existence and the macroscopic model exhibits finite time blow up.