BZ Seminar in Analysis: Michael Sigal (University of Toronto and ETH Zurich)

Blowup Dynamics in the Keller-Segel Model of Chemotaxis

The Keller-Segel equations model chemotaxis of bio-organisms. In a reduced form, considered in this talk, they are related to Vlasov equation for self-gravitating systems and are used in social sciences in descriptions of crime patterns. It is relatively easy to show that in the critical dimension 2 and for the mass of initial conditions greater than 8π, the solutions break down in finite time. Understanding the mechanism of this breakdown turned out to be a subtle problem defying solution for a long time. Preliminary results indicate that the solutions “blowup”. This blowup is supposed to describe the chemotactic aggregation of the organisms and understanding its universal features would allow comparison of theoretical results with experimental observations. In this talk I discuss recent results on dynamics of solutions of the (reduced) Keller-Segel equations in the critical dimension 2, which include a formal derivation and partial rigorous results on the blowup dynamics of solutions.

(The talk is based on joint work with S. I. Dejak, D. Egli and P.M. Lushnikov.)

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