February  2009, 25(1): 109-121. doi: 10.3934/dcds.2009.25.109

The two-dimensional Keller-Segel model after blow-up

1. 

CEREMADE UMR CNRS no. 7534, Université Paris-Dauphine, Place de Lattre de Tassigny, 75776 Paris 16, France

2. 

Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria

Received  June 2007 Revised  May 2008 Published  June 2009

In the two-dimensional Keller-Segel model for chemotaxis of biological cells, blow-up of solutions in finite time occurs if the total mass is above a critical value. Blow-up is a concentration event, where point aggregates are created. In this work global existence of generalized solutions is proven, allowing for measure valued densities. This extends the solution concept after blow-up. The existence result is an application of a theory developed by Poupaud, where the cell distribution is characterized by an additional defect measure, which vanishes for smooth cell densities. The global solutions are constructed as limits of solutions of a regularized problem.
   A strong formulation is derived under the assumption that the generalized solution consists of a smooth part and a number of smoothly varying point aggregates. Comparison with earlier formal asymptotic results shows that the choice of a solution concept after blow-up is not unique and depends on the type of regularization.
   This work is also concerned with local density profiles close to point aggregates. An equation for these profiles is derived by passing to the limit in a rescaled version of the regularized model. Solvability of the profile equation can also be obtained by minimizing a free energy functional.
Citation: Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109
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