Cahiers du CEREMADE 

Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : In the twodimensional KellerSegel model for chemotaxis of biological cells, blowup of solutions in finite time occurs if the total mass is above a critical value. Blowup 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 blowup. 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 blowup 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. 





200728 

29062007 

Université
de PARIS  DAUPHINE Place du Maréchal de Lattre De Tassigny  75775 PARIS CEDEX 16  FRANCE Téléphone : +33 (0)1 44054923  fax : +33 (0)1 44054599 