Workshop on Biomechanics and Chemotaxis, Mon, 10 Dec, 2007
Speaker: Jean Dolbeault
Abstract
The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. Based on estimates for a free energy functional, existence of weak solutions in the two-dimensional euclidean space can be established below the critical mass, above which any solution blows-up in finite time.
In the super-critical case, blow-up of solutions occurs in finite time. Blow-up is a concentration event, where point aggregates are created. Global existence of generalized solutions can still be 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 F. 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 asymptotic results obtained by Velázquez shows that the choice of a solution concept after blow-up is not unique and depends on the type of regularization.
An equation for local density profiles close to point aggregates can be 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.
[ A joint work with C. Schmeiser, based on a result of F. Poupaud ]
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