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

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.

[1]  J. J. L. Velázquez,et al.  Point Dynamics in a Singular Limit of the Keller--Segel Model 1: Motion of the Concentration Regions , 2004, SIAM J. Appl. Math..

[2]  J. J. L. Velázquez,et al.  Point Dynamics in a Singular Limit of the Keller--Segel Model 2: Formation of the Concentration Regions , 2004, SIAM J. Appl. Math..

[3]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[4]  C. Schmeiser,et al.  Global existence for chemotaxis with finite sampling radius , 2006 .

[5]  Juan J. L. Velázquez.  Well-posedness of a model of point dynamics for a limit of the Keller-Segel system , 2004 .

[6]  W. Jäger,et al.  On explosions of solutions to a system of partial differential equations modelling chemotaxis , 1992 .

[7]  José A. Carrillo,et al.  Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in R 2 , 2007 .

[8]  F. Poupaud,et al.  Diagonal Defect Measures, Adhesion Dynamics and Euler Equation , 2002 .

[9]  Benoît Perthame,et al.  Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions , 2006 .

[10]  C. Patlak.  Random walk with persistence and external bias , 1953 .

[11]  Manuel del Pino,et al.  Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions☆ , 2002 .

[12]  Thomas Hillen,et al.  Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding , 2001, Adv. Appl. Math..

[13]  B. Perthame,et al.  Kinetic Models for Chemotaxis and their Drift-Diffusion Limits , 2004 .

[14]  Michael Loss,et al.  Competing symmetries, the logarithmic HLS inequality and Onofri's inequality onsn , 1992 .

[15]  Benoît Perthame,et al.  Optimal critical mass in the two dimensional Keller–Segel model in R2 , 2004 .

[16]  M. A. Herrero,et al.  A blow-up mechanism for a chemotaxis model , 1997 .

[17]  P. Laurençot,et al.  The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in a disc , 2006 .

[18]  P. Laurençot,et al.  The 8π‐problem for radially symmetric solutions of a chemotaxis model in the plane , 2006 .

[19]  Christian Schmeiser,et al.  The Keller-Segel Model with Logistic Sensitivity Function and Small Diffusivity , 2005, SIAM J. Appl. Math..

[20]  Marion Kee,et al.  Analysis , 2004, Machine Translation.