• français
    • English
  • English 
    • français
    • English
  • Login
JavaScript is disabled for your browser. Some features of this site may not work without it.
BIRD Home

Browse

This CollectionBy Issue DateAuthorsTitlesSubjectsJournals BIRDResearch centres & CollectionsBy Issue DateAuthorsTitlesSubjectsJournals

My Account

Login

Statistics

View Usage Statistics

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

Thumbnail
Date
2009
Link to item file
http://hal.archives-ouvertes.fr/hal-00158767/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
Keller-Segel; chemotaxis; blow-up; aggregation; measure valued solutions; defect measure
Journal issue
Discrete and Continuous Dynamical Systems. Series A
Volume
25
Number
1
Publication date
2009
Article pages
109-121
Publisher
American Institute of Mathematical Sciences
DOI
http://dx.doi.org/10.3934/dcds.2009.25.109
URI
https://basepub.dauphine.fr/handle/123456789/1099
Collections
  • CEREMADE : Publications
Metadata
Show full item record
Author
Schmeiser, Christian
Dolbeault, Jean
Type
Article accepté pour publication ou publié
Abstract (EN)
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.

  • Accueil Bibliothèque
  • Site de l'Université Paris-Dauphine
  • Contact
SCD Paris Dauphine - Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16

 Content on this site is licensed under a Creative Commons 2.0 France (CC BY-NC-ND 2.0) license.