Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system

Jan Haskovec*, Christian Schmeiser

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

The analysis of a stochastic interacting particle scheme for the approximation of measure solutions of the parabolic-elliptic Keller-Segel system in 2D is continued. In previous work it has been shown that solutions of a regularized scheme converge to solutions of the regularized Keller-Segel system, when the number of particles tends to infinity. In the present work, the regularization is eliminated in the particle model, which requires an application of the framework of time dependent measures with diagonal defects, developed by Poupaud. The subsequent many particle limit of the BBGKY hierarchy can be solved using measure solutions of the Keller-Segel system and the molecular chaos assumption. However, a uniqueness result for the limiting hierarchy and therefore a proof of propagation of chaos is missing. Finally, the dynamics of strong measure solutions, i.e., sums of smooth distributions and Delta measures, of the particle model is discussed formally for the cases of 2 and 3 particles. The blow-up behavior for more than 2 particles is not completely understood.

Original languageEnglish (US)
Pages (from-to)940-960
Number of pages21
JournalCommunications in Partial Differential Equations
Volume36
Issue number6
DOIs
StatePublished - Jun 1 2011

Keywords

  • BBGKY hierarchy
  • Defect measure
  • Keller-Segel model
  • Measure valued solutions
  • Stochastic interacting particle systems

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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