Coalescing particle systems and applications to nonlinear Fokker–Planck equations

Gleb Zhelezov, Ibrahim Fatkullin

Research output: Contribution to journalArticlepeer-review


We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.
Original languageEnglish
Pages (from-to)463-490
Number of pages29
JournalCommunications in Mathematical Sciences
Issue number2
Publication statusPublished - 15 May 2018


Dive into the research topics of 'Coalescing particle systems and applications to nonlinear Fokker–Planck equations'. Together they form a unique fingerprint.

Cite this