A study of blow-ups in the Keller-Segel model of chemotaxis

TL;DR

This paper develops a numerical method to accurately simulate singular solutions in the Keller-Segel chemotaxis model and compares these results with analytical predictions about singularity formation and interactions.

Contribution

It introduces a composite particle-grid numerical method with adaptive time stepping for resolving singularities in the Keller-Segel model and validates it against analytical results.

Findings

Successfully resolves singular solutions in 2D simulations.

Confirms analytical predictions of singularity formation.

Provides insights into singularity interactions in chemotaxis models.

Abstract

We study the Keller-Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to accurately resolve singular solutions. The numerical findings (in two dimensions) are then compared with analytical predictions regarding formation and interaction of singularities obtained via analysis of the stochastic differential equations associated with the Keller-Segel model.