Chemotaxis: from kinetic equations to aggregate dynamics

TL;DR

This paper derives a nonlocal conservation law from a kinetic chemotaxis model, analyzing measure-valued solutions, blow-up phenomena, and aggregate dynamics, and introduces a numerical scheme for simulation.

Contribution

It develops a rigorous framework for the hydrodynamic limit of chemotaxis models, including existence, uniqueness, and numerical methods for measure-valued solutions.

Findings

Finite time blow-up leads to measure-valued solutions.

Existence of solutions via duality methods.

Numerical scheme based on particle methods.

Abstract

The hydrodynamic limit for a kinetic model of chemotaxis is investigated. The limit equation is a non local conservation law, for which finite time blow-up occurs, giving rise to measure-valued solutions and discontinuous velocities. An adaptation of the notion of duality solutions, introduced for linear equations with discontinuous coefficients, leads to an existence result. Uniqueness is obtained through a precise definition of the nonlinear flux as well as the complete dynamics of aggregates, i.e. combinations of Dirac masses. Finally a particle method is used to build an adapted numerical scheme.