Congestion in a macroscopic model of self-driven particles modeling gregariousness

TL;DR

This paper studies a macroscopic model of self-driven particles with a maximal density constraint, focusing on the transition between compressible and incompressible regions to understand congestion phenomena in biological systems.

Contribution

It introduces a regularization approach for the compressible-incompressible transition in a macroscopic particle model, addressing non-conservativity issues.

Findings

Analysis of the transition between compressible and incompressible regions.

Development of a perturbation method to regularize the transition.

Insights into congestion dynamics in biological systems.

Abstract

We analyze a macroscopic model with a maximal density constraint which describes short range repulsion in biological systems. This system aims at modeling finite-size particles which cannot overlap and repel each other when they are too close. The parts of the fluid where the maximal density is reached behave like incompressible fluids while lower density regions are compressible. This paper investigates the transition between the compressible and incompressible regions. To capture this transition, we study a one-dimensional Riemann problem and introduce a perturbation problem which regularizes the compressible-incompressible transition. Specific difficulties related to the non-conservativity of the problem are discussed.