A stiff limit of non-homogeneous conservation laws for crowd motion modeling

TL;DR

This paper introduces a new conservation law-based model for crowd motion that limits agent speed in saturated areas, deriving a novel PDE through asymptotic analysis and providing theoretical and numerical insights.

Contribution

It develops a new crowd motion model based on a limit of conservation laws, proving existence, deriving entropy inequalities, and analyzing qualitative behavior.

Findings

Established uniform BV estimates for the density

Proved existence of solutions to the new PDE

Provided numerical illustrations in 1D and 2D

Abstract

We propose a new approach for crowd motion models where the density constraint can only slow down the motion of each agent, with no effect on those agents who are not in a saturated area or who have no saturated density ''in front'' of them. This is done by means of a limit of conservation laws inspired by the equations used for traffic as in Follow the leader-type models. We study the asymptotics of the solutions of these conservation laws in a certain asymptotic regime, and obtain a PDE at the limit of a whole new type. One of the main goals of the paper is to prove uniform BV estimates on the density, and thus strong compactness to prove the existence of solutions to this limit equation. We also discuss the qualitative behavior of solutions, provide numerical illustrations both in dimension 1 and 2, and establish the new entropy inequalities associated with this limit equation.