A macroscopic crowd motion model of gradient flow type

TL;DR

This paper develops a macroscopic gradient flow model for crowd movement during emergency evacuations, incorporating density constraints and analyzing convergence of a discrete scheme.

Contribution

It introduces a novel macroscopic gradient flow framework for crowd dynamics with density constraints, extending previous microscopic models.

Findings

Formulates a gradient flow model in Wasserstein space for crowd motion.

Addresses non-convexity and constraints in the functional governing the flow.

Provides analysis of convergence for a discrete approximation scheme.

Abstract

A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknwon is the evolution of the density . If a gradient structure is given, say U is the opposite of the gradient of D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad-hoc study of the convergence of a discrete scheme.