Calculus of Variations and Geometric Measure Theory

B. Maury - A. Roudneff-Chupin - F. Santambrogio

A macroscopic crowd motion model of gradient flow type

created by santambro on 18 Sep 2009
modified on 22 Jul 2011


Published Paper

Inserted: 18 sep 2009
Last Updated: 22 jul 2011

Journal: M3AS
Volume: 20
Number: 10
Pages: 1787-1821
Year: 2010

revised version, with Section 6 on extensions and limits of the model


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 a density $\rho(t,x)$. If a gradient structure is given, say $U=-\nabla D$ 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.

Keywords: Wasserstein distance, Gradient Flow, crowd motion, continuity equation