# Crowd movements: gradient flow in Wasserstein spaces under density constraints.

##
Filippo Santambrogio
(Institut Camille Jordan, Université Claude Bernard - Lyon 1)

created by depascal on 11 Oct 2009

14 oct 2009

**Abstract.**

Next Seminar:
Prof. Filippo Santambrogio (Univ. de Paris Dauphine)
Wednesday 14 october 2009
At 17pm Aula Seminari (Dipartimento di Matematica)

Crowd movements: gradient flow in Wasserstein spaces under density constraints.

A crowd is located in a room and everybody wants to exit, thus moving towards, say, a single small exit. Yet, the density $\rho$ of the crowd cannot exceed a fixed bound ($\rho\leq 1$). The actual velocity that every agent may realize is hence obtained in the following way: take the speed they would like (the unit vector pointing to the door), and project the whole velocity field in $L^2(\rho)$ on the cone of feasible velocities, i.e. those who have a positive divergence on the set where the density already saturates the constraint.

This evolution is actually a gradient flow in the space $W_2$ of probability densities for the functional which associates to any $\rho$ satisfying the density constraint the mean value $\int D d\rho$, where $D$ is the distance function to the exit, and $+\infty$ to any probability violating the same constraint.

This gradient flow approach allows to give existence results for a problem that had previously been studied in a microscopical way (with people represented by small disks, and the density constraint replaced by a non-overlap condition).