23.05.2018 16:00 Prof. Dr. Filippo Santambrogio (Paris-Sud+TUM/JvN):
Crowd motion and evolution PDE with density constraintsMI HS 3 (Boltzmannstr. 3, 85748 Garching)

I will explain a general model to deal with the evolution of a population density ρ which is advected by a velocity field u, but is subject to a non-overcrowding constraint ρ≤1. This model (rather, a meta-model) mainly refers to the motion of a crowd of pedestrians, but can be adapted to many different situations according to how u is given or depends on ρ. Since in general u will not preserve the density constraint, the main assumption is that motion will be advected by the projection of u onto the cone of feasible velocities. This takes its inspiration from granular contact models, when the crowd is described by a collection of particles. I will present the equations, the main ideas to prove existence of solutions (in particular, using tools from optimal transport and gradient flows), and to simulate them. We will see how this continuous PDE model provides results which are stinkingly qualitatively similar to the simulations obtained by granular models, but could require a much smaller complexity. The talk summarizes joint works with several colleagues in Orsay as well as numerical methods developed both by us and by the INRIA team MOKAPLAN, and will try not to be exhaustive but just focus on the main features of the theory.