A diffusive model for macroscopic crowd motion with density constraints, (2015), preprint available at http://cvgmt.sns.it/paper/2644

Abstract. In the spirit of the macroscopic crowd motion models with hard congestion (i.e. a strong density constraint ρ ≤ 1) introduced by Maury et al. some years ago, we analyze a variant of the same models where diffusion of the agents is also take into account. From the modeling point of view, this means that individuals try to follow a given spontaneous velocity, but are subject to a Brownian diffusion, and have to adapt to a density constraint which introduces a pressure term affecting the movement. From the PDE point of view, this corresponds to a modified Fokker-Planck equation, where with an additional gradient of a pressure (only living in the saturated zone {ρ = 1}) in the drift. The paper proves existence and some estimates, based on optimal transport techniques. 1