Advection-diffusion equations with density constraints

Abstract

In the spirit of the macroscopic crowd motion models with hard congestion (i.e., a strong density constraint $\rho \le 1$) introduced by Maury et al. some years ago, we analyze a variant of the same models where diffusion of the agents is also taken 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 point of view of PDEs, this corresponds to a modified Fokker–Planck equation, with an additional gradient of a pressure (only living in the saturated zone $\left\{\rho =1\right\}$) in the drift. We prove existence and some estimates, based on optimal transport techniques.