Advection-diffusion equations with density constraints

TL;DR

This paper studies a modified crowd motion model incorporating diffusion and density constraints, leading to a PDE with a pressure term, and proves existence and estimates using optimal transport methods.

Contribution

It introduces a new PDE model combining diffusion with density constraints and provides existence results using optimal transport techniques.

Findings

Proved existence of solutions for the modified PDE model.

Derived estimates for solutions based on optimal transport.

Extended previous models to include diffusion effects.

Abstract

In the spirit of the macroscopic crowd motion models with hard congestion (i.e. a strong density constraint $\rho\leq 1$) introduced by Maury {\it 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 PDE point of view, this corresponds to a modified Fokker-Planck equation, with an additional gradient of a pressure (only living in the saturated zone $\{\rho=1\}$) in the drift. The paper proves existence and some estimates, based on optimal transport techniques.