Calculus of Variations and Geometric Measure Theory

A. R. Mészáros - F. Santambrogio

Advection-diffusion equations with density constraints

created by mészáros on 08 Mar 2015
modified by santambro on 12 Feb 2016


Accepted Paper

Inserted: 8 mar 2015
Last Updated: 12 feb 2016

Journal: Analysis and PDEs
Year: 2016

new version (the title has changed according to a referee's suggestion)


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 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, 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.