Vol. 9, No. 3, 2016

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Advection-diffusion equations with density constraints

Alpár Richárd Mészáros and Filippo Santambrogio

Vol. 9 (2016), No. 3, 615–644

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 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 {ρ = 1}) in the drift. We prove existence and some estimates, based on optimal transport techniques.

diffusive crowd motion model, Fokker–Planck equation, density constraint, optimal transportation
Mathematical Subject Classification 2010
Primary: 35K61, 49J40, 49J45
Received: 10 March 2015
Revised: 27 October 2015
Accepted: 9 February 2016
Published: 17 June 2016
Alpár Richárd Mészáros
Department of Mathematics
University of California
520 Portola Plaza
Los Angeles, CA 90095
United States
Filippo Santambrogio
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud, CNRS, Université Paris-Saclay
91405 Orsay Cedex