Advection-diffusion equations with density constraints

Mészáros, Alpár Richárd; Santambrogio, Filippo

doi:10.2140/apde.2016.9.615

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Advection-diffusion equations with density constraints

Alpár Richárd Mészáros and Filippo Santambrogio

Vol. 9 (2016), No. 3, 615–644

DOI: 10.2140/apde.2016.9.615

Abstract

In the spirit of the macroscopic crowd motion models with hard congestion (i.e., a strong density constraint $ρ \leq 1$ ) 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.

Keywords

diffusive crowd motion model, Fokker–Planck equation, density constraint, optimal transportation

Mathematical Subject Classification 2010

Primary: 35K61, 49J40, 49J45

Milestones

Received: 10 March 2015

Revised: 27 October 2015

Accepted: 9 February 2016

Published: 17 June 2016

Authors

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 France

Vol. 9, No. 3, 2016