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Advection-diffusion
equations with density constraints
Alpár Richárd Mészáros and Filippo Santambrogio |
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Vol. 9 (2016), No. 3, 615–644
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Abstract |
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In the spirit of the macroscopic crowd motion models with hard congestion (i.e., a strong density constraint ) 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 ) 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
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Mathematical Subject Classification 2010
Primary: 35K61, 49J40, 49J45
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Milestones
Received: 10 March 2015
Revised: 27 October 2015
Accepted: 9 February 2016
Published: 17 June 2016
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Authors |
| Alpár Richárd Mészáros | |
| Department of Mathematics University of California 520 Portola Plaza Los Angeles, CA 90095 United States |
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| Filippo Santambrogio | |
| Laboratoire de Mathématiques
d’Orsay Université Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay Cedex France |
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