Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 3 (2016), 615-644.

Advection-diffusion equations with density constraints

Alpár Richárd Mészáros and Filippo Santambrogio

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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.

Article information

Anal. PDE, Volume 9, Number 3 (2016), 615-644.

Received: 10 March 2015
Revised: 27 October 2015
Accepted: 9 February 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 49J40: Variational methods including variational inequalities [See also 47J20] 49J45: Methods involving semicontinuity and convergence; relaxation

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


Mészáros, Alpár Richárd; Santambrogio, Filippo. Advection-diffusion equations with density constraints. Anal. PDE 9 (2016), no. 3, 615--644. doi:10.2140/apde.2016.9.615.

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