Analysis & PDE
- Anal. PDE
- Volume 9, Number 3 (2016), 615-644.
Advection-diffusion equations with density constraints
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.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.apde/1510843264