## Analysis & PDE

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

### Advection-diffusion equations with density constraints

#### Abstract

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

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

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

https://projecteuclid.org/euclid.apde/1510843264

Digital Object Identifier
doi:10.2140/apde.2016.9.615

Mathematical Reviews number (MathSciNet)
MR3518532

Zentralblatt MATH identifier
1342.35157

#### Citation

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

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