Advection-diffusion equations with density constraints

Mészáros, Alpár Richárd; Santambrogio, Filippo

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.

Citation

Mészáros, A. R., & Santambrogio, F. (2016). Advection-diffusion equations with density constraints. Analysis & PDE, 9(3), 615-644. https://doi.org/10.2140/apde.2016.9.615

• Journal Article Type: Article
• Acceptance Date: Feb 9, 2016
• Online Publication Date: Jun 17, 2016
• Publication Date: 2016
• Deposit Date: Oct 1, 2019
• Journal: Analysis and PDE
• Print ISSN: 2157-5045
• Electronic ISSN: 1948-206X
• Publisher: Mathematical Sciences Publishers (MSP)
• Peer Reviewed: Peer Reviewed
• Volume: 9
• Issue: 3
• Pages: 615-644
• DOI: https://doi.org/10.2140/apde.2016.9.615
• Public URL: https://durham-repository.worktribe.com/output/1290228
• Related Public URLs: https://arxiv.org/abs/1503.02311