Calculus of Variations and Geometric Measure Theory

F. Santambrogio

Crowd motion and evolution PDEs under density constraints

created by santambro on 07 Jan 2018
modified on 04 Apr 2018


Accepted Paper

Inserted: 7 jan 2018
Last Updated: 4 apr 2018

Journal: proceedings of the 2017 SMAI conference
Year: 2018

This survey corresponds to some talks I recently gave in different seminars and workshops, and in particular at the SMAI 2017 French meeting.


This is a survey about the theory of density-constrained evolutions in the Wasserstein space developed by B. Maury, the author, and their collaborators as a model for crowd motion. Connections with microscopic models and other PDEs are presented, as well as several time-discretization schemes based on variational techniques, together with the main theorems guaranteeing their convergence as a tool to prove existence results. Then, a section is devoted to the uniqueness question, and a last one to different numerical methods inspired by optimal transport.

Keywords: Optimal transport, Gradient flows, hele-shaw, Entropic regularization, pedestrian movement, sweeping process, splitting schemes, augmented lagrangian, semidiscrete transport