Communications in Mathematical Sciences

Volume 16 (2018)

Number 4

Anisotropic challenges in pedestrian flow modeling

Pages: 1067 – 1093



Elliot Cartee (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.)

Alexander Vladimirsky (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.)


Macroscopic models of crowd flow incorporating individual pedestrian choices present many analytic and computational challenges. Anisotropic interactions are particularly subtle, both in terms of describing the correct “optimal” direction field for the pedestrians and ensuring that this field is uniquely defined. We develop sufficient conditions, which establish a range of “safe” densities and parameter values for each model. We illustrate our approach by analyzing several established intracrowd and inter-crowd models. For the two-crowd case, we also develop sufficient conditions for the uniqueness of Nash Equilibria in the resulting non-zero-sum game.


pedestrian dynamics, conservation laws, Hamilton–Jacobi equations, Hughes’ model, anisotropic speed profiles, minimum time problem, non-zero-sum differential games, Nash equilibrium

2010 Mathematics Subject Classification

35L65, 49N90, 65N22, 91D10

Both authors’ work was supported in part by the National Science Foundation grants DMS-1016150 and DMS-1738010. The first author’s work was also supported by the National Science Foundation grants DMS-0739164 and DMS-1645643.

Received 19 June 2017

Accepted 27 March 2018

Published 31 October 2018