On the Hughes' model for pedestrian flow: The one-dimensional case

Type
Article

Authors
Di Francesco, Marco
Markowich, Peter A.
Pietschmann, Jan-Frederik
Wolfram, Marie-Therese

KAUST Grant Number
KUK-I1-007-43

Date
2011-02

Abstract
In this paper we investigate the mathematical theory of Hughes' model for the flow of pedestrians (cf. Hughes (2002) [17]), consisting of a non-linear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task. For such an approximated system we prove existence and uniqueness of entropy solutions (in one space dimension) in the sense of Kružkov (1970) [22], in which the boundary conditions are posed following the approach of Bardos et al. (1979) [7]. We use BV estimates on the density ρ and stability estimates on the potential Π in order to prove uniqueness. Furthermore, we analyze the evolution of characteristics for the original Hughes' model in one space dimension and study the behavior of simple solutions, in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves. The characteristic calculus is supported by numerical simulations. © 2010 Elsevier Inc.

Citation
Di Francesco M, Markowich PA, Pietschmann J-F, Wolfram M-T (2011) On the Hughes’ model for pedestrian flow: The one-dimensional case. Journal of Differential Equations 250: 1334–1362. Available: http://dx.doi.org/10.1016/j.jde.2010.10.015.

Acknowledgements
This publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST), by the Leverhulme Trust through the research grant entitled Kinetic and mean field partial differential models for socio-economic processes (PI Peter Markowich) and by the Royal Society through the Wolfson Research Merit Award of Peter Markowich. PM is also grateful to the Humboldt foundation for their support. MDF is partially supported by the Italian MIUR under the PRIN program 'Nonlinear Systems of Conservation Laws and Fluid Dynamics'. Furthermore, the authors thank Martin Burger and the Institute for Computational and Applied Mathematics at the University of Munster for their kind hospitality and stimulating discussions.

Publisher
Elsevier BV

Journal
Journal of Differential Equations

DOI
10.1016/j.jde.2010.10.015

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