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

Handle URI:
http://hdl.handle.net/10754/599056
Title:
On the Hughes' model for pedestrian flow: The one-dimensional case
Authors:
Di Francesco, Marco; Markowich, Peter A.; Pietschmann, Jan-Frederik; Wolfram, Marie-Therese
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.
Publisher:
Elsevier BV
Journal:
Journal of Differential Equations
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Feb-2011
DOI:
10.1016/j.jde.2010.10.015
Type:
Article
ISSN:
0022-0396
Sponsors:
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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorDi Francesco, Marcoen
dc.contributor.authorMarkowich, Peter A.en
dc.contributor.authorPietschmann, Jan-Frederiken
dc.contributor.authorWolfram, Marie-Thereseen
dc.date.accessioned2016-02-25T13:52:01Zen
dc.date.available2016-02-25T13:52:01Zen
dc.date.issued2011-02en
dc.identifier.citationDi 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.en
dc.identifier.issn0022-0396en
dc.identifier.doi10.1016/j.jde.2010.10.015en
dc.identifier.urihttp://hdl.handle.net/10754/599056en
dc.description.abstractIn 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.en
dc.description.sponsorshipThis 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.en
dc.publisherElsevier BVen
dc.subjectCharacteristicsen
dc.subjectEikonal equationen
dc.subjectElliptic couplingen
dc.subjectEntropy solutionsen
dc.subjectPedestrian flowen
dc.subjectScalar conservation lawsen
dc.titleOn the Hughes' model for pedestrian flow: The one-dimensional caseen
dc.typeArticleen
dc.identifier.journalJournal of Differential Equationsen
dc.contributor.institutionUniversita degli Studi dell'Aquila, L'Aquila, Italyen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionUniversitat Wien, Vienna, Austriaen
kaust.grant.numberKUK-I1-007-43en
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