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


Reference:

Di Francesco, M., Markowich, P.A., Pietschmann, J.-F. and Wolfram, M.-T., 2011. On the Hughes' model for pedestrian flow: the one-dimensional case. Journal of Differential Equations, 250 (3), pp. 1334-1362.

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    Official URL:

    http://dx.doi.org/10.1016/j.jde.2010.10.015

    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.

    Details

    Item Type Articles
    CreatorsDi Francesco, M., Markowich, P.A., Pietschmann, J.-F. and Wolfram, M.-T.
    DOI10.1016/j.jde.2010.10.015
    DepartmentsFaculty of Science > Mathematical Sciences
    Publisher Statementcrowd.pdf: NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Differential Equations. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Differential Equations, vol 250, issue 3, 2011, DOI 10.1016/j.jde.2010.10.015
    RefereedYes
    StatusPublished
    ID Code32322

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