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

Marco Di Francesco, P.A. Markowich, J.-F. Pietschmann, M.-T. Wolfram

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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.
LanguageEnglish
Pages1334-1362
Number of pages29
JournalJournal of Differential Equations
Volume250
Issue number3
Early online date26 Oct 2010
DOIs
StatusPublished - 1 Feb 2011

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Pedestrian Flow
Eikonal Equation
Rarefaction Wave
Entropy Solution
Stability Estimates
Shock Waves
Conservation Laws
Calculus
Existence and Uniqueness
Uniqueness
Boundary conditions
Numerical Simulation
Modeling
Model
Estimate

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Di Francesco, M., Markowich, P. A., Pietschmann, J-F., & Wolfram, M-T. (2011). On the Hughes' model for pedestrian flow: the one-dimensional case. DOI: 10.1016/j.jde.2010.10.015

On the Hughes' model for pedestrian flow: the one-dimensional case. / Di Francesco, Marco; Markowich, P.A.; Pietschmann, J.-F.; Wolfram, M.-T.

In: Journal of Differential Equations, Vol. 250, No. 3, 01.02.2011, p. 1334-1362.

Research output: Contribution to journalArticle

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, vol. 250, no. 3, pp. 1334-1362. DOI: 10.1016/j.jde.2010.10.015
Di Francesco M, Markowich PA, Pietschmann J-F, Wolfram M-T. On the Hughes' model for pedestrian flow: the one-dimensional case. Journal of Differential Equations. 2011 Feb 1;250(3):1334-1362. Available from, DOI: 10.1016/j.jde.2010.10.015
Di Francesco, Marco ; Markowich, P.A. ; Pietschmann, J.-F. ; Wolfram, M.-T./ On the Hughes' model for pedestrian flow: the one-dimensional case. In: Journal of Differential Equations. 2011 ; Vol. 250, No. 3. pp. 1334-1362
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