NONLOCAL CROWD DYNAMICS MODELS FOR SEVERAL POPULATIONS

doi:10.1016/S0252-9602(12)60011-3

Abstract

Abstract:

This paper develops the basic analytical theory related to some recently intro-duced crowd dynamics models. Where well posedness was known only locally in time, it is here extended to all of R⁺ . The results on the stability with respect to the equations are improved. Moreover, here the case of several populations is considered, obtaining the well posedness of systems of multi-D non-local conservation laws. The basic analytical tools are provided by the classical Kruˇzkov theory of scalar conservation laws in several space dimensions.

Key words: hyperbolic conservation laws, nonlocal flow, pedestrian traffic

CLC Number:

35L65

Rinaldo M. Colombo, Magali L′ecureux-Mercier. NONLOCAL CROWD DYNAMICS MODELS FOR SEVERAL POPULATIONS[J].Acta mathematica scientia,Series B, 2012, 32(1): 177-196.

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References

[1] Appert-Rolland C, Degond P, Motsch S. Two-way multi-lane traffic model for pedestrians in corridors.
Networks and Heterogeneous Media, 2011, 6(3): 351–381

[2] Bellomo N, Dogb′e C. On the modelling of tra?c and crowds - a survey of models, speculations, and
perspectives. SIAM Review, 2011, 53(3): 409–463

[3] Bressan A, Guerra G. Shift-di?erentiability of the flow generated by a conservation law. Discrete Contin
Dynam Systems, 1997, 3(1): 35–58

[4] Colombo R M, Facchi G, Maternini G, Rosini M D. On the continuum modeling of crowds//Proceedings of
Hyp2008 -the twelfth International Conference on HyperbolicProblemsheld inthe Universityof Maryland,
College Park, June 2008. Hyperbolic Problems: Theory, Numerics and Applications. Acer Math Soc, 2009:
517–526

[5] Colombo R M, Garavello M, L′ecureux-Mercier M. A class of non-local models for pedestrian traffic. Preprint, 2010

[6] Colombo R M, Herty M, Mercier M. Control of the continuity equation with a non local flow. ESAIM: Control Optim Calcu Var, 2011, 17(2): 353–379

[7] Colombo R M, Mercier M, Rosini M D. Stability and total variation estimates on general scalar balance laws. Commun Math Sci, 2009, 7(1): 37–65

[8] Crippa G, L′ecureux-Mercier M. Existence and uniqueness of measure solutions for a system of continuity
equations with non-local flow. Preprint, 2011

[9] Cristiani E, Piccoli B, Tosin A. Multiscale modeling of granular flows with application to crowd dynamics.
Multiscale Modeling & Simulation, 2011, 9(1): 155–182

[10] Daamen W, Hoogendoorn S. Experimental research of pedestrian walking behavior//Transportation Re-
search Board annual meeting 2003. National Academy Press, 2007: 1–16

[11] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Volume 325 of Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 2nd ed. Berlin: Springer-Verlag, 2005

[12] Helbing D, Moln′ar P, Farkas I, Bolay K. Self-organizing pedestrian movement. Environment and Planning
B: Planning and Design, 2001, 28: 361–383

[13] Kruˇzkov S N. First order quasilinear equations with several independent variables. Mat Sb (N S), 1970,
81(123): 228–255

[14] L′ecureux-Mercier M. Improved stability estimates on general scalar balance laws. To appear on J of
Hyperbolic Differential Equations, 2011

[15] Levy R, Shearer M. The motion of a thin liquid film driven by surfactant and gravity. SIAM J Appl Math, 2006, 66(5): 1588–1609