×

Existence of nonclassical solutions in a pedestrian flow model. (English) Zbl 1169.35360

Summary: The main result of this note is the existence of nonclassical solutions to the Cauchy problem for a conservation law modeling pedestrian flow. From the physical point of view, the main assumption of this model was recently experimentally confirmed in [D. Helbing, A. Johansson and H.Z. Al-Abideen, Phys. Rev. E 75, No. 4, 046109 (2007)]. Furthermore, the present model describes the fall in a door through-flow due to the rise of panic, as well as the Braess’ paradox. From the analytical point of view, this model is an example of a conservation law in which nonclassical solutions have a physical motivation and a global existence result for the Cauchy problem, with large data, is available.

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35Q35 PDEs in connection with fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amadori, D.; Colombo, R. M., Continuous dependence for 2×2 conservation laws with boundary, J. Differential Equations, 138, 2, 229-266 (1997) · Zbl 0884.35091
[2] Baiti, P.; Bressan, A., The semigroup generated by a Temple class system with large data, Differential Integral Equations, 10, 3, 401-418 (1997) · Zbl 0890.35083
[3] Bianchini, S., The semigroup generated by a Temple class system with non-convex flux function, Differential Integral Equations, 13, 10-12, 1529-1550 (2000) · Zbl 1043.35110
[4] Bressan, A., The one-dimensional Cauchy problem, (Hyperbolic Systems of Conservation Laws. Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20 (2000), Oxford University Press: Oxford University Press Oxford)
[5] Bressan, A., The front tracking method for systems of conservation laws, (Dafermos, C. M.; Feireisl, E., Handbook of Partial Differential Equations, Evolutionary Equations, Vol. I (2004), Elsevier) · Zbl 1074.35001
[6] Bressan, A.; Colombo, R. M., The semigroup generated by 2×2 conservation laws, Arch. Ration. Mech. Anal., 133, 1, 1-75 (1995) · Zbl 0849.35068
[7] Colombo, R. M., Wave front tracking in systems of conservation laws, Appl. Math., 49, 6, 501-537 (2004) · Zbl 1099.35063
[8] Colombo, R. M.; Corli, A., On a class of hyperbolic balance laws, J. Hyperbolic Differ. Equ., 1, 4, 725-745 (2004) · Zbl 1066.35055
[9] Colombo, R. M.; Goatin, P., A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234, 2, 654-675 (2007) · Zbl 1116.35087
[10] Colombo, R. M.; Groli, A., On the initial boundary value problem for Temple systems, Nonlinear Anal., 56, 4, 569-589 (2004) · Zbl 1058.35150
[11] Colombo, R. M.; Risebro, N. H., Continuous dependence in the large for some equations of gas dynamics, Comm. Partial Differential Equations, 23, 9-10, 1693-1718 (1998) · Zbl 0927.35082
[12] Colombo, R. M.; Rosini, M. D., Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28, 13, 1553-1567 (2005) · Zbl 1108.90016
[13] Crasta, G.; Piccoli, B., Viscosity solutions and uniqueness for systems of inhomogeneous balance laws, Discrete Contin. Dyn. Syst., 3, 4, 477-502 (1997) · Zbl 0949.35089
[14] Dafermos, C. M., (Hyperbolic Conservation Laws in Continuum Physics. Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325 (2005), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1078.35001
[15] Dubois, F.; Lefloch, P., Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71, 1, 93-122 (1988) · Zbl 0649.35057
[16] Helbing, D.; Farkas, I.; Vicsek, T., Simulating dynamical features of escape panic, Nature, 407, September (2000)
[17] Helbing, D.; Johansson, A.; Al-Abideen, H. Z., Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75, 4, 046109 (2007)
[18] Hoogendoorn, S.; Bovy, P. H.L., Simulation of pedestrian flows by optimal control and differential games, Optimal Control Appl. Methods, 24, 3, 153-172 (2003) · Zbl 1073.93553
[19] Hoogendoorn, S.; Bovy, P. H.L., Pedestrian route-choice and activity scheduling theory and models, Transp. Res. B, 38, 169-190 (2004)
[20] Hughes, R. L., A continuum theory for the flow of pedestrians, Transp. Res. B, 36, 507-535 (2002)
[21] Hughes, R. L., The flow of human crowds, Annual Rev. Fluid Mechanics, 35, 169-182 (2003), Palo Alto, CA · Zbl 1125.92324
[22] Lefloch, P. G., The theory of classical and nonclassical shock eaves, (Hyperbolic Systems of Conservation Laws. Hyperbolic Systems of Conservation Laws, Lectures in Mathematics ETH Zürich (2002), Birkhäuser Verlag: Birkhäuser Verlag Basel)
[23] M.D. Rosini, Nonclassical interactions portrait in a pedestrian flow model, J. Differential Equations (in press). http://www.dmf.unicatt.it/cgi-bin/preprintserv/semmat/Quad2008n05; M.D. Rosini, Nonclassical interactions portrait in a pedestrian flow model, J. Differential Equations (in press). http://www.dmf.unicatt.it/cgi-bin/preprintserv/semmat/Quad2008n05 · Zbl 1171.35074
[24] Tajima, Y.; Nagatani, T., Scaling behavior of crowd flow outside a hall, Physica A, 292, 545-554 (2001) · Zbl 0972.90011
[25] Takimoto, K.; Nagatani, T., Spatio-temporal distribution of escape time in evacuation process, Physica A, 320, 611-621 (2003) · Zbl 1010.90008
[26] Yu, W.; Johansson, A., Modeling crowd turbulence by many-particle simulations, Phy. Rev. E, 76, 4, 046105 (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.