[1] Bardos C, Leroux A Y, Nedelec J C. First order quasilinear equations with boundary conditions. Comm
Partial Di?er Equ, 1979, 4: 1017–1034
[2] Bellomo N, Dogb′e C. On the modelling crowd dynamics from scaling to hyperbolic macroscopic models.
Math Models Methods Appl Sci, 2008, 18(suppl): 1317–1345
[3] Betancourt F, B¨urger R, Karlsen K H, Tory E M. On nonlocal conservation laws modelling sedimentation.
Nonlinearity, 2011, 24: 855–885
[4] Bradley G. A proposed mathematical model for computer prediction of crowd movements and their as-
sociated risks//Smith R A, Dickie J F, eds. Engineering for Crowd Safety. Amsterdam: Elsevier, 1993:303–311
[5] Bressan A. Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem. Oxford
University Press, 2000
[6] Bressan A, Colombo R M. P.D.E. models of pedestrian flow. Unpublished, 2007
[7] Burger M, Markowich P A, Pietschmann J -F. Continuous limit of a crowd motion and herding model:
analysis and numerical simulations. Preprint, 2011
[8] Buttazzo G, Jimenez C, Oudet E. An optimization problem for mass transportation with congested dy-
namics. SIAM J Control Optim, 2009, 48: 1961–1976
[9] Cannarsa P, Sinestrari C. Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control.
Vlume 58 of Progress in Nonlinear Di?erential Equations and Their Applications. Birkh¨auser, 2004
[10] Colombo R M, Garavello M, Mercier M. A class of non local models for pedestrian traffc. Preprint, 2011
[11] Colombo R M, Rosini M D. Pedestrian flows and non-classical shocks. Math Methods Appl Sci, 2005,
28(13): 1553–1567
[12] Colombo R M, Rosini M D. Existence of nonclassical solutions in a pedestrian flow model. Nonlinear Anal
Real World Appl, 2009, 10(5): 2716–2728
[13] Cristiani E, Piccoli B, Tosin A. Multiscale modeling of granular flows with applications to crowd dynamics.
Multiscale Modeling and Simulations, 2011, 9: 155–182
[14] Dafermos C M. Polygonal approximations of solutions of the initial value problem for a conservation law.
J Math Anal Appl, 1972, 38: 33–41
[15] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. 3rd ed. Springer-Verlag, 2010
[16] Di Francesco M, Markowich P A, Pietschmann J -F, Wolfram M T. On the Hughes’ model for pedestrian
flow: The one-dimensional case. J Differ Equ, 2011, 250: 1334–1362
[17] Dogb′e C. Modeling crowd dynamics by the mean-field limit approach. Math Comp Model, 2010, 52:
1506–1520
[18] Dubois F, LeFloch P. Boundary conditions for nonlinear hyperbolic systems of conservation laws. J Di?er
Equ, 1988, 71(1): 93–122
[19] Fukui M, Ishibashi Y. Self-organized phase transitions in CA-models for pedestrians. J Phys Soc Japan,
1999, 8: 2861–2863
[20] Helbing D. Tra?c and related self-driven many-particle systems. Rev Mod Phys, 2001, 73(4): 1067–1141
[21] Helbing D, Farkas I J, Molnar P, Vicsek T. Simulation of pedestrian crowds in normal and evacuation
situations//Schreckenberg M, Sharma S D, eds. Pedestrian and Evacuation Dynamics. Berlin: Springer,
2002: 21–58
[22] Henderson L F. The statistics of crowd fluids. Nature, 1971, 229: 381–383
[23] Henderson L F. On the fluids mechanics of human crowd motion. Transp Res, 1974, 8: 509–515
[24] Hughes R L. A continuum theory for the flow of pedestrians. Transportation Research Part B: Method-
ological, 2002, 36(6): 507–535
[25] Hughes R L. The flow of human crowds. Annu Rev Fluid Mech, 2003, 35: 169–182
[26] Klausen R A, Risebro N H. Stability of conservation laws with discontinuous coe?cients. J Differ Equ,
1999, 157: 41–60
[27] Kruˇzkov S N. First order quasilinear equations in several independent variables. Math USSR Sb, 1970,
10: 217–243
[28] Kruˇzkov S N. Generalized solutions of the Hamilton-Jacobi equations of eikonal type. i. formulation of
the problems; existence, uniqueness and stability theorems; some properties of the solutions. Math USSR
Sb, 1975, 27(3): 406–446
[29] Lax P D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. CBMS
Regional Conference Series in Applied Mathematics, No 11. Philadelphia: SIAM, 1973
[30] Lighthill M J, Whitham G B. On kinematic waves. ii. a theory of traffc flow on long crowded roads.
Royal Society of London Proceedings Series A, 1955, 229: 317–345
[31] Maury B, Roudneff-Chupin A, Santambrogio F. A macroscopic crowd motion model of the gradient-flow
type. Math Model Meth Appl Sci, 2010, 20: 1787–1821
[32] Muramatsu M, Nagatani T. Jamming transition in two-dimensional pedestrian traffc. Physica A, 2000,
275: 281–291
[33] Oleininik O A. Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear
equation. Uspehi Mat Nauk, 1959, 14(2(86)): 165–170
[34] Piccoli B, Tosin A. Pedestrian flows in bounded domains with obstacles. Contin Mech Thermodyn, 2009,
21(2): 85–107
[35] Schadschneider A, Klingsch W, Kluepfel H, Kretz T, Rogsch C, Seyfried A. Encyclopedia of Complexity
and System Science. Meyers R A, ed. Vol 3, Chapter Evacuation Dynamics: Empirical Results, Modeling
and Applications. Springer, 2009
[36] Whitham G. Linear and Nonlinear Waves. New York: Wiley & Sons, 1974
|