Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics
Pages: 323 - 339, Issue 1, January 2015

doi:10.3934/dcds.2015.35.323      Abstract        References        Full text (470.5K)           Related Articles

Yongki Lee - Department of Mathematics, Iowa State University, Ames, IA 50011, United States (email)
Hailiang Liu - Department of Mathematics, Iowa State University, Ames, IA 50011, United States (email)

1 F. Betancourt, R. Burger, K. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity., 24 (2011), 855-885.       
2 M. Burger, Y. Dolak and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Commun. Math. Sci., 6 (2008), 1-28.       
3 A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.       
4 C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, 2005.       
5 A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Scientific, River Edge, NJ, 1999, 23-37.       
6 Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308.       
7 S. Engelberg, H. Liu and E. Tadmor, Critical Thresholds in Euler-Poisson equations, Indiana Univ. Math. J., 50 (2001), 109-157.       
8 K. Hamer, Non-linear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168.
9 D. D. Holm and A. N. W. Hone, A class of equations with peakon and pulson solutions (with a appendix by Braden H and Byatt-Smith), J. Nonlinear Math. Phys, 12 (2005), 380-394.       
10 J. K. Hunter, Numerical solutions of some nonlinear dispersive wave equations, Lect. Appl. Math, 26 (1990), 301-316.       
11 A. Kurganov and A. Polizzi, Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics, Netw. Heterog. Media., 4 (2009), 431-451.       
12 G. Kynch, A theory of sedimentation, Trans. Fraday Soc., 48 (1952), 166-176.
13 D. Li and T. Li, Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Netw. Heterog. Media, 6 (2001), 681-694.       
14 H. Liu, Wave breaking in a class of nonlocal dispersive wave equations, Journal of Nonlinear Math Phys., 13 (2006), 441-466.       
15 T. Li and H. Liu, Critical thresholds in hyperbolic relaxation systems, J. Differential Equations, 247 (2009), 33-48.       
16 H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435-466.       
17 H. Liu and E. Tadmor, Critical thresholds in 2D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910.       
18 M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc., London, Ser. A, 229 (1955), 317-345.       
19 H. L. Liu and E. Tadmor, Critical Thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal. 33 (2001), 930-945.       
20 E. J. Parkes and V. O. Vakhneko, The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method, Chaos Solitons Fractals., 13 (2002), 1819-1826.       
21 B. Perthame and A. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.       
22 P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.       
23 P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A, 40 (1989), 7193-7196.       
24 R. Seliger, A note on the breaking of waves, Proc. Roy. Soc. Ser. A, 303 (1968), 493-496.
25 A. Sopasakis and M. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), 921-944.       
26 V. O. Vakhnenko, Solitons in a nonlinear model medium, J. Phys., 25 (1992), 4181-4187.       
27 G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1974.       
28 K. Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Q. Appl. Math, 57 (1999), 573-600.       

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