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Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics
1. | Department of Mathematics, Iowa State University, Ames, IA 50011, United States |
References:
[1] |
F. Betancourt, R. Burger, K. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity., 24 (2011), 855-885.
doi: 10.1088/0951-7715/24/3/008. |
[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.
doi: 10.4310/CMS.2008.v6.n1.a1. |
[3] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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.
doi: 10.1137/040612841. |
[7] |
S. Engelberg, H. Liu and E. Tadmor, Critical Thresholds in Euler-Poisson equations, Indiana Univ. Math. J., 50 (2001), 109-157.
doi: 10.1512/iumj.2001.50.2177. |
[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.
doi: 10.1093/qjmam/24.2.155. |
[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.
doi: 10.2991/jnmp.2005.12.s1.31. |
[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.
doi: 10.3934/nhm.2009.4.431. |
[12] |
G. Kynch, A theory of sedimentation, Trans. Fraday Soc., 48 (1952), 166-176.
doi: 10.1039/tf9524800166. |
[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.
doi: 10.3934/nhm.2011.6.681. |
[14] |
H. Liu, Wave breaking in a class of nonlocal dispersive wave equations, Journal of Nonlinear Math Phys., 13 (2006), 441-466.
doi: 10.2991/jnmp.2006.13.3.8. |
[15] |
T. Li and H. Liu, Critical thresholds in hyperbolic relaxation systems, J. Differential Equations, 247 (2009), 33-48.
doi: 10.1016/j.jde.2009.03.032. |
[16] |
H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435-466.
doi: 10.1007/s002200200667. |
[17] |
H. Liu and E. Tadmor, Critical thresholds in 2D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910.
doi: 10.1137/S0036139902416986. |
[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.
doi: 10.1098/rspa.1955.0089. |
[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.
doi: 10.1137/S0036141001386908. |
[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.
doi: 10.1016/S0960-0779(01)00200-4. |
[21] |
B. Perthame and A. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.
doi: 10.1090/S0002-9947-08-04656-4. |
[22] |
P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[23] |
P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A, 40 (1989), 7193-7196.
doi: 10.1103/PhysRevA.40.7193. |
[24] |
R. Seliger, A note on the breaking of waves, Proc. Roy. Soc. Ser. A, 303 (1968), 493-496.
doi: 10.1098/rspa.1968.0063. |
[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.
doi: 10.1137/040617790. |
[26] |
V. O. Vakhnenko, Solitons in a nonlinear model medium, J. Phys., 25 (1992), 4181-4187.
doi: 10.1088/0305-4470/25/15/025. |
[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. |
show all references
References:
[1] |
F. Betancourt, R. Burger, K. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity., 24 (2011), 855-885.
doi: 10.1088/0951-7715/24/3/008. |
[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.
doi: 10.4310/CMS.2008.v6.n1.a1. |
[3] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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.
doi: 10.1137/040612841. |
[7] |
S. Engelberg, H. Liu and E. Tadmor, Critical Thresholds in Euler-Poisson equations, Indiana Univ. Math. J., 50 (2001), 109-157.
doi: 10.1512/iumj.2001.50.2177. |
[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.
doi: 10.1093/qjmam/24.2.155. |
[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.
doi: 10.2991/jnmp.2005.12.s1.31. |
[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.
doi: 10.3934/nhm.2009.4.431. |
[12] |
G. Kynch, A theory of sedimentation, Trans. Fraday Soc., 48 (1952), 166-176.
doi: 10.1039/tf9524800166. |
[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.
doi: 10.3934/nhm.2011.6.681. |
[14] |
H. Liu, Wave breaking in a class of nonlocal dispersive wave equations, Journal of Nonlinear Math Phys., 13 (2006), 441-466.
doi: 10.2991/jnmp.2006.13.3.8. |
[15] |
T. Li and H. Liu, Critical thresholds in hyperbolic relaxation systems, J. Differential Equations, 247 (2009), 33-48.
doi: 10.1016/j.jde.2009.03.032. |
[16] |
H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435-466.
doi: 10.1007/s002200200667. |
[17] |
H. Liu and E. Tadmor, Critical thresholds in 2D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910.
doi: 10.1137/S0036139902416986. |
[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.
doi: 10.1098/rspa.1955.0089. |
[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.
doi: 10.1137/S0036141001386908. |
[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.
doi: 10.1016/S0960-0779(01)00200-4. |
[21] |
B. Perthame and A. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.
doi: 10.1090/S0002-9947-08-04656-4. |
[22] |
P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[23] |
P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A, 40 (1989), 7193-7196.
doi: 10.1103/PhysRevA.40.7193. |
[24] |
R. Seliger, A note on the breaking of waves, Proc. Roy. Soc. Ser. A, 303 (1968), 493-496.
doi: 10.1098/rspa.1968.0063. |
[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.
doi: 10.1137/040617790. |
[26] |
V. O. Vakhnenko, Solitons in a nonlinear model medium, J. Phys., 25 (1992), 4181-4187.
doi: 10.1088/0305-4470/25/15/025. |
[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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