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January  2015, 35(1): 323-339. doi: 10.3934/dcds.2015.35.323

Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics

Yongki Lee 1, and  Hailiang Liu 1,

1. 

Department of Mathematics, Iowa State University, Ames, IA 50011, United States

Received  January 2014 Revised  February 2014 Published  August 2014

We investigate a class of nonlocal conservation laws with the nonlinear advection coupling both local and nonlocal mechanism, which arises in several applications such as the collective motion of cells and traffic flows. It is proved that the $C^1$ solution regularity of this class of conservation laws will persist at least for a short time. This persistency may continue as long as the solution gradient remains bounded. Based on this result, we further identify sub-thresholds for finite time shock formation in traffic flow models with Arrhenius look-ahead dynamics.
Keywords: traffic flows., shock formation, critical threshold, Nonlocal conservation laws, well-posedness.
Mathematics Subject Classification: Primary: 35L65; Secondary: 35L6.
Citation: Yongki Lee, Hailiang Liu. Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 323-339. doi: 10.3934/dcds.2015.35.323
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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