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September  2013, 8(3): 773-781. doi: 10.3934/nhm.2013.8.773

## Qualitative analysis of some PDE models of traffic flow

 1 Department of Mathematics, University of Iowa, Iowa City, IA 52242

Received  February 2012 Revised  June 2013 Published  October 2013

We review our previous results on partial differential equation(PDE) models of traffic flow. These models include the first order PDE models, a nonlocal PDE traffic flow model with Arrhenius look-ahead dynamics, and the second order PDE models, a discrete model which captures the essential features of traffic jams and chaotic behavior. We study the well-posedness of such PDE problems, finite time blow-up, front propagation, pattern formation and asymptotic behavior of solutions including the stability of the traveling fronts. Traveling wave solutions are wave front solutions propagating with a constant speed and propagating against traffic.
Citation: Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks & Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773
##### References:
 [1] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916.  doi: 10.1137/S0036139997332099.  Google Scholar [2] M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation,, Phys. Rev. E, 51 (1995), 1035.   Google Scholar [3] N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives,, SIAM Rev., 53 (2011), 409.  doi: 10.1137/090746677.  Google Scholar [4] N. Bellomo and V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow,, C. R. Mecanique, 333 (2005), 843.  doi: 10.1016/j.crme.2005.09.004.  Google Scholar [5] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams,, Arch. Rational Mech. Anal., 187 (2008), 185.  doi: 10.1007/s00205-007-0061-9.  Google Scholar [6] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle and J. Royer, A traffic-flow model with constraints for the modelling of traffic jams,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1269.  doi: 10.1142/S0218202508003030.  Google Scholar [7] V. Coscia, On a closure of mass conservation equation and stability analysis in the mathematical theory of vehicular traffic flow,, C. R. Mecanique, 332 (2004), 585.  doi: 10.1016/j.crme.2004.03.016.  Google Scholar [8] C. Daganzo, Requiem for second-order approximations of traffic flow,, Transportation Research, 29 (1995), 277.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar [9] P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation,, Kinetic Related Models, 1 (2008), 279.  doi: 10.3934/krm.2008.1.279.  Google Scholar [10] D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Physical Review E, 51 (1995), 3154.  doi: 10.1103/PhysRevE.51.3164.  Google Scholar [11] D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phy., 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar [12] D. Helbing, Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models,, Eur. Phys. J. B, 69 (2009), 539.   Google Scholar [13] S. Jin and Jian-Guo Liu, Relaxation and diffusion enhanced dispersive waves,, Proceedings: Mathematical and Physical Sciences, 446 (1994), 555.   Google Scholar [14] W. L. Jin and H. M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model,, Transportation Research, 37 (2003), 207.  doi: 10.1016/S0191-2615(02)00008-5.  Google Scholar [15] B.S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Physical Review E, 50 (1994), 54.  doi: 10.1103/PhysRevE.50.54.  Google Scholar [16] A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60 (2000), 1749.  doi: 10.1137/S0036139999356181.  Google Scholar [17] R. D. Kühne, Macroscopic freeway model for dense traffic-stop-start waves and incident detection,, Ninth International Symposium on Transportation and Traffic Theory, (1984), 21.   Google Scholar [18] A. Kurganov and A. Polizzi, Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics,, Netw. Heterog. Media, 4 (2009), 431.  doi: 10.3934/nhm.2009.4.431.  Google Scholar [19] D. A. Kurtze and D. C. Hong, Traffic jams, granular flow, and soliton selection,, Phys. Rev. E, 52 (1995), 218.  doi: 10.1103/PhysRevE.52.218.  Google Scholar [20] H. Y. Lee, H.-W. Lee and D. Kim, Steady-state solutions of hydrodynamic traffic models,, Phys. Rev. E, 69 (2004).  doi: 10.1103/PhysRevE.69.016118.  Google Scholar [21] Dong Li and Tong Li, Shock formation in a traffic flow model with arrhenius look-ahead dynamics,, Networks and Heterogeneous Media, 6 (2011), 681.  doi: 10.3934/nhm.2011.6.681.  Google Scholar [22] Tong Li, Global solutions and zero relaxation limit for a traffic flow model,, SIAM J. Appl. Math., 61 (2000), 1042.  doi: 10.1137/S0036139999356788.  Google Scholar [23] Tong Li, $L^1$ stability of conservation laws for a traffic flow model,, Electron. J. Diff. Eqns., 2001 ().   Google Scholar [24] Tong Li, Well-posedness theory of an inhomogeneous traffic flow model,, Discrete and Continuous Dynamical Systems, 2 (2002), 401.  doi: 10.3934/dcdsb.2002.2.401.  Google Scholar [25] Tong Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow,, J. Diff. Eqns., 190 (2003), 131.  doi: 10.1016/S0022-0396(03)00014-7.  Google Scholar [26] Tong Li, Mathematical modelling of traffic flows,, Hyperbolic problems: Theory, (2003), 695.   Google Scholar [27] Tong Li, Modelling traffic flow with a time-dependent fundamental diagram,, Math. Methods Appl. Sci., 27 (2004), 583.  doi: 10.1002/mma.470.  Google Scholar [28] Tong Li, Nonlinear dynamics of traffic jams,, Physica D, 207 (2005), 41.  doi: 10.1016/j.physd.2005.05.011.  Google Scholar [29] Tong Li, Instability and formation of clustering solutions of traffic flow,, Bulletin of the Institute of Mathematics, 2 (2007), 281.   Google Scholar [30] Tong Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion,, SIAM J. Math. Anal., 40 (2008), 1058.  doi: 10.1137/070690638.  Google Scholar [31] Tong Li and Hailiang Liu, Stability of a traffic flow model with nonconvex relaxation,, Comm. Math. Sci., 3 (2005), 101.   Google Scholar [32] Tong Li and Hailiang Liu, Critical thresholds in hyperbolic relaxation systems,, J. Diff. Eqns., 247 (2009), 33.  doi: 10.1016/j.jde.2009.03.032.  Google Scholar [33] Tong Li and Hailiang Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds,, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 511.  doi: 10.3934/dcds.2009.24.511.  Google Scholar [34] Tong Li and Hailiang Liu, Critical thresholds in a relaxation model for traffic flows,, Indiana Univ. Math. J., 57 (2008), 1409.  doi: 10.1512/iumj.2008.57.3215.  Google Scholar [35] Tong Li and Yaping Wu, Linear and nonlinear exponential stability of traveling waves for hyperbolic systems with relaxation,, Comm. Math. Sci., 7 (2009), 571.   Google Scholar [36] Tong Li and H. M. Zhang, The mathematical theory of an enhanced nonequilibrium traffic flow model,, Network and Spatial Economics, 1&2 (2001), 167.   Google Scholar [37] M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc., A229 (1955), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar [38] T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331.  doi: 10.1088/0034-4885/65/9/203.  Google Scholar [39] K. Nagel, Particle hopping models and traffic flow theory,, Phys. Rev. E, 53 (1996), 4655.   Google Scholar [40] O. A. Oleinik, Discontinuous solutions of non-linear differential equations,, (Russian) Uspehi. Mat. Nauk., 12 (1957), 3.   Google Scholar [41] H. J. Payne, Models of freeway traffic and control,, "Simulation Councils Proc. Ser. : Mathematical Models of Public Systems,", 1 (1971), 51.   Google Scholar [42] I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic,", American Elsevier Publishing Company Inc., (1971).   Google Scholar [43] P. I. Richards, Shock waves on highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar [44] 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.  doi: 10.1137/040617790.  Google Scholar [45] Lina Wang, Yaping Wu and Tong Li, Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion,, Physica D, 240 (2011), 971.  doi: 10.1016/j.physd.2011.02.003.  Google Scholar [46] G. B. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar [47] Lei Yu, Tong Li and Zhong-Ke Shi, Density waves in a traffic flow model with reactive-time delay,, Physica A, 389 (2010), 2607.  doi: 10.1016/j.physa.2010.03.009.  Google Scholar [48] Lei Yu, Tong Li and Zhong-Ke Shi, The effect of diffusion in a new viscous continuum model,, Physics Letters, 374 (2010), 2346.  doi: 10.1016/j.physleta.2010.03.056.  Google Scholar [49] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transportation Research, 36 (2002), 275.  doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar [50] H. M. Zhang, Driver memory, traffic viscosity and a viscous vehicular traffic flow model,, Transportation Research, 37 (2003), 27.   Google Scholar

show all references

##### References:
 [1] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916.  doi: 10.1137/S0036139997332099.  Google Scholar [2] M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation,, Phys. Rev. E, 51 (1995), 1035.   Google Scholar [3] N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives,, SIAM Rev., 53 (2011), 409.  doi: 10.1137/090746677.  Google Scholar [4] N. Bellomo and V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow,, C. R. Mecanique, 333 (2005), 843.  doi: 10.1016/j.crme.2005.09.004.  Google Scholar [5] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams,, Arch. Rational Mech. Anal., 187 (2008), 185.  doi: 10.1007/s00205-007-0061-9.  Google Scholar [6] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle and J. Royer, A traffic-flow model with constraints for the modelling of traffic jams,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1269.  doi: 10.1142/S0218202508003030.  Google Scholar [7] V. Coscia, On a closure of mass conservation equation and stability analysis in the mathematical theory of vehicular traffic flow,, C. R. Mecanique, 332 (2004), 585.  doi: 10.1016/j.crme.2004.03.016.  Google Scholar [8] C. Daganzo, Requiem for second-order approximations of traffic flow,, Transportation Research, 29 (1995), 277.  doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar [9] P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation,, Kinetic Related Models, 1 (2008), 279.  doi: 10.3934/krm.2008.1.279.  Google Scholar [10] D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Physical Review E, 51 (1995), 3154.  doi: 10.1103/PhysRevE.51.3164.  Google Scholar [11] D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phy., 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar [12] D. Helbing, Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models,, Eur. Phys. J. B, 69 (2009), 539.   Google Scholar [13] S. Jin and Jian-Guo Liu, Relaxation and diffusion enhanced dispersive waves,, Proceedings: Mathematical and Physical Sciences, 446 (1994), 555.   Google Scholar [14] W. L. Jin and H. M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model,, Transportation Research, 37 (2003), 207.  doi: 10.1016/S0191-2615(02)00008-5.  Google Scholar [15] B.S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Physical Review E, 50 (1994), 54.  doi: 10.1103/PhysRevE.50.54.  Google Scholar [16] A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60 (2000), 1749.  doi: 10.1137/S0036139999356181.  Google Scholar [17] R. D. Kühne, Macroscopic freeway model for dense traffic-stop-start waves and incident detection,, Ninth International Symposium on Transportation and Traffic Theory, (1984), 21.   Google Scholar [18] A. Kurganov and A. Polizzi, Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics,, Netw. Heterog. Media, 4 (2009), 431.  doi: 10.3934/nhm.2009.4.431.  Google Scholar [19] D. A. Kurtze and D. C. Hong, Traffic jams, granular flow, and soliton selection,, Phys. Rev. E, 52 (1995), 218.  doi: 10.1103/PhysRevE.52.218.  Google Scholar [20] H. Y. Lee, H.-W. Lee and D. Kim, Steady-state solutions of hydrodynamic traffic models,, Phys. Rev. E, 69 (2004).  doi: 10.1103/PhysRevE.69.016118.  Google Scholar [21] Dong Li and Tong Li, Shock formation in a traffic flow model with arrhenius look-ahead dynamics,, Networks and Heterogeneous Media, 6 (2011), 681.  doi: 10.3934/nhm.2011.6.681.  Google Scholar [22] Tong Li, Global solutions and zero relaxation limit for a traffic flow model,, SIAM J. Appl. Math., 61 (2000), 1042.  doi: 10.1137/S0036139999356788.  Google Scholar [23] Tong Li, $L^1$ stability of conservation laws for a traffic flow model,, Electron. J. Diff. Eqns., 2001 ().   Google Scholar [24] Tong Li, Well-posedness theory of an inhomogeneous traffic flow model,, Discrete and Continuous Dynamical Systems, 2 (2002), 401.  doi: 10.3934/dcdsb.2002.2.401.  Google Scholar [25] Tong Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow,, J. Diff. Eqns., 190 (2003), 131.  doi: 10.1016/S0022-0396(03)00014-7.  Google Scholar [26] Tong Li, Mathematical modelling of traffic flows,, Hyperbolic problems: Theory, (2003), 695.   Google Scholar [27] Tong Li, Modelling traffic flow with a time-dependent fundamental diagram,, Math. Methods Appl. Sci., 27 (2004), 583.  doi: 10.1002/mma.470.  Google Scholar [28] Tong Li, Nonlinear dynamics of traffic jams,, Physica D, 207 (2005), 41.  doi: 10.1016/j.physd.2005.05.011.  Google Scholar [29] Tong Li, Instability and formation of clustering solutions of traffic flow,, Bulletin of the Institute of Mathematics, 2 (2007), 281.   Google Scholar [30] Tong Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion,, SIAM J. Math. Anal., 40 (2008), 1058.  doi: 10.1137/070690638.  Google Scholar [31] Tong Li and Hailiang Liu, Stability of a traffic flow model with nonconvex relaxation,, Comm. Math. Sci., 3 (2005), 101.   Google Scholar [32] Tong Li and Hailiang Liu, Critical thresholds in hyperbolic relaxation systems,, J. Diff. Eqns., 247 (2009), 33.  doi: 10.1016/j.jde.2009.03.032.  Google Scholar [33] Tong Li and Hailiang Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds,, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 511.  doi: 10.3934/dcds.2009.24.511.  Google Scholar [34] Tong Li and Hailiang Liu, Critical thresholds in a relaxation model for traffic flows,, Indiana Univ. Math. J., 57 (2008), 1409.  doi: 10.1512/iumj.2008.57.3215.  Google Scholar [35] Tong Li and Yaping Wu, Linear and nonlinear exponential stability of traveling waves for hyperbolic systems with relaxation,, Comm. Math. Sci., 7 (2009), 571.   Google Scholar [36] Tong Li and H. M. Zhang, The mathematical theory of an enhanced nonequilibrium traffic flow model,, Network and Spatial Economics, 1&2 (2001), 167.   Google Scholar [37] M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc., A229 (1955), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar [38] T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331.  doi: 10.1088/0034-4885/65/9/203.  Google Scholar [39] K. Nagel, Particle hopping models and traffic flow theory,, Phys. Rev. E, 53 (1996), 4655.   Google Scholar [40] O. A. Oleinik, Discontinuous solutions of non-linear differential equations,, (Russian) Uspehi. Mat. Nauk., 12 (1957), 3.   Google Scholar [41] H. J. Payne, Models of freeway traffic and control,, "Simulation Councils Proc. Ser. : Mathematical Models of Public Systems,", 1 (1971), 51.   Google Scholar [42] I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic,", American Elsevier Publishing Company Inc., (1971).   Google Scholar [43] P. I. Richards, Shock waves on highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar [44] 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.  doi: 10.1137/040617790.  Google Scholar [45] Lina Wang, Yaping Wu and Tong Li, Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion,, Physica D, 240 (2011), 971.  doi: 10.1016/j.physd.2011.02.003.  Google Scholar [46] G. B. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar [47] Lei Yu, Tong Li and Zhong-Ke Shi, Density waves in a traffic flow model with reactive-time delay,, Physica A, 389 (2010), 2607.  doi: 10.1016/j.physa.2010.03.009.  Google Scholar [48] Lei Yu, Tong Li and Zhong-Ke Shi, The effect of diffusion in a new viscous continuum model,, Physics Letters, 374 (2010), 2346.  doi: 10.1016/j.physleta.2010.03.056.  Google Scholar [49] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transportation Research, 36 (2002), 275.  doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar [50] H. M. Zhang, Driver memory, traffic viscosity and a viscous vehicular traffic flow model,, Transportation Research, 37 (2003), 27.   Google Scholar
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