American Institute of Mathematical Sciences

Advanced
May  2013, 12(3): 1501-1526. doi: 10.3934/cpaa.2013.12.1501

Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models

John M. Hong 1, , Cheng-Hsiung Hsu 2, , Bo-Chih Huang 1, and  Tzi-Sheng Yang 3,

1. 

Department of Mathematics, National Central University, Chung-Li 32001, Taiwan, Taiwan
2. 

Department of Mathematics, National Central University, Chung-Li 32054
3. 

Department of Mathematics, Tunghai University, Taichung 40704, Taiwan

Received  November 2011 Revised  July 2012 Published  September 2012

The purpose of this work is to study the existence and stability of stationary waves for viscous traffic flow models. From the viewpoint of dynamical systems, the steady-state problem of the systems can be formulated as a singularly perturbed problem. Using the geometric singular perturbation method, we establish the existence of stationary waves for both the inviscid and viscous systems. The inviscid stationary waves contain smooth waves and discontinuous transonic waves. Both waves admit viscous profiles for the viscous systems. Then we consider the linearized eigenvalue problem of the systems along smooth stationary waves. Applying the technique of center manifold reduction, we show that any one of the supersonic smooth stationary waves is spectrally unstable.
Keywords: invariant manifold theory., shock waves, conservation laws, Traffic flows, geometric singular perturbations.
Mathematics Subject Classification: Primary: 35L65, 35L67; Secondary: 34B16, 34E20, 37D1.
Citation: John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501
References:
[1]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems,, Ann. of Math., 161 (2005), 223. doi: 10.4007/annals.2005.161.223. Google Scholar

[2]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63 (2002), 259. doi: 10.1137/S0036139900380955. Google Scholar

[3]

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

[4]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,", 2$^{nd}$ edition, 325 (2005). doi: 10.1007/s11012-008-9160-4. Google Scholar

[5]

J. M. Del Castillo, P. Pintado and F. G. Benitez, A formulation of reaction time of traffic flow models,, in, (1993), 387. Google Scholar

[6]

N. Fenichel, Persistence and smoothness of invariant manifolds and flows,, Indiana Univ. Math. J., 21 (1971), 193. doi: 10.1512/iumj.1971.21.21017. Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Diff. Eqns., 31 (1979), 53. doi: 10.1016/0022-0396(79)90152-9. Google Scholar

[8]

L. R. Foy, Steady state solution of hyperbolic systems of conservation laws with viscosity terms,, Comm. Pure Appl. Math., 17 (1964), 177. doi: 10.1002/cpa.3160170204. Google Scholar

[9]

J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws,, Arch. Rational Mech. Anal., 121 (1992), 235. doi: 10.1007/BF00410614. Google Scholar

[10]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads,, SIAM J. Math. Anal., 26 (1995), 999. doi: 10.1137/S0036141093243289. Google Scholar

[11]

J. M. Hong, C.-H. Hsu and B-C. Huang, Existence and uniqueness of generalized stationary waves for viscous gas flow through a nozzle with discontinuous cross section,, J. Diff. Eqns., 253 (2012), 1088. doi: 10.1016/j.jde.2012.04.021. Google Scholar

[12]

J. M. Hong, C.-H. Hsu and W. Liu, Viscous standing asymptotic states of isentropic compressible flows through a nozzle,, Arch. Ration. Mech. Anal., 196 (2010), 575. doi: 10.1007/s00205-009-0245-6. Google Scholar

[13]

J. M. Hong, C.-H. Hsu and W. Liu, Inviscid and viscous stationary waves of gas flow through contracting-expanding nozzles,, J. Diff. Eqns., 248 (2010), 50. doi: 10.1016/j.jde.2009.06.016. Google Scholar

[14]

J. M. Hong, C.-H. Hsu, Y.-C. Lin and W. Liu, Linear stability of the sub-to-super inviscid transonic stationary wave for gas flow in a nozzle of varying area,, preprint., (). Google Scholar

[15]

C. K. R. T. Jones, Geometric singular perturbation theory,, in, 1609 (1994), 44. doi: 10.1007/BFb0095239. Google Scholar

[16]

R. D. Kühne, Freeway control and incident detection using a stochastic continuum theory of traffic flow,, in, (1989), 287. Google Scholar

[17]

R. D. Kühne and R. Beckschulte, Non-linearity stochastics of unstable traffic flow,, in, (1993), 367. Google Scholar

[18]

M. J. Lighthill and G. B. Whittam, On kinematic waves: II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. London. Ser. A., 229 (1995), 317. doi: 10.1098/rspa.1955.0089. Google Scholar

[19]

T. 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

[20]

T. Li and H.-L. 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

[21]

W. Liu, Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws,, Discrete Contin. Dynam. Syst., 10 (2004), 871. doi: 10.3934/dcds.2004.10.871. Google Scholar

[22]

H. J. Payne, Models of freeway traffic and control,, in, 1 (1971), 51. Google Scholar

[23]

P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar

[24]

S. Schecter, Undercompressive shock waves and the Dafermos regularization,, Nonlinearity, 15 (2002), 1361. doi: 10.1088/0951-7715/15/4/318. Google Scholar

[25]

S. Schecter and P. Szmolyan, Composite waves in the Dafermos regularization,, J. Dynam. Differential Equations, 16 (2004), 847. doi: 10.1007/s10884-004-6698-2. Google Scholar

[26]

P. Szmolyan and M. Wechselberger, Canards in $R^3$, , J. Diff. Eqns., 177 (2001), 419. doi: 10.1006/jdeq.2001.4001. Google Scholar

[27]

B. Whitham, "Linear and nonlinear waves,", Pure and Applied Mathematics, (1974). doi: 10.1002/9781118032954. Google Scholar

[28]

H. M. Zhang, A theory of nonequilibrium traffic flow,, Transportation Research-B., 32 (1998), 485. doi: 10.1016/S0191-2615(98)00014-9. Google Scholar

[29]

H. M. Zhang, Structural properties of solutions arising from a nonequilibrium traffic flow theory,, Transportation Research-B., 34 (2000), 583. doi: 10.1016/S0191-2615(99)00041-7. Google Scholar

[30]

H. M. Zhang, Driver memory, traffic viscosity and a viscous vehicular traffic flow model,, Transportation Research-B., 37 (2003), 27. doi: 10.1016/S0191-2615(01)00043-1. Google Scholar

show all references

References:
[1]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems,, Ann. of Math., 161 (2005), 223. doi: 10.4007/annals.2005.161.223. Google Scholar

[2]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63 (2002), 259. doi: 10.1137/S0036139900380955. Google Scholar

[3]

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

[4]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,", 2$^{nd}$ edition, 325 (2005). doi: 10.1007/s11012-008-9160-4. Google Scholar

[5]

J. M. Del Castillo, P. Pintado and F. G. Benitez, A formulation of reaction time of traffic flow models,, in, (1993), 387. Google Scholar

[6]

N. Fenichel, Persistence and smoothness of invariant manifolds and flows,, Indiana Univ. Math. J., 21 (1971), 193. doi: 10.1512/iumj.1971.21.21017. Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Diff. Eqns., 31 (1979), 53. doi: 10.1016/0022-0396(79)90152-9. Google Scholar

[8]

L. R. Foy, Steady state solution of hyperbolic systems of conservation laws with viscosity terms,, Comm. Pure Appl. Math., 17 (1964), 177. doi: 10.1002/cpa.3160170204. Google Scholar

[9]

J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws,, Arch. Rational Mech. Anal., 121 (1992), 235. doi: 10.1007/BF00410614. Google Scholar

[10]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads,, SIAM J. Math. Anal., 26 (1995), 999. doi: 10.1137/S0036141093243289. Google Scholar

[11]

J. M. Hong, C.-H. Hsu and B-C. Huang, Existence and uniqueness of generalized stationary waves for viscous gas flow through a nozzle with discontinuous cross section,, J. Diff. Eqns., 253 (2012), 1088. doi: 10.1016/j.jde.2012.04.021. Google Scholar

[12]

J. M. Hong, C.-H. Hsu and W. Liu, Viscous standing asymptotic states of isentropic compressible flows through a nozzle,, Arch. Ration. Mech. Anal., 196 (2010), 575. doi: 10.1007/s00205-009-0245-6. Google Scholar

[13]

J. M. Hong, C.-H. Hsu and W. Liu, Inviscid and viscous stationary waves of gas flow through contracting-expanding nozzles,, J. Diff. Eqns., 248 (2010), 50. doi: 10.1016/j.jde.2009.06.016. Google Scholar

[14]

J. M. Hong, C.-H. Hsu, Y.-C. Lin and W. Liu, Linear stability of the sub-to-super inviscid transonic stationary wave for gas flow in a nozzle of varying area,, preprint., (). Google Scholar

[15]

C. K. R. T. Jones, Geometric singular perturbation theory,, in, 1609 (1994), 44. doi: 10.1007/BFb0095239. Google Scholar

[16]

R. D. Kühne, Freeway control and incident detection using a stochastic continuum theory of traffic flow,, in, (1989), 287. Google Scholar

[17]

R. D. Kühne and R. Beckschulte, Non-linearity stochastics of unstable traffic flow,, in, (1993), 367. Google Scholar

[18]

M. J. Lighthill and G. B. Whittam, On kinematic waves: II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. London. Ser. A., 229 (1995), 317. doi: 10.1098/rspa.1955.0089. Google Scholar

[19]

T. 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

[20]

T. Li and H.-L. 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

[21]

W. Liu, Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws,, Discrete Contin. Dynam. Syst., 10 (2004), 871. doi: 10.3934/dcds.2004.10.871. Google Scholar

[22]

H. J. Payne, Models of freeway traffic and control,, in, 1 (1971), 51. Google Scholar

[23]

P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar

[24]

S. Schecter, Undercompressive shock waves and the Dafermos regularization,, Nonlinearity, 15 (2002), 1361. doi: 10.1088/0951-7715/15/4/318. Google Scholar

[25]

S. Schecter and P. Szmolyan, Composite waves in the Dafermos regularization,, J. Dynam. Differential Equations, 16 (2004), 847. doi: 10.1007/s10884-004-6698-2. Google Scholar

[26]

P. Szmolyan and M. Wechselberger, Canards in $R^3$, , J. Diff. Eqns., 177 (2001), 419. doi: 10.1006/jdeq.2001.4001. Google Scholar

[27]

B. Whitham, "Linear and nonlinear waves,", Pure and Applied Mathematics, (1974). doi: 10.1002/9781118032954. Google Scholar

[28]

H. M. Zhang, A theory of nonequilibrium traffic flow,, Transportation Research-B., 32 (1998), 485. doi: 10.1016/S0191-2615(98)00014-9. Google Scholar

[29]

H. M. Zhang, Structural properties of solutions arising from a nonequilibrium traffic flow theory,, Transportation Research-B., 34 (2000), 583. doi: 10.1016/S0191-2615(99)00041-7. Google Scholar

[30]

H. M. Zhang, Driver memory, traffic viscosity and a viscous vehicular traffic flow model,, Transportation Research-B., 37 (2003), 27. doi: 10.1016/S0191-2615(01)00043-1. Google Scholar

[1]

Tatsien Li, Libin Wang. Global exact shock reconstruction for quasilinear hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 597-609. doi: 10.3934/dcds.2006.15.597
[2]

C. M. Khalique, G. S. Pai. Conservation laws and invariant solutions for soil water equations. Conference Publications, 2003, 2003 (Special) : 477-481. doi: 10.3934/proc.2003.2003.477
[3]

Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51
[4]

Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657
[5]

K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091
[6]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i
[7]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008
[8]

Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411
[9]

Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157
[10]

James K. Knowles. On shock waves in solids. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 573-580. doi: 10.3934/dcdsb.2007.7.573
[11]

Hermano Frid. Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 585-593. doi: 10.3934/dcds.1995.1.585
[12]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation. Networks & Heterogeneous Media, 2016, 11 (2) : 281-300. doi: 10.3934/nhm.2016.11.281
[13]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[14]

Mauro Garavello. A review of conservation laws on networks. Networks & Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[15]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[16]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[17]

Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251
[18]

Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469
[19]

Yuri Gaididei, Anders Rønne Rasmussen, Peter Leth Christiansen, Mads Peter Sørensen. Oscillating nonlinear acoustic shock waves. Evolution Equations & Control Theory, 2016, 5 (3) : 367-381. doi: 10.3934/eect.2016009
[20]

Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences