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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), 223-342. 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), 259-278. doi: 10.1137/S0036139900380955.  Google Scholar

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A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar

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C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," 2nd edition, Grundlehren der Mathematischen Wissenschaften (German) [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2005. doi: 10.1007/s11012-008-9160-4.  Google Scholar

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N. Fenichel, Persistence and smoothness of invariant manifolds and flows, Indiana Univ. Math. J., 21 (1971), 193-226. doi: 10.1512/iumj.1971.21.21017.  Google Scholar

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N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqns., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

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L. R. Foy, Steady state solution of hyperbolic systems of conservation laws with viscosity terms, Comm. Pure Appl. Math., 17 (1964), 177-188. 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), 235-265. 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), 999-1017. 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), 1088-1110. 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), 575-597. 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), 50-76. 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

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R. D. Kühne, Freeway control and incident detection using a stochastic continuum theory of traffic flow, in "Proceedings of the 1st International Conference on Applied Advanced Technology in Transportation Engineering," San Diego, CA, 1989, 287-292. Google Scholar

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R. D. Kühne and R. Beckschulte, Non-linearity stochastics of unstable traffic flow, in "Transportation and Traffic Theory"(C. F. Daganzo eds.), Elsevier Science Publishers, (1993), 367-386. Google Scholar

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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), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

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T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM. J. Math. Anal., 40 (2008), 1058-1075. doi: 10.1137/070690638.  Google Scholar

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T. Li and H.-L. Liu, Critical thresholds in a relaxation model for traffic flows, Indiana Univ. Math. J., 57 (2008), 1409-1430. 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), 871-884. doi: 10.3934/dcds.2004.10.871.  Google Scholar

[22]

H. J. Payne, Models of freeway traffic and control, in "Mathematical Models of Public Systems" (G. A. Bekey eds.), Simulation Councils Proc. Ser., 1 (1971), 51-60. Google Scholar

[23]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

[24]

S. Schecter, Undercompressive shock waves and the Dafermos regularization, Nonlinearity, 15 (2002), 1361-1377. 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), 847-867. doi: 10.1007/s10884-004-6698-2.  Google Scholar

[26]

P. Szmolyan and M. Wechselberger, Canards in $R^3$, J. Diff. Eqns., 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001.  Google Scholar

[27]

B. Whitham, "Linear and nonlinear waves," Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. doi: 10.1002/9781118032954.  Google Scholar

[28]

H. M. Zhang, A theory of nonequilibrium traffic flow, Transportation Research-B., 32 (1998), 485-498. 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), 583-603. 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), 27-41. 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), 223-342. 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), 259-278. 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), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar

[4]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," 2nd edition, Grundlehren der Mathematischen Wissenschaften (German) [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 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 "Transportation and Traffic Theory" (C. F. Daganzo eds.), Elsevier, Amsterdam, (1993), 387-405. Google Scholar

[6]

N. Fenichel, Persistence and smoothness of invariant manifolds and flows, Indiana Univ. Math. J., 21 (1971), 193-226. 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), 53-98. 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), 177-188. 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), 235-265. 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), 999-1017. 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), 1088-1110. 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), 575-597. 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), 50-76. 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 "Dynamical Systems"(R. Johnson eds.), Lecture Notes in Math., 1609, Springer-Verlag, Berlin, 1994, 44-118. doi: 10.1007/BFb0095239.  Google Scholar

[16]

R. D. Kühne, Freeway control and incident detection using a stochastic continuum theory of traffic flow, in "Proceedings of the 1st International Conference on Applied Advanced Technology in Transportation Engineering," San Diego, CA, 1989, 287-292. Google Scholar

[17]

R. D. Kühne and R. Beckschulte, Non-linearity stochastics of unstable traffic flow, in "Transportation and Traffic Theory"(C. F. Daganzo eds.), Elsevier Science Publishers, (1993), 367-386. 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), 317-345. 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), 1058-1075. 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), 1409-1430. 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), 871-884. doi: 10.3934/dcds.2004.10.871.  Google Scholar

[22]

H. J. Payne, Models of freeway traffic and control, in "Mathematical Models of Public Systems" (G. A. Bekey eds.), Simulation Councils Proc. Ser., 1 (1971), 51-60. Google Scholar

[23]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

[24]

S. Schecter, Undercompressive shock waves and the Dafermos regularization, Nonlinearity, 15 (2002), 1361-1377. 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), 847-867. doi: 10.1007/s10884-004-6698-2.  Google Scholar

[26]

P. Szmolyan and M. Wechselberger, Canards in $R^3$, J. Diff. Eqns., 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001.  Google Scholar

[27]

B. Whitham, "Linear and nonlinear waves," Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. doi: 10.1002/9781118032954.  Google Scholar

[28]

H. M. Zhang, A theory of nonequilibrium traffic flow, Transportation Research-B., 32 (1998), 485-498. 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), 583-603. 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), 27-41. doi: 10.1016/S0191-2615(01)00043-1.  Google Scholar

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