- Previous Article
- CPAA Home
- This Issue
-
Next Article
Asymptotic analysis of continuous opinion dynamics models under bounded confidence
Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[24] |
S. Schecter, Undercompressive shock waves and the Dafermos regularization, Nonlinearity, 15 (2002), 1361-1377.
doi: 10.1088/0951-7715/15/4/318. |
[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. |
[26] |
P. Szmolyan and M. Wechselberger, Canards in $R^3$, J. Diff. Eqns., 177 (2001), 419-453.
doi: 10.1006/jdeq.2001.4001. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[24] |
S. Schecter, Undercompressive shock waves and the Dafermos regularization, Nonlinearity, 15 (2002), 1361-1377.
doi: 10.1088/0951-7715/15/4/318. |
[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. |
[26] |
P. Szmolyan and M. Wechselberger, Canards in $R^3$, J. Diff. Eqns., 177 (2001), 419-453.
doi: 10.1006/jdeq.2001.4001. |
[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. |
[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. |
[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. |
[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. |
[1] |
Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks and Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028 |
[2] |
Tatsien Li, Libin Wang. Global exact shock reconstruction for quasilinear hyperbolic systems of conservation laws. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 597-609. doi: 10.3934/dcds.2006.15.597 |
[3] |
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 |
[4] |
Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51 |
[5] |
Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro. Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media, 2022, 17 (1) : 101-128. doi: 10.3934/nhm.2021025 |
[6] |
Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657 |
[7] |
K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091 |
[8] |
Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i |
[9] |
Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411 |
[10] |
Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008 |
[11] |
Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157 |
[12] |
Hermano Frid. Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 585-593. doi: 10.3934/dcds.1995.1.585 |
[13] |
James K. Knowles. On shock waves in solids. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 573-580. doi: 10.3934/dcdsb.2007.7.573 |
[14] |
Avner Friedman. Conservation laws in mathematical biology. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081 |
[15] |
Mauro Garavello. A review of conservation laws on networks. Networks and Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565 |
[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] |
Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159 |
[18] |
Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251 |
[19] |
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 |
[20] |
Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation. Networks and Heterogeneous Media, 2016, 11 (2) : 281-300. doi: 10.3934/nhm.2016.11.281 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]