參考文獻 |
[1] 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), pp. 259-278.
[2] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J.
Appl. Math. 60 (2000), pp. 916-938.
[3] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems.
Ann. of Math. 161 (2005), pp. 223-342.
[4] S. R. Chakravarthy and S. Osher, Numerical experiments with the Osher upwind scheme
for the Euler equations, AIAA J. 21 (1983), no. 9, pp. 1241-1248.
[5] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd edition.
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], 325, Springer-Verlag, Berlin, 2005.
[6] G. Dal Maso, P. G. LeFloch, and F. Murat, Definition and weak stability of nonconservative
products, J. Math. Pures Appl. 74 (1995), pp. 483-548.
[7] J. M. Del Castillo, P. Pintado and F. G. Benitez, A formulation of reaction time of traffic
flow models, In: Daganzo, C.F. (Ed.), Transportation and Traffic Theory, Elsevier,
Amsterdam, 1993, pp. 387-405.
[8] F. Dumortier, R. Roussarie, Canard cycles and center manifolds, Memoirs of the Amer.
Math. Soc., Providence, 577 (1996).
[9] P. Embid, J. Goodman, and A. Majda, Multiple steady states for 1-D transonic flow,
SIAM J. Sci. Stat. Comput. 5 (1984), no. 1, pp. 21-41.
[10] N. Fenichel, Persistence and smoothness of invariant manifolds and flows, Indiana Univ.
Math. J. 21 (1971/1972), pp. 193-226.
[11] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,
J. Diff. Eqns. 31 (1979), no. 1, pp. 53-98.
[12] L. R. Foy, Steady state solution of hyperbolic systems of conservation laws with viscosity
terms, Comm. Pure Appl. Math. 17 (1964), pp. 177-188.
[13] J. Glimm, Solutions in the large for nonlinar hyperbolic systems of equations, Comm.
Pure Appl. Math. 18, (1956), pp. 697-715.
[14] J. Hadamard, Sur l’’iteration et les solutions asymptotiques des equations differentielles,
Bull. Soc. Math. France. 29 (1901), pp. 224-228.
[15] H. Holden and N. H. Risebro, A mathematical model of traffc flow on a network of
unidirectional roads, SIAM J. Math. Anal. 26, pp. 999-1017.
[16] James R. Holton, An introduction to dynamical meteorology, Elsevier Academic Press
200 Burlington, (2004).
[17] J. M. Hong, An extension of Glimm’’s method to inhomogeneous strictly hyperbolic
systems of conservation laws by `weaker than weak’’ solutions of the Riemann problem,
J. Diff. Eqns. 222 (2006), no. 2, pp. 515-549.
[18] John 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), pp. 1088-1110.
[19] John M. Hong, C.-H. Hsu, B.-C. Huang and T.-S. Yang, Goemetric singular perturbation
approach to the existence and instability of stationary waves for viscous traffc flow
models, Accepted by Commun. Pur. Appl. Anal, (2012).
[20] John 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), pp. 50-76..
[21] 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), no. 2, pp.
575-597.
[22] J. M. Hong, C.-H. Hsu and W. Liu, Sub-to-super transonic steady states and their linear
stabilities for gas flows, (2010), submitted.
[23] J. M. Hong and B. Temple, A bound on the total variation of the conserved quantities for
solutions of a general resonant nonlinear balance law, SIAM J. Appl. Math. 64 (2004),
no. 3, pp. 819-857.
[24] J. M. Hong and B. Temple, The generic solution of the Riemann problem in a neighborhood
of a point of resonance for systems of nonlinear balance laws, Methods Appl.
Anal. 10 (2003), no. 2, pp. 279-294.
[25] M. Hirsch, C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583,
Springer-Verlag, New York, 1976.
[26] S.-B. Hsu, and T.-P. Liu, Nonlinear singular Sturm-Liouville problems and an application
to transonic flow through a nozzle, Comm. Pure Appl. Math. 43 (1990), no. 1, pp.
31-61.
[27] E. Isaacson and B. Temple, Convergence of the 2 x 2 Godunov method for a general
resonant nonlinear balance law, SIAM J. Appl. Math. 55 (1995), no. 3, pp. 625-640.
[28] C.K.R.T. Jones, Geometric singular perturbation theory, Dynamical Systems (Montecatini
Terme, 1994). Lecture Notes in Math. 1609, Springer-Verlag, Berlin, 1995, pp.
44-118.
[29] B. Keyfitz, H. Kranzer, A system of non-strictly hyperbolic conservation laws arising in
elasticity theory, Arch. Ration. Mech. Anal. 72 (1980), pp. 219-241.
[30] M. Krupa, P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic
points - fold and canard points in two dimensions, SIAM J. Math. Anal. 33
(2001), no. 2, pp. 286-314.
[31] R. D. K:uhne and R. Beckschulte, Non-linearity stochastics of unstable traffc flow. In:
Daganzo, C.F. (Ed.), Transportation and Traffc Theory, Elsevier Science Publishers,
(1993), pp. 367-386.
[32] R. D. K:uhne, Freeway control and incident detection using a stochastic continuum theory
of traffc flow, Proceedings of the 1st International Conference on Applied Advanced
Technology in Transportation Engineering, San Diego, CA, (1989), pp. 287-292.
[33] R. D. K:uhne and R. Beckschulte, Non-linearity stochastics of unstable traffc flow. In:
Daganzo, C.F. (Ed.), Transportation and Traffc Theory, Elsevier Science Publishers,
(1993), pp. 367-386.
[34] P. D. Lax, Hyperbolic system of conservation laws, II, Comm. Pure Appl. Math. 10
(1957), pp. 537-566.
[35] P. LeFLock, Entropy weak solutions to nonlinear hyperbolic system under nonconservative
form, Comm. Part. Diff. Eq. 13 (1988), pp. 669-727.
[36] P. LeFloch and T.-P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative
form, Forum Math. 5 (1993), no. 3, pp. 261-280.
[37] P. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of nonconservative
products, SIAM J. Math. Anal. 30 (1999), no. 6, pp. 1309-1342.
[38] T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and
diffusion, SIAM. J. Math. Anal. 40 (2008), pp. 1058-1075.
[39] T. Li and H.-L. Liu, Critical thresholds in a relaxation model for traffc flows, Indiana
Univ. Math. J. 57, pp. 1409-1430.
[40] H. W. Liepmann and A. Roshlo, Elementary of Gas Dynamics, GALCIT Aeronautical
Series, New York: Wiely, 1957.
[41] M. J. Lighthill and G. B. Whittam, On kinematic waves: II. A theory of traffc flow on
long crowded roads, Proceedings of the Royal Society-A 229 (1995), pp. 317-345.
[42] X.-B. Lin and S. Schecter, Stability of self-similar solutions of the Dafermos regularization
of a system of conservation laws, SIAM J. Math. Anal. 35 (2004), no. 4, pp.
884-921.
[43] T.-P. Liu, Quasilinear hyperbolic system, Comm. Math. Phys. 68 (1979), no. 2, pp.
141-172.
[44] T. P. Liu Transonic gas flow in a duct of varying area, Arch. Ration. Mech. Anal. 80
(1982), no. 1, pp. 1-18.
[45] W. Liu, Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems
of conservation laws, Discrete Contin. Dyn. Syst. 10 (2004), no. 4, pp. 871-884.
[46] H. J. Payne, Models of freeway traffc and control. In Mathematical Models of Public
Systems. In: Bekey, G.A. (Ed.),. Simulation Councils Proc. Ser., vol. 1, 1971, pp. 51-60.
[47] P. I. Richards, Shock waves on the highway, Operations Research 4 (1956), pp. 42-51.
[48] K. Sakamoto, Invariant manifolds in singular perturbation problems for ordinary differential
equations, Proc. Roy. Soc. Ed. 116A (1990), pp.45-78.
[49] S. Schecter, Undercompressive shock waves and the Dafermos regularization, Nonlin-
earity 15 (2002), no. 4, pp. 1361-1377.
[50] S. Schecter, Eigenvalues of self-similar solutions of the Dafermos regularization of a
system of conservation laws via geometric singular perturbation theory, J. Dynam. Dif-
ferential Equations 18 (2006), no. 1, pp. 53-101.
[51] S. Schecter and P. Szmolyan, Composite waves in the Dafermos regularization, J. Dy-
nam. Differential Equations 16 (2004), no. 3, pp. 847-867.
[52] D. Serre, Systems of conservation laws. 1. Hyperbolicity, entropies, shock waves, Translated
from the 1996 French original by I. N. Sneddon. Cambridge University Press,
Cambridge, 1999.
[53] D. Serre, Systems of conservation laws. 2. Geometric structures, oscillations, and initial-
boundary value problems, Translated form the 1996 French original by I.N. Sneddon.
Cambridge University Press, Cambridge, 2000.
[54] G. R. Shubin, A. B. Stephens and H. Glaz, Steady shock tracking and Newton’’s method
applied to one-dimensional duct flow, J. Comput. Phys. 39 (1980), no. 2, pp. 364-374.
[55] D. H. Smith, Non-uniqueness and multi-shock solutions for transonic flows, IMA J.
Appl. Math. 71 (2006), no. 1, pp. 120-132.
[56] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer-Verlag,
Berlin, New York, 1983.
[57] Darrell F. Strobel, Titan’’s hydrodynamically escaping atmosphere, Icarus 193 (2008),
pp. 588-594.
[58] Darrell F. Strobel, N2 escape rates from Pluto’’s atmosphere, Icarus 193 (2008), pp.
612-619.
[59] P. Szmolyan and M. Wechselberger, Canards in R^3, J. Diff. Eqns. 177 (2001), no. 2,
pp. 419-453.
[60] Feng Tian, Owen B. Toon, Alexander A. Pavlov, and H. De Sterck, Transonic hydrodynamic
escape of hydrogen from extrasolar planetary atmospheres, The Astrophysical
Journal 621 (2005), pp. 1049-1060.
[61] Feng Tian, Owen B. Toon, Alexander A. Pavlov, and H. De Sterck, A hydrogen-rich
early Earth atmosphere. Scicence 308 (2005), pp. 1014-1017.
[62] B. Temple, Global solution of the Cauchy problem for a class of 2x2 nonstrictly hyperbolic
conservation laws, Adv. Appl. Math. 3 (1982), pp. 335-375.
[63] B. Whitham, Linear and nonlinear waves, New York, John Wiley, 1974.
[64] S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer-
Verlag, New York, 1994.
[65] H. M. Zhang, A theory of nonequilibrium traffc flow, Transportation Research-B. 32
(1998), pp. 485-498.
[66] H. M. Zhang, Structural properties of solutions arising from a nonequilibrium traffc
flow theory,, Transportation Research-B. 34 (2000), pp. 583-603.
[67] H. M. Zhang, Driver memory, traffc viscosity and a viscous vehicular traffc flow model.
Transportation Research-B. 37 (2003), pp. 27-41.
[68] M. Zingale et all, Mapping initial hydrostatic models in Godunov codes, The Astrophys-
ical Journal Supplement Series 143 (2002), pp. 539-565.
|