臺灣博碩士論文加值系統

在這篇論文裡，我們研究具有隨著時間而震盪的通量項及來源項的非線性平衡律，此系統應用於許多著名的方程，如描述氣體在周期性變化管子內的運動行為，行駛中的車輛切換車道時的動作，淺水波與河床交互作用的關係。而處理此類方程是結合了我們處理通量項及來源項的新方法與一般化的格林方法，不但證明了解的存在性，更進一步找出了熵條件去證明滿足為熵解，利用了推廣的格林方法與拉克斯方法，我們處理波的交互作用能更加精準，亦證明其穩定性。最後也給了一些超音波與次音波間的作用關係，及道路寬所影響的傳遞波之行為。

We study the Cauchy problem for general nonlinear hyperbolic balance laws assuming time-oscillating fluxes and sources. Such nonlinear balance laws arise in, for instance, the nozzle flows of gas dynamics with time periodic ducts, traffic models incorporating lane changing effects model and shallow water equations with time-dependent river’s bottom. The global existence of weak solutions is established by a new version of the generalized Glimm method which incorporates asymptotic expansions of the fluxes and sources. We prove existence of weak solutions and demonstrate that they are indeed entropy solutions satisfying the entropy inequality. The approximate solutions of the perturbed Riemann problem, the building blocks of the generalized Glimm scheme, are constructed by a modified Lax method, and a generalized version of the wave interaction estimates are provided for the proof of stability. The consistency of the scheme is established by proving the weak convergence of the residuals and source terms, thereby extending the methods introduced in [12].

1 Introduction
1.1 Nonlinear Balance Laws
1
1
2 The Cauchy Problem for Systems of Nonlinear Balance Laws 17
2.1 Perturbed Riemann problem and error estimate . . . . . . . . . . . . . . . . . . 17
2.2 Generalized Glimm scheme and wave interaction estimate . . . . . . . . . . . . . 24
2.3 \月veak convergence and global existence theorem . . . . . . . . . . . . . . . . . . 31
3 Gas dynamical model of transonic flows 39
3.1 The Generalized Glimm Scheme for Gas dynamical model ............ 39
3.2 Convergence of the GGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
3.2.1 Wave interaction estimates . . . . . . . . . . . . . . . . . . . . . . . . .. 42
3.2.2 The stability of the GGS . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
3.3 Applications...................................... 49
4 Hyperbolic Balance Laws with time-oscillation fluxes and sources 53
4.1 Approximate Riemann solver and its error estimate . . . . . . . . . . . . . . .. 53
4.2 Generalized Glimm scheme and wave interaction estimate . . . . . . . . . . . .. 62
4.3 Weak convergence and global existence theorem . . . . . . . . . . . . . . . . .. 72
5 Traveling Wave Solutions to Nonlinear Balance Laws in Traffic Flows 81
5.1 Tr、aveling wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Reformulation of the stability problem . . . . . . . . . . . . . . . . . . . . . .. 92
5.3 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
5.4 Conclusions 102
Bibliography 104

[1] D. Amadori, L. Gosse, G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic
systems of balance laws, Arch. Rational Mech. Anal. 162 (2002) pp. 327-366.
[2] Y. Chang, J. M. Hong, C.-H. Hsu, Globally Lipschitz continuous solutions to a class of
quasilinear wave equations, J. Diff. Equ. 236 (2007), pp. 504-531.
[3] G. Q. Chen, C. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation
terms and entropy, Comm. Pure Appl. Math. 47 (1993), pp. 787-830.
[4] C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Series of Comprehensive
Studies in Mathematics, Vol. 325, Springer.
[5] C. M. Dafermos, L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and
dissipation, Indiana U. Math. J. 31 (1982), pp. 471-491.
[6] G. Dal Maso, P. LeFloch, F. Murat, Definition and Weak Stability of Nonconservative
Products, J. Math. Pures Appl. 74 (1995), pp. 483-548.
[7] P. Dionne, Sur les probl`emes de Cauchy hyperboliques bien pos´es, J. d’Analyse Math. 10
(1962), pp. 1-90.
[8] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm.
Pure Appl. Math. 18 (1965), pp. 697-715.
[9] J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws,
Arch. Rational Mech. Anal. 95 (1986), pp. 325-344.
[10] J. Groah, J. Smoller, B. Temple, Shock Wave Interactions in General Relativity, Monographs
in Mathematics, Springer, Berlin, New York, 2007.
[11] D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phy. 73
(2001), pp. 1067-1141.
[12] 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.
Equ. 222 (2006), pp. 515-549.
[13] J. M. Hong, C.-H. Hsu, Bo-Chih Huang, and Tzi-Sheng Yang, Geometric singular perturbation
approach to the existence and instability of stationary waves for viscous traffic flow
models, preprint.
[14] J. M. Hong, C.-H. Hsu, Y.-C. Su, Global solutions for initial-boundary value problem of
quasilinear wave equations, J. Diff. Equ. 245 (2008), pp. 223-248.
[15] J. M. Hong, P. G. LeFloch, A version of Glimm method based on generalized Riemann
problems, J. Portugal Math., Vol. 64, (2007) pp. 199-236.
[16] J. M. Hong, 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., Vol. 64, No.
3 (2004) pp. 819-857.
[17] T. J. R. Hughes, T. Kato, and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic
systems with applications to nonlinear elastodynamics and general relativity, Arch.
Rational Mech. Anal. 63 (1976), pp. 273-294.
[18] S.-B. Hsu, Ordinary differential equations with applications, World Scientific, 2006.
[19] E. Isaacson, B. Temple, Nonlinear resonant in inhomogenous systems of conservation laws,
Cotemp. Math., 108, 1990.
[20] E. Isaacson, B. Temple, Convergence of 2 × 2 Godunov method for a general resonant
nonlinear balance law, SIAM. J. Appl. Math. Vol. 55, No 3 (1995) pp. 625-640.
[21] Wen-Long Jin, A kinematic wave theory of lane-changing traffic flow, to appear in Transportation
research, Part B.
[22] S. Jin and M. A. Katsoulakis, Hyperbolic systems with supercharacteristic relaxations and
roll waves, SIAM J. Appl. Math. 61 (2000), pp. 273-292.
[23] S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary
space dimensions, Comm. Pure Appl. Math. 48 (1995), pp. 555-563.
[24] W. L. Jin and H. M. Zhang, The formation and structure of vehicle clusters in the Payne-
Whitham traffic flow model, Transportation Research, B. 37 (2003), pp. 207-223.
[25] C. Jones, R. Gardner and T. Kapitula, Stability of traveling waves for non-convex scalar
viscous conservation laws, Comm. Pure Appl. Math. 46 (1993), pp. 505-526.
[26] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational
Mech. Anal. 58 (1975), pp. 181-205.
[27] 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.
[28] S. N. Kruzkov, First order quasilinear equations with several space variables, Mat. USSR
Sb., 10 (1970), pp. 217-243.
[29] W. Knospe, L. Santen, A. Schadschneider and M. Schreckenberg, Single-vehicle data of
highway traffic: Microscopic description of traffic phases, Phys. Rev. E 65, 056133 (2002),
pp. 1-16.
[30] P. D. Lax, Hyperbolic system of conservation laws, II, Comm. Pure Appl. Math., 10 (1957),
pp. 537-566.
[31] P. D. Lax, Hyperbolic system of conservation laws and mathematical theory of shock
waves., Conf. Board Math. Sci., 11, SIAM, 1973.
[32] P. G. LeFloch, Entropy Weak Solutions to Nonlinear Hyperbolic Systems Under Nonconservative
Form, Comm. Part. Diff. Eq., 13 (1988), pp 669-727.
[33] P. G. LeFloch, T.-P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative
form, Forum Math. 5 (1993), pp. 261-280.
[34] T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising
from traffic flow, J. Differential Equations 190 (2003), pp. 131-149.
[35] T. Li, Nonlinear dynamics of traffic jams, Physica D 207 (2005), pp. 41-51.
[36] T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and
diffusion, SIAM J. Math. Anal. 40 (2008), pp. 1058-1075.
[37] T.-P. Liu, Quasilinear hyperbolic systems, Comm. Math. Phys., 68 (1979), pp. 141-172.
[38] T.-P. Liu, Admissible solutions of hyperbolic conservation laws, Mem. Amer. Math. Soc.
30 (1981).
[39] T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987),
pp. 153-175.
[40] H. L. Liu, Asymptotic stability of shock profiles for nonconvex convection-diffusion equation,
Appl. Math. Lett. 10 (1997), pp. 129-134.
[41] H. L. Liu, Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes,
Analysis and numerics for conservation laws, pp. 453-478, Springer, Berlin, 2005.
[42] T. Li and H. L. Liu, Stability of a traffic flow model with nonconvex relaxation, Comm.
Math. Sci. 3 (2005), pp. 101-118.
[43] H. L. Liu and J. Wang, Asymptotic stability of traveling wave solutions of a hyperbolic
system with relaxation terms, Beijing Math. 2 (1996), pp. 119-130.
[44] M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on
long crowded roads, Proceedings of the Royal Society A 229 (1955), pp. 317-345.
[45] H. L. Liu, C. W. Woo and T. Yang, Decay rate for traveling waves of a relaxation model,
J. Differential Equations 134 (1997), pp. 343-367.
[46] H. L. Liu, J. Wang and T. Yang, Stability for a relaxation model with a nonconvex flux,
SIAM J. Math. Anal. 29 (1998), pp. 18-29.
[47] C. Lattanzio and P. Marcati, The zero relaxation limit for the hydrodynamic Whitham
traffic flow model, J. Differential Equations 141 (1997), pp. 150-178.
[48] M. Luskin and B. Temple, The existence of a global weak solution to the non-linear waterhammer
problem, Comm. Pure Appl. Math. 35 (1982), pp. 697-735.
[49] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space
Variables, Applied Mathematical Sciences 53, Springer-Verlag, New York, 1984.
[50] A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves of scalar viscous
conservation laws with non-convex nonlinearity, Comm. Math. Phys. 165 (1994), pp. 83-96.
[51] M. Mei, Stability of shock profiles for non-convex scalar conservation laws, Math. Models
Method Appl. Sci. 5 (1995), pp. 279-296.
[52] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ.
Math. d’Orsay 78-02, D´epartment de Math´ematique, Universit´e Paris-sud, Orsay, 1978.
[53] T. Nishida, J. Smoller, Mixed problems for nonlinear conservation laws, J. Diff. Equ. 23
(1977), pp. 244-269.
[54] O. A. Oleinik, Discontinuous solutions of non-linear different equations, Uspekhi Math.
Nauk(N.S.), 12 (1957), pp. 3-73. (Trans. Amer. Math. Soc., Ser. 2, 26, pp. 172-195.)
[55] H. J. Payne, Models of freeway traffic and control, Simulation Councils Proceedings Series:
Mathematical Models of Public Systems, vol. 1, 51-61, G.A. Bekey (Ed.), 1971.
[56] P. I. Richards, Shock waves on the highway, Operations Research 4 (1956), pp. 42-51.
[57] D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math. 22
(1976), pp. 312-355.
[58] J. Schauder, Das Anfangswertproblem einer quasilinearen hyperbolischen Differentialgleichung
zweiter Ordnung in beliebiger Anzahl von unabh¨angigen Vera¨nderlichen, Fund.
Math. 24 (1935), pp. 213-246.
[59] S. Schochet, The instant-response limit in Whitham’s nonlinear traffic-flow model: uniform
well-posedness and global existence, Asymptotic Analysis 1 (1988), pp. 263-282.
[60] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer-Verlag,
Berlin, New York, 1994.
[61] S. L. Sobolev, Sur les ´equations aux d´eriv´ees partielles hyperboliques non-lin´eaires, Edizioni
Cremonese, Rome, 1961.
[62] B. Temple, Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic
conservation laws, Adv. Appl. Math., 3 (1982), pp. 335-375.
[63] T.-Q. Tang, C.-Y. Li, H.-J. Huang, and H.-Y. Shang, Macro modeling and analysis of
traffic flow with road width, J. Cent. South Univ. Technol. 18 (2011), pp. 1757-1764.
[64] G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
[65] Y. Yamada, Quasilinear wave equations and related nonlinear evolution equations, Nagoya
Math. J. 84 (1981), pp. 31-83.
[66] H. M. Zhang, A theory of nonequilibrium traffic flow, Transportation Res. Part B 32
(1998), pp. 485-498.
[67] H. M. Zhang, Driver memory, traffic viscosity and a viscous vehicular traffic flow model,
Transportation Res. Part B 37 (2003), pp. 27-41.
[68] C. J. Zhu, Asymptotic behavior of solutions for p-system with relaxation, J. Differential
Equations 180 (2002) pp. 273-306