一类改进的 Aw-Rascle-Zhang 模型的柯西问题

Abstract

Abstract:

This paper studies the Cauchy problem for an improved Aw-Rascle-Zhang traffic flow model exhibiting non-genuine nonlinearity. The Riemann solution structure for this model contains not only shocks, rarefaction waves, and contact discontinuities, but also composite waves. A method based on a modified Glimm scheme is developed within the framework of the space of functions of bounded variation. The key aspect of this method lies in constructing a wave interaction functional using the variation of Riemann invariants, thereby controlling the total variation of the solution during its time evolution. Therefore we establish the existence of a global weak solution for the Cauchy problem with large initial data.

Key words: cauchy problem, improved Aw-Rascle-Zhang model, modified Glimm's scheme, BV solutions, large initial data.

CLC Number:

O175.23

Tingting Chen, Weifeng Jiang, Tong Li, Yibo Lai. The Cauchy Problem for an Improved Aw-Rascle-Zhang Model[J].Acta mathematica scientia,Series A, 2026, 46(3): 1025-1027.

Trendmd

Figures/Tables 5

References 40

[1]	Aw A, Rascle M. Resurrection of “second order” models of traffic flow. SIAM J Appl Math, 2000, 60 : 916-938 doi: 10.1137/S0036139997332099
[2]	Betancourt F, Burger R, Chalons C, et al. A random sampling method for a family of Temple-class systems of conservation laws. Numer Math, 2018, 138 : 37-73 doi: 10.1007/s00211-017-0900-z
[3]	Bianchini S. The semigroup generated by a Temple class system with non-convex flux function. Diferential Integral Equations, 2000, 13 (10-12): 1529-1550
[4]	Chalons C, Goatin P. Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling. Interfaces Free Bound, 2008, 10 (2): 197-221 doi: 10.4171/ifb
[5]	Chalons C, Goatin P. Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling. Commun Math Sci, 2007, 5 (3): 533-551 doi: 10.4310/CMS.2007.v5.n3.a2
[6]	Chen G, Huang F, Wang T. Isothermal limit of entropy solutions of the Euler equations for isentropic gas dynamics. SIAM J Math Anal, 2024, 56 : 1300-1320 doi: 10.1137/23M1549948
[7]	Chen G, LeFloch P G. Existence Theory for the Isentropic Euler Equations. Arch Ration Mech Anal, 2003, 166 : 81-98 doi: 10.1007/s00205-002-0229-2
[8]	Chen T, Jiang W, Li T. On the stability of the improved Aw-Rascle-Zhang model with Chaplygin pressure. Nonlinear Anal Real world Appl, 2021, 62 : Art 103351
[9]	Chen T, Qu A, Wang Z. Existence and uniqueness of the global $L^1$ solution of the Euler equations for Chaplygin gas. Acta Math Sci, 2021, {\bf 41B}(3): 941-958
[10]	Cheng H. Two-dimensional Aw-Rascle model with Chaplygin pressures: Riemann solutions consisting of contact discontinuities. Rocky Mountain J Math, 2022, 52 : 1267-1288
[11]	Colombo R M. A $2\times2$ hyperbolic traffic flow model. Math Comput Model, 2002, 35 (5/6): 683-688 doi: 10.1016/S0895-7177(02)80029-2
[12]	Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Berlin Heidelberg: Springer, 2016
[13]	Fan S, Herty M, Seibold B. Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model. Netw Heterog Media, 2014, 9 (2): 239-268 doi: 10.3934/nhm.2014.9.239
[14]	Garavello M, Goatin P. The Aw-Rascle traffic model with locally constrained flow. J Math Anal Appl, 2011, 378 (2): 634-648 doi: 10.1016/j.jmaa.2011.01.033
[15]	Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18 (4): 687-715
[16]	Goatin P. The Aw-Rascle vehicular traffic flow model with phase transitions. Math Comput Model, 2006, 44 (3/4): 287-303 doi: 10.1016/j.mcm.2006.01.016
[17]	Godvik M, Hanche-Olsen H. Existence of solutions for the Aw-Rascle traffic flow model with vacuum. J Hyperbolic Differ Equ, 2008, 5 (1): 45-63 doi: 10.1142/S0219891608001428
[18]	Gottlich S, Herty M, Moutari S, Weissen J. Second-order traffic flow models on networks. SIAM J Appl Math, 2021, 81 (1): 258-281 doi: 10.1137/20M1339908
[19]	Helbing . Traffic and related self-driven many-particle systems. Rev Modern Phy, 2001, 73 (4): 1067-1141 doi: 10.1103/RevModPhys.73.1067
[20]	Jiang W, Wang Z. Developing an Aw-Rascle model of traffic flow. J Eng Math, 2016, 97 : 135-146 doi: 10.1007/s10665-015-9801-2
[21]	Jiang W, Chen T, Li T, Wang Z. The wave interactions of an improved Aw-Rascle-Zhang model with a non-genuinely nonlinear field. Discrete Contin Dyn Syst Ser B, 2023, 28 : 1528-1552 doi: 10.3934/dcdsb.2022134
[22]	Jiang W, Chen T, Li T, Wang Z. The Riemann problem with delta initial data for the non-isentropic improved Aw-Rascle-Zhang model. Acta Math Sci, 2023, 43B (1): 237-258
[23]	Jin W L, Zhang H M. The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model. Transp Res Part B Methodol, 2003, 37 (3): 207-223 doi: 10.1016/S0191-2615(02)00008-5
[24]	Kerner B S, Rehborn H. Experimental properties of phase transitions in traffic flow. Phys Rev Lett, 1997, 79 (20): Art 4030
[25]	Lebacque J P, Mammar S, Haj-Salem H. The Aw-Rascle and Zhang's model: vacuum problems, existence and regularity of the solutions of the Riemann problem. Transp Res Part B Methodol, 2007, 41 (7): 710-721 doi: 10.1016/j.trb.2006.11.005
[26]	Li S, Wang Q, Yang H. Riemann problem for the Aw-Rascle model of traffic flow with general pressure. Bull Malays Math Sci Soc, 2020, 43 : 3757-3775 doi: 10.1007/s40840-020-00892-0
[27]	Li T. Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow. J Differ Equ, 2003, 190 (1): 131-149 doi: 10.1016/S0022-0396(03)00014-7
[28]	Liu T P. The deterministic version of the Glimm scheme. Comm Math Phys, 1975, 57 : 135-148 doi: 10.1007/BF01625772
[29]	Liu T P, Yang T. Weak solutions of general systems of hyperbolic conservation laws. Comm Math Phys, 2002, 230 (2): 289-327 doi: 10.1007/s00220-002-0705-4
[30]	Liu Y, Sun W. Wave interactions and stability of Riemann solutions of the Aw-Rascle model for generalized Chaplygin gas. Acta Appl Math, 2018, 154 : 95-109 doi: 10.1007/s10440-017-0135-0
[31]	Lu Y. Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type. J Funct Anal, 2011, 261 (10): 2797-2815 doi: 10.1016/j.jfa.2011.07.008
[32]	Moutari S, Rascle M. A hybrid Lagrangian model based on the Aw-Rascle traffic flow model. SIAM J Appl Math, 2007, 68 (2): 413-436 doi: 10.1137/060678415
[33]	Nishida T. Global solution for an initial boundary value problem of a quasilinear hyperbolic system. Proc Japan Acad, 1968, 44 (7): 642-646
[34]	Qu A, Yuan H. Measure solutions of one-dimensional piston problem for compressible Euler equations of Chaplygin gas. J Math Anal Appl, 2020, 481 (1): Art 123486
[35]	Shen C, Sun M. Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Aw-Rascle model. J Differ Equations, 2010, 249 (12): 3024-3051 doi: 10.1016/j.jde.2010.09.004
[36]	Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1994
[37]	Sun M. Interactions of elementary waves for the Aw-Rascle model. SIAM J Appl Math, 2009, 69 : 1542-1558 doi: 10.1137/080731402
[38]	Temple B. Global solution of the Cauchy problem for a class of $2\times2$ nonstrictly hyperbolic conservation laws. Adv App Math, 1982, 3 (3): 335-375 doi: 10.1016/S0196-8858(82)80010-9
[39]	Zhang H. A non-equilibrium traffic model devoid of gas-like behavior. Transportation Res Part B, 2002, 36 (3): 275-290 doi: 10.1016/S0191-2615(00)00050-3
[40]	Zhang Q, Sheng W. Interaction of elementary waves for the Aw-Rascle traffic flow model with variable lane width. Z Angew Math Phys, 2021, 72 (5): Art 175

The Cauchy Problem for an Improved Aw-Rascle-Zhang Model

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 5

References 40

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Liutao Guo, Chaojiang Xu. Analytical Smoothing Effect on Cauchy Problem of Nonlinear Hyperbolic Schrödinger Equation [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1463-1476.
[2]	Jian Weigang, Long Wei. Tauberian Theorem for Asymptotically Periodic Functions and Its Application to Abstract Cauchy Problems [J]. Acta mathematica scientia,Series A, 2023, 43(6): 1699-1709.
[3]	Xiao Su,Shubin Wang. The Cauchy Problem for the Forth Order p-Generalized Benney-Luke Equation [J]. Acta mathematica scientia,Series A, 2022, 42(6): 1744-1753.
[4]	Xinhai He,Mei Liu,Han Yang. Existence and Uniqueness of Global Solutions for a Class of Semilinear Time Fractional Diffusion-Wave Equations [J]. Acta mathematica scientia,Series A, 2022, 42(6): 1705-1718.
[5]	Hongwu Zhang. One Regularization Method for a Cauchy Problem of Semilinear Elliptic Equation [J]. Acta mathematica scientia,Series A, 2022, 42(1): 45-57.
[6]	Xiangying Chen,Guowang Chen. Cauchy Problem for a Generalized System of Coupled Boussinesq Type Equations Arising from Nonlinear Layered Lattice Model [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1552-1567.
[7]	Wang Li. The Inverse Problem for the Supersonic Plane Flow Past a Curved Wedge [J]. Acta mathematica scientia,Series A, 2018, 38(4): 679-686.
[8]	Xie Ou, Meng Zehong, Zhao Zhenyu, You Lei. A Truncation Method Based on Hermite Functions Expansion for a Cauchy Problem of the Laplace Equation [J]. Acta mathematica scientia,Series A, 2017, 37(3): 457-468.
[9]	Deng Jiqin, Deng Ziming. Existence and Uniqueness of Solutions for Nonlocal Cauchy Problem for Fractional Evolution Equations [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1157-1164.
[10]	CHEN Xiang-Ying. Global Existence of Solution of Cauchy Problem for A Generalized IMBq Equation with Weak Damping [J]. Acta mathematica scientia,Series A, 2015, 35(1): 68-82.
[11]	SHEN Ji-Hong, ZHANG Ming-You, YANG Yan-Bing, LIU Bo-Wei, XU Run-Zhang. Global Well-posedness of Cauchy Problem for Damped Multidimensional Generalized Boussinesq Equations [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1173-1187.
[12]	SHU Xiao-Bao, DAI Bin-Xiang. S-asymptotically ω-periodic Solutions of Semi-linear Neutral Fractional Differential Equations [J]. Acta mathematica scientia,Series A, 2014, 34(1): 16-26.
[13]	XU Hong-Mei, XU Ying-Ying. Pointwise Estimates of Solutions of Generalized Benjamin-Bona-Mahony Equations in Multi Dimensions [J]. Acta mathematica scientia,Series A, 2013, 33(6): 1112-1121.
[14]	YUAN Hong-Jun, TONG Li-Ning. BV Solutions for the Quasilinear Hyperbolic Equation with Nonlinear Source and Finite Radon Measure as Initial Conditions [J]. Acta mathematica scientia,Series A, 2010, 30(1): 54-70.
[15]	ZHANG Quan-Ju. The Cauchy Problems of a class of Equations \|for the Tension of Extensible Beam [J]. Acta mathematica scientia,Series A, 2003, 23(6): 704-710.