THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE NON-ISENTROPIC IMPROVED AW-RASCLE-ZHANG MODEL*

  • Weifeng Jiang ,
  • Tingting Chen ,
  • Tong Li ,
  • Zhen Wang
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  • 1. College of Science, Key Laboratory of Intelligent Manufacturing Quality Big Data Tracing and Analysis of Zhejiang Province, China Jiliang University, Hangzhou 310018, China;
    2. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China;
    3. Department of Mathematics, The University of Iowa, IA 52246, USA;
    4. Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China
Weifeng Jiang, E-mail:casujiang89@gmail.com; Tingting Chen, E-mail: chenting0617@163.com;Tong Li, E-mail: tong-li@uiowa.edu

Received date: 2021-06-16

  Revised date: 2022-06-12

  Online published: 2023-03-01

Supported by

*Natural Science Foundation of Zhejiang (LQ18A010004), Matematical Analysis, The First class courses in Zhejiang Province (210052), the Fundamental Research Funds for the Provincial Universities of Zhejiang (210039). The research of Zhen Wang was supported by the National Natural Science Foundation of China (11771442).

Abstract

In this paper, we study the Radon measure initial value problem for the non-isentropic improved Aw-Rascle-Zhang model. For arbitrary convex $F(u)$ in this model we construct the Riemann solutions by elementary waves and $\delta$-shock waves using the method of generalized characteristic analysis. We obtain the solutions constructively for initial data containing the Dirac measure by taking the limit of the solutions for that with three piecewise constants. Moreover, we analyze different kinds of wave interactions, including the interactions of the $\delta$-shock waves with elementary waves.

Cite this article

Weifeng Jiang , Tingting Chen , Tong Li , Zhen Wang . THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE NON-ISENTROPIC IMPROVED AW-RASCLE-ZHANG MODEL*[J]. Acta mathematica scientia, Series B, 2023 , 43(1) : 237 -258 . DOI: 10.1007/s10473-023-0114-7

References

[1] Aw A, Rascle M.Resurrection of “second order” models of traffic flow. SIAM J Appl Math, 2000, 60: 916-938
[2] Zhang H.A non-equilibrium traffic model devoid of gas-like behavior. Transportation Res Part B, 2002, 36: 275-290
[3] Jiang W, Wang Z.Developing an Aw-Rascle model of traffic flow. J Eng Math, 2016, 97: 135-146
[4] Chaplygin S A.On gas jets. Sci Mem Moscow Univ Math Phys, 1904, 21: 1-121
[5] Serre D.Multidimensional shock interaction for a Chaplygin gas. Arch Rational Mech Anal, 2009, 191: 539-577
[6] Li T.Nonlinear dynamics of traffic jams. Second International Multi-Symposiums on Computer and Com- putational Sciences: IMSCCS, 2007, 2007: 550-555. doi: 10.1109/IMSCCS.2007.60
[7] Brenier Y.Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations. J Math Fluid Mech, 2005, 7: 326-331
[8] Wang Z, Zhang Q.The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Acta Math Sci, 2012, 32B(3): 825-841
[9] Qu A, Wang Z.Stability of the Riemann solutions for a Chaplygin gas. J Math Anal Appl, 2014, 409(1): 347-361
[10] Chen T, Qu A, Wang Z.Existence and uniqueness of the global L1 solution of the Euler equations for Chaplygin gas. Acta Math Sci, 2021, 41B: 941-958
[11] Lu X, Xu M, ChenW, et al. Adaptive-AR model with drivers’ prediction for traffic simulation. International Journal of Computer Games Technology, 2013, 8 pages
[12] Garavello M, Piccoli B.Traffic flow on a road network using the Aw-Rascle model. Commun Partial Differ Equ, 2006, 31: 243-275
[13] Greenberg J M.Extensions and amplifications of a traffic model of Aw and Rascle. SIAM J Appl Math, 2001, 62: 729-745
[14] Herty M, Rascle M.Coupling conditions for a class of second-order models for traffic flow. SIAM J Math Appl, 2006, 38: 595-616
[15] Klar A, Greenberg J M, Rascle M.Congestion on multilane highways. SIAM J Appl Math, 2003, 63: 818-833
[16] Moutari S, Rascle M.A hybrid Lagrangian model based on the Aw-Rascle traffic flow model. SIAM J Appl Math, 2007, 68: 413-436
[17] Li T.Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow. J Differential Equations, 2003, 190: 131-149
[18] Li T.Nonlinear dynamics of traffic jams. Physica D, 2005, 207: 41-51
[19] Lebacque J, Mammar S, Salem H.The Aw-Rascle and Zhang’s model:Vacuum problems, existence and regularity of the solutions of the Riemann problem. Transportation Res Part B, 2007, 41: 710-721
[20] Berthelin F, Degond P, Delitala M, et al.A model for the formation and evolution of traffic jams. Arch Rational Mech Anal, 2008, 187: 185-220
[21] Shen C, Sun M.Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Aw-Rascle model. J Differential Equations, 2010, 249: 3024-3051
[22] Sun M.Interactions of elementary waves for the Aw-Rascle model. SIAM J Appl Math, 2009, 69(6): 1542-1558
[23] Lu Y.Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type. J Funct Anal, 2011, 261: 2797-2815
[24] Shao Z, Huang M.Interactions of delta shock waves for the Aw-Rascle traffic model with split delta functions. J Appl Anal Comput, 2017, 7: 119-133
[25] Chen T, Jiang W, Li T.On the stability of the improved Aw-Rascle-Zhang model with Chaplygin pressure. Nonlinear Anal RWA, 2021, 62: 103351
[26] Wang G.The Riemann problem for Aw-Rascle traffic flow with negative pressure. Chin Ann Math Ser A, 2014, 35: 73-82
[27] Latora V, Baranger M, Rapisarda A, et al.The rate of entropy increase at the edge of chaos. Physics Letters A, 2000, 273(1/2): 97-103
[28] Jiang W, Wang Z.The comparison of the Riemann solutions in gas dynamics. J Math Anal Appl, 2015, 428(2): 1252-1264
[29] Song Y, Guo L.General limiting behavior of Riemann solutions to the non-isentropic Euler equations for modified Chaplygin gas. J Math Phys, 2020, 61(4): 1-18
[30] Tong M, Shen C, Lin X.The asymptotic limits of Riemann solutions for the isentropic extended Chaplygin gas dynamic system with the vanishing pressure. Bound Value Probl, 2018, 144: 1-20
[31] Pang Y.Delta shock wave in the compressible Euler equations for a Chaplygin gas. J Math Anal Appl, 2017, 448: 245-261
[32] Pang Y, Hu M.The non-self-similar Riemann solutions to a compressible fluid described by the generalized Chaplygin gas. Internat [J] Non-Linear Mech, 2018, 107: 56-63
[33] Pang Y.Delta shock wave with Dirac delta function in multiple components for the system of generalized Chaplygin gas dynamics. Bound Value Probl, 2016, 1: 1-20
[34] Ding Q, Guo L. The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas. Adv Math Phys, 2019, 12 pages
[35] Pan L, Han X.The Aw-Rascle traffic model with Chaplygin pressure. J Math Anal Appl, 2013, 401: 379-387
[36] Wang Y, Chen Y, Lai J. Fuzzy prediction for traffic flow based on delta test. Math Probl Eng, 2016, 13 pages
[37] Yang H, Sun W.The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws. Nonlinear Anal, 2007, 67(11): 3041-3049
[38] Guo L, Li T, Pan L, et al.The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term. Nonlinear Anal RWA, 2018, 41: 588-606
[39] Shao Z.The Riemann problem with delta initial data for the Aw-Rascle traffic model with Chaplygin pressure. Acta Math Sci, 2014, 34A(6): 1353-1371
[40] Chen Y, Chen T, Wang Z.The existence of the measure solution for the non-isentropic Chaplygin gas. Acta Math Sci, 2020, 40A(4): 833-841
[41] Li H, Shao Z.Vanishing pressure limit of Riemann solutions to the Aw-Rascle model for generalized Chaplygin gas. Acta Math Sci, 2017, 37A(5): 917-930
[42] Smoller J.Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1994
[43] Sheng W, Zhang T.The Riemann problem for the transportation equations in gas dynamics. Mem Amer Math Soc, 1999, 137
[44] Qu A, Yuan H.Measure solutions of one-dimensional piston problem for compressible Euler equations of Chaplygin gas. J Math Anal Appl, 2020, 481: 123486
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