Advanced Search
Article Contents
Article Contents

Existence of solutions to a boundary value problem for a phase transition traffic model

Abstract Full Text(HTML) Figure(2) Related Papers Cited by
  • We consider the initial boundary value problem for the phase transition traffic model introduced in [9], which is a macroscopic model based on a 2×2 system of conservation laws. We prove existence of solutions by means of the wave-front tracking technique, provided the initial data and the boundary conditions have finite total variation.

    Mathematics Subject Classification: Primary: 35L65; Secondary: 90B20.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The free phase $F$ and the congested phase $C$ resulting from (1.1) in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$

    Figure 2.  Wave interactions in a road. Above, from left to right, the cases $2-1/1-2$ and $\mathcal{LW}-\mathcal{PT}/\mathcal{PT}-2$. Below, from left to right, the cases $1-1/1$ and $\mathcal{PT}-1/\mathcal{PT}$

  • [1] D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1-42.  doi: 10.1007/PL00001406.
    [2] D. Amadori and R. M. Colombo, Continuous dependence for 2×2 conservation laws with boundary, J. Differential Equations, 138 (1997), 229-266.  doi: 10.1006/jdeq.1997.3274.
    [3] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099.
    [4] S. BlandinD. WorkP. GoatinB. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127.  doi: 10.1137/090754467.
    [5] A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem.
    [6] R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.  doi: 10.1137/S0036139901393184.
    [7] R. M. Colombo, Phase transitions in hyperbolic conservation laws, in Progress in analysis, Vol. I, II (Berlin, 2001), World Sci. Publ., River Edge, NJ, 2003,1279-1287.
    [8] R. M. Colombo and F. Marcellini, A mixed ODE-PDE model for vehicular traffic, Mathematical Methods in the Applied Sciences, 38 (2015), 1292-1302.  doi: 10.1002/mma.3146.
    [9] R. M. ColomboF. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666.  doi: 10.1137/090752468.
    [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.
    [11] F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122.  doi: 10.1016/0022-0396(88)90040-X.
    [12] M. Garavello, Boundary value problem for a phase transition model, Netw. Heterog. Media, 11 (2016), 89-105.  doi: 10.3934/nhm.2016.11.89.
    [13] M. Garavello and F. Marcellini, The godunov method for a 2-phase model, preprint, arXiv: 1703.05135.
    [14] M. Garavello and F. Marcellini, The riemann problem at a junction for a phase-transition traffic model, Discrete Contin. Dyn. Syst. Ser. A, to appear.
    [15] M. Garavello and B. Piccoli, Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006, Conservation laws models.
    [16] M. Garavello and B. Piccoli, Coupling of Lighthill-Whitham-Richards and phase transition models, J. Hyperbolic Differ. Equ., 10 (2013), 577-636.  doi: 10.1142/S0219891613500215.
    [17] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.  doi: 10.1016/j.mcm.2006.01.016.
    [18] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, 2nd edition, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.
    [19] J. P. LebacqueX. LouisS. MammarB. Schnetzlera and H. Haj-Salem, Modélisation du trafic autoroutier au second ordre, Comptes Rendus Mathematique, 346 (2008), 1203-1206.  doi: 10.1016/j.crma.2008.09.024.
    [20] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.
    [21] F. Marcellini, Free-congested and micro-macro descriptions of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 543-556.  doi: 10.3934/dcdss.2014.7.543.
    [22] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.
    [23] H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.
  • 加载中



Article Metrics

HTML views(419) PDF downloads(150) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint