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

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  • 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.

    Citation:

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  • 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 $\mathcal{PT}-1/\mathcal{PT}$

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    [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.
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    [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.
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