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Networks and Heterogeneous Media

2016, Volume 11, Issue 1: 89-105. doi: 10.3934/nhm.2016.11.89
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Boundary value problem for a phase transition model

  • 1.
    Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 55, 20125 Milano
  • Received: 01 April 2015 Revised: 01 September 2015

  • Primary: 35L65; Secondary: 90B20.

  • We consider the boundary value problem for the phase transition (PT) model, introduced in [4] and in [7]. By using the wave-front tracking technique, we prove existence of solutions when the initial and boundary conditions have finite total variation.

    Citation: Mauro Garavello. Boundary value problem for a phase transition model[J]. Networks and Heterogeneous Media, 2016, 11(1): 89-105. doi: 10.3934/nhm.2016.11.89

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  • We consider the boundary value problem for the phase transition (PT) model, introduced in [4] and in [7]. By using the wave-front tracking technique, we prove existence of solutions when the initial and boundary conditions have finite total variation.


    [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
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    [8] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 2005. doi: 10.1007/3-540-29089-3
    [9] 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
    [10] 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
    [11] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, volume 152 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-3-642-56139-9
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    [14] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3

  • This article has been cited by:

    1. Francesca Marcellini, Existence of solutions to a boundary value problem for a phase transition traffic model, 2017, 12, 1556-181X, 259, 10.3934/nhm.2017011
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