# American Institute of Mathematical Sciences

March  2016, 11(1): 89-105. doi: 10.3934/nhm.2016.11.89

## Boundary value problem for a phase transition model

 1 Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 55, 20125 Milano, Italy

Received  April 2015 Revised  September 2015 Published  January 2016

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. Networks and Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89
##### References:
 [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 $2\times 2$ 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. Blandin, D. Work, P. Goatin, B. 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, volume 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 (electronic). doi: 10.1137/S0036139901393184. [7] R. M. Colombo, F. 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. [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. [12] 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. [13] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [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.

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##### References:
 [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 $2\times 2$ 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. Blandin, D. Work, P. Goatin, B. 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, volume 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 (electronic). doi: 10.1137/S0036139901393184. [7] R. M. Colombo, F. 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. [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. [12] 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. [13] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [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.
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