# American Institue of Mathematical Sciences

2017, 12(2): 259-275. doi: 10.3934/nhm.2017011

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

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

Received  September 2016 Revised  January 2017 Published  May 2017

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.

Citation: Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011
##### 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, 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). [4] S. Blandin, D. Work, P. Goatin, B. Piccoli, 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, 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. Colombo, F. Marcellini, 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. [11] F. Dubois, 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, 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. [19] J. P. Lebacque, X. Louis, S. Mammar, B. Schnetzlera, 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, 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.

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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, 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). [4] S. Blandin, D. Work, P. Goatin, B. Piccoli, 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, 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. Colombo, F. Marcellini, 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. [11] F. Dubois, 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, 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. [19] J. P. Lebacque, X. Louis, S. Mammar, B. Schnetzlera, 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, 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.
The free phase $F$ and the congested phase $C$ resulting from (1.1) in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$
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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