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

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

Francesca Marcellini

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.

Keywords: Hyperbolic systems of conservation laws, continuum traffic models, wave-front tracking, boundary value problem.
Mathematics Subject Classification: Primary: 35L65; Secondary: 90B20.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.  doi: 10.1137/S0036139901393184.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Garavello and F. Marcellini, The godunov method for a 2-phase model, preprint, arXiv: 1703.05135. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. P. Lebacque, X. Louis, S. Mammar, B. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.  doi: 10.1137/S0036139901393184.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Garavello and F. Marcellini, The godunov method for a 2-phase model, preprint, arXiv: 1703.05135. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. P. Lebacque, X. Louis, S. Mammar, B. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[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.  Google Scholar

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 $(\rho, \rho v)$
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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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