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

Boundary value problem for a phase transition model

Mauro Garavello 1,

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.
Keywords: hyperbolic conservation laws, Cauchy problem, Phase transition model, wave-front tracking technique..
Mathematics Subject Classification: Primary: 35L65; Secondary: 90B2.
Citation: Mauro Garavello. Boundary value problem for a phase transition model. Networks & 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.  Google Scholar

[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.  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, volume 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 (electronic). doi: 10.1137/S0036139901393184.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  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 $2\times 2$ 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, volume 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 (electronic). doi: 10.1137/S0036139901393184.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

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