Boundary value problem for a phase transition model

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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

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

Abstract
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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.

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. 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. 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. 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. 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, (2000).
[6]	R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2002), 708. 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. 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, (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. 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. 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, (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. doi: 10.1098/rspa.1955.0089.
[13]	P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42. 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. doi: 10.1016/S0191-2615(00)00050-3.

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. 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. 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. 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. 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, (2000).
[6]	R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2002), 708. 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. 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, (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. 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. 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, (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. doi: 10.1098/rspa.1955.0089.
[13]	P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42. 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. doi: 10.1016/S0191-2615(00)00050-3.

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