# American Institute of Mathematical Sciences

June  2012, 32(6): 1915-1938. doi: 10.3934/dcds.2012.32.1915

## The Cauchy problem at a node with buffer

 1 Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale “A. Avogadro”, viale T. Michel 11, 15121 Alessandria, Italy 2 INRIA Sophia Antipolis - Méditerranée, EPI OPALE, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France

Received  January 2011 Revised  October 2011 Published  February 2012

We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by an arbitrary number of incoming and outgoing arcs connected together by a node with a buffer. Similar to [15], we define the solution to the Riemann problem at the node and we prove existence and well posedness of solutions to the Cauchy problem, by using the wave-front tracking technique and the generalized tangent vectors.
Citation: Mauro Garavello, Paola Goatin. The Cauchy problem at a node with buffer. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 1915-1938. doi: 10.3934/dcds.2012.32.1915
##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56 (electronic).  Google Scholar [2] A. Bressan, A contractive metric for systems of conservation laws with coinciding shock and rarefaction curves, J. Differential Equations, 106 (1993), 332-366. doi: 10.1006/jdeq.1993.1111.  Google Scholar [3] A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar [4] A. Bressan and R. M. Colombo, The semigroup generated by $2\times 2$ conservation laws, Arch. Rational Mech. Anal., 133 (1995), 1-75. doi: 10.1007/BF00375350.  Google Scholar [5] A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Mem. Amer. Math. Soc., 146 (2000), viii+134 pp.  Google Scholar [6] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886 (electronic). doi: 10.1137/S0036141004402683.  Google Scholar [7] R. M. Colombo, P. Goatin and B. Piccoli, Road network with phase transitions, J. Hyperbolic Differ. Equ., 7 (2010), 85-106. doi: 10.1142/S0219891610002025.  Google Scholar [8] C. D'apice, R. Manzo and B. Piccoli, Packet flow on telecommunication networks, SIAM J. Math. Anal., 38 (2006), 717-740 (electronic). doi: 10.1137/050631628.  Google Scholar [9] M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275.  Google Scholar [10] M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar [11] M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. H. Poincaré, 26 (2009), 1925-1951. Google Scholar [12] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, preprint, 2010. Google Scholar [13] S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks, Commun. Math. Sci., 4 (2006), 315-330.  Google Scholar [14] M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations, SIAM J. Math. Anal., 39 (2007), 160-173. doi: 10.1137/060659478.  Google Scholar [15] M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826 (electronic). Google Scholar [16] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Netw. Heterog. Media, 1 (2006), 275-294 (electronic). doi: 10.3934/nhm.2006.1.275.  Google Scholar [17] M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616. doi: 10.1137/05062617X.  Google Scholar [18] H. Holden and N. H. Risebro, "Front Tracking for Hyperbolic Conservation Laws," Applied Mathematical Sciences, 152, Springer-Verlag, New York, 2002.  Google Scholar [19] 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.  Google Scholar [20] A. Marigo and B. Piccoli, A fluid dynamic model for $T$-junctions, SIAM J. Math. Anal., 39 (2008), 2016-2032. doi: 10.1137/060673060.  Google Scholar [21] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [22] D. Sun, I. S. Strub and A. M. Bayen, Comparison of the performance of four Eulerian network flow models for strategic air traffic management, Netw. Heterog. Media, 2 (2007), 569-595. doi: 10.3934/nhm.2007.2.569.  Google Scholar

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##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56 (electronic).  Google Scholar [2] A. Bressan, A contractive metric for systems of conservation laws with coinciding shock and rarefaction curves, J. Differential Equations, 106 (1993), 332-366. doi: 10.1006/jdeq.1993.1111.  Google Scholar [3] A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar [4] A. Bressan and R. M. Colombo, The semigroup generated by $2\times 2$ conservation laws, Arch. Rational Mech. Anal., 133 (1995), 1-75. doi: 10.1007/BF00375350.  Google Scholar [5] A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Mem. Amer. Math. Soc., 146 (2000), viii+134 pp.  Google Scholar [6] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886 (electronic). doi: 10.1137/S0036141004402683.  Google Scholar [7] R. M. Colombo, P. Goatin and B. Piccoli, Road network with phase transitions, J. Hyperbolic Differ. Equ., 7 (2010), 85-106. doi: 10.1142/S0219891610002025.  Google Scholar [8] C. D'apice, R. Manzo and B. Piccoli, Packet flow on telecommunication networks, SIAM J. Math. Anal., 38 (2006), 717-740 (electronic). doi: 10.1137/050631628.  Google Scholar [9] M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275.  Google Scholar [10] M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar [11] M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. H. Poincaré, 26 (2009), 1925-1951. Google Scholar [12] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, preprint, 2010. Google Scholar [13] S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks, Commun. Math. Sci., 4 (2006), 315-330.  Google Scholar [14] M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations, SIAM J. Math. Anal., 39 (2007), 160-173. doi: 10.1137/060659478.  Google Scholar [15] M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826 (electronic). Google Scholar [16] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Netw. Heterog. Media, 1 (2006), 275-294 (electronic). doi: 10.3934/nhm.2006.1.275.  Google Scholar [17] M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616. doi: 10.1137/05062617X.  Google Scholar [18] H. Holden and N. H. Risebro, "Front Tracking for Hyperbolic Conservation Laws," Applied Mathematical Sciences, 152, Springer-Verlag, New York, 2002.  Google Scholar [19] 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.  Google Scholar [20] A. Marigo and B. Piccoli, A fluid dynamic model for $T$-junctions, SIAM J. Math. Anal., 39 (2008), 2016-2032. doi: 10.1137/060673060.  Google Scholar [21] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [22] D. Sun, I. S. Strub and A. M. Bayen, Comparison of the performance of four Eulerian network flow models for strategic air traffic management, Netw. Heterog. Media, 2 (2007), 569-595. doi: 10.3934/nhm.2007.2.569.  Google Scholar
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