# American Institute of Mathematical Sciences

June  2015, 10(2): 255-293. doi: 10.3934/nhm.2015.10.255

## Conservation law models for traffic flow on a network of roads

 1 Penn State University Mathematics Dept., University Park, State College, PA 16802 2 Department of Mathematics, Penn State University, University Park, PA 16802, United States

Received  June 2014 Revised  January 2015 Published  April 2015

The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads.
If the queue sizes within the buffer are known, then the initial-boundary value problems become decoupled and can be independently solved along each incoming road. Three variational problems are introduced, related to different kind of boundary conditions. From the value functions, one recovers the traffic density along each incoming or outgoing road by a Lax type formula.
Conversely, if these value functions are known, then the queue sizes can be determined by balancing the boundary fluxes of all incoming and outgoing roads. In this way one obtains a contractive transformation, whose fixed point yields the unique solution of the Cauchy problem for traffic flow in an neighborhood of the intersection.
The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves well-posedness for general $L^\infty$ data, and continuity w.r.t. weak convergence of the initial densities.
Citation: Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks and Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255
##### References:
 [1] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1. [2] C. I. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear equations with boundary conditions, Comm. P.D.E., 4 (1979), 1017-1034. doi: 10.1080/03605307908820117. [3] A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2. [4] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM J. Math. Anal., 43 (2011), 2384-2417. doi: 10.1137/110825145. [5] A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks, Networks & Heter. Media, 8 (2013), 627-648. doi: 10.3934/nhm.2013.8.627. [6] A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues, to appear in Networks Heter. Media. [7] A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction, to appear in Discr. Cont. Dyn. Syst. [8] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [9] C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains. A Continuous Approach, SIAM, Philadelphia, PA, 2010. doi: 10.1137/1.9780898717600. [10] L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, Providence, RI, 2010. [11] T. Friesz, Dynamic Optimization and Differential Games, Springer, New York, 2010. doi: 10.1007/978-0-387-72778-3. [12] T. Friesz and K. Han, Dynamic Network User Equilibrium, Springer-Verlag, to appear. [13] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst., 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915. [14] M. Garavello and B. Piccoli, Source-destination flow on a road network, Commun. Math. Sci., 3 (2005), 261-283. doi: 10.4310/CMS.2005.v3.n3.a1. [15] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, Springfield, Mo., 2006. [16] M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincaré, 26 (2009), 1925-1951. doi: 10.1016/j.anihpc.2009.04.001. [17] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, in Advances in Dynamic Network Modeling in Complex Transportation Systems (eds. S V. Ukkusuri and K. Ozbay), Complex Networks and Dynamic Systems, 2, Springer, New York, 2013, 143-161. doi: 10.1007/978-1-4614-6243-9_6. [18] M. Herty, C. Kirchner, S. Moutari and M. Rascle, Multicommodity flows on road networks, Commun. Math. Sci., 6 (2008), 171-187. doi: 10.4310/CMS.2008.v6.n1.a8. [19] 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. [20] M. Herty, J. P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826. doi: 10.3934/nhm.2009.4.813. [21] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002. [22] P. D. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 10 (1957), 537-556. doi: 10.1002/cpa.3160100406. [23] P. Le Floch, Explicit formula for scalar non-linear conservation laws with boundary condition, Math. Methods Appl. Sciences, 10 (1988), 265-287. doi: 10.1002/mma.1670100305. [24] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London: Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [25] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [26] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Second edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

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##### References:
 [1] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1. [2] C. I. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear equations with boundary conditions, Comm. P.D.E., 4 (1979), 1017-1034. doi: 10.1080/03605307908820117. [3] A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2. [4] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM J. Math. Anal., 43 (2011), 2384-2417. doi: 10.1137/110825145. [5] A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks, Networks & Heter. Media, 8 (2013), 627-648. doi: 10.3934/nhm.2013.8.627. [6] A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues, to appear in Networks Heter. Media. [7] A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction, to appear in Discr. Cont. Dyn. Syst. [8] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [9] C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains. A Continuous Approach, SIAM, Philadelphia, PA, 2010. doi: 10.1137/1.9780898717600. [10] L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, Providence, RI, 2010. [11] T. Friesz, Dynamic Optimization and Differential Games, Springer, New York, 2010. doi: 10.1007/978-0-387-72778-3. [12] T. Friesz and K. Han, Dynamic Network User Equilibrium, Springer-Verlag, to appear. [13] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst., 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915. [14] M. Garavello and B. Piccoli, Source-destination flow on a road network, Commun. Math. Sci., 3 (2005), 261-283. doi: 10.4310/CMS.2005.v3.n3.a1. [15] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, Springfield, Mo., 2006. [16] M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincaré, 26 (2009), 1925-1951. doi: 10.1016/j.anihpc.2009.04.001. [17] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, in Advances in Dynamic Network Modeling in Complex Transportation Systems (eds. S V. Ukkusuri and K. Ozbay), Complex Networks and Dynamic Systems, 2, Springer, New York, 2013, 143-161. doi: 10.1007/978-1-4614-6243-9_6. [18] M. Herty, C. Kirchner, S. Moutari and M. Rascle, Multicommodity flows on road networks, Commun. Math. Sci., 6 (2008), 171-187. doi: 10.4310/CMS.2008.v6.n1.a8. [19] 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. [20] M. Herty, J. P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826. doi: 10.3934/nhm.2009.4.813. [21] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002. [22] P. D. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 10 (1957), 537-556. doi: 10.1002/cpa.3160100406. [23] P. Le Floch, Explicit formula for scalar non-linear conservation laws with boundary condition, Math. Methods Appl. Sciences, 10 (1988), 265-287. doi: 10.1002/mma.1670100305. [24] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London: Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [25] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [26] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Second edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.
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