# American Institute of Mathematical Sciences

September  2015, 35(9): 4149-4171. doi: 10.3934/dcds.2015.35.4149

## Continuous Riemann solvers for traffic flow at a junction

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

Received  May 2014 Revised  August 2014 Published  April 2015

The paper studies a class of conservation law models for traffic flow on a family of roads, near a junction. A Riemann Solver is constructed, where the incoming and outgoing fluxes depend Hölder continuously on the traffic density and on the drivers' turning preferences. However, various examples show that, if junction conditions are assigned in terms of Riemann Solvers, then the Cauchy problem on a network of roads can be ill posed, even for initial data having small total variation.
Citation: Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149
##### References:
 [1] A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem,, Oxford University Press, (2000).   Google Scholar [2] 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.  doi: 10.4171/EMSS/2.  Google Scholar [3] A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks,, Networks & Heter. Media, 8 (2013), 627.  doi: 10.3934/nhm.2013.8.627.  Google Scholar [4] A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads,, Networks & Heter. Media, ().   Google Scholar [5] A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues,, Networks & Heter. Media, ().   Google Scholar [6] A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws,, Nonlinear Analysis, 34 (1998), 637.  doi: 10.1016/S0362-546X(97)00590-7.  Google Scholar [7] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 36 (2005), 1862.  doi: 10.1137/S0036141004402683.  Google Scholar [8] C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law,, J. Math. Anal. Appl., 38 (1972), 33.  doi: 10.1016/0022-247X(72)90114-X.  Google Scholar [9] C. Daganzo, Fundamentals of Transportation and Traffic Operations,, Pergamon-Elsevier, (1997).   Google Scholar [10] L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (2010).   Google Scholar [11] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst., 32 (2012), 1915.  doi: 10.3934/dcds.2012.32.1915.  Google Scholar [12] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models,, in AIMS Series on Applied Mathematics, (2006).   Google Scholar [13] M. Garavello and B. Piccoli, Conservation laws on complex networks,, Ann. Inst. H. Poincaré, 26 (2009), 1925.  doi: 10.1016/j.anihpc.2009.04.001.  Google Scholar [14] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modeling intersections of vehicular traffic flow,, Netw. Heter. Media., 1 (2006), 275.  doi: 10.3934/nhm.2006.1.275.  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.  doi: 10.3934/nhm.2009.4.813.  Google Scholar [16] 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.  doi: 10.1051/cocv/2012002.  Google Scholar [17] 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.  doi: 10.1098/rspa.1955.0089.  Google Scholar [18] P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar [19] J. Smoller, Shock Waves and Reaction-Diffusion Equations,, $2^{nd}$ edition, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

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##### References:
 [1] A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem,, Oxford University Press, (2000).   Google Scholar [2] 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.  doi: 10.4171/EMSS/2.  Google Scholar [3] A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks,, Networks & Heter. Media, 8 (2013), 627.  doi: 10.3934/nhm.2013.8.627.  Google Scholar [4] A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads,, Networks & Heter. Media, ().   Google Scholar [5] A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues,, Networks & Heter. Media, ().   Google Scholar [6] A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws,, Nonlinear Analysis, 34 (1998), 637.  doi: 10.1016/S0362-546X(97)00590-7.  Google Scholar [7] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 36 (2005), 1862.  doi: 10.1137/S0036141004402683.  Google Scholar [8] C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law,, J. Math. Anal. Appl., 38 (1972), 33.  doi: 10.1016/0022-247X(72)90114-X.  Google Scholar [9] C. Daganzo, Fundamentals of Transportation and Traffic Operations,, Pergamon-Elsevier, (1997).   Google Scholar [10] L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (2010).   Google Scholar [11] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst., 32 (2012), 1915.  doi: 10.3934/dcds.2012.32.1915.  Google Scholar [12] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models,, in AIMS Series on Applied Mathematics, (2006).   Google Scholar [13] M. Garavello and B. Piccoli, Conservation laws on complex networks,, Ann. Inst. H. Poincaré, 26 (2009), 1925.  doi: 10.1016/j.anihpc.2009.04.001.  Google Scholar [14] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modeling intersections of vehicular traffic flow,, Netw. Heter. Media., 1 (2006), 275.  doi: 10.3934/nhm.2006.1.275.  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.  doi: 10.3934/nhm.2009.4.813.  Google Scholar [16] 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.  doi: 10.1051/cocv/2012002.  Google Scholar [17] 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.  doi: 10.1098/rspa.1955.0089.  Google Scholar [18] P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar [19] J. Smoller, Shock Waves and Reaction-Diffusion Equations,, $2^{nd}$ edition, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar
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