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Continuous Riemann solvers for traffic flow at a junction
A model problem for Mean Field Games on networks
1. | "Sapienza" Università di Roma, Dip. di Scienze di Base e Applicate per l'Ingegneria, via Scarpa 16, 0161 Roma |
2. | "Sapienza" Università di Roma, Dipartimento di Matematica, P.le A. Moro 5, 00185 Roma, Italy |
3. | Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, 35121 Padova |
References:
[1] |
Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math. volume 2074, Springer, Berlin, (2013), 1-47.
doi: 10.1007/978-3-642-36433-4_1. |
[2] |
J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.
doi: 10.1016/0022-0396(88)90158-1. |
[3] |
J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion, Comm. Partial Differential Equations, 21 (1996), 255-279.
doi: 10.1080/03605309608821184. |
[4] |
M. Burger, M. Di Francesco, P. Markowich and M.-T. Wolfram, Mean field games with linear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333, arXiv:1304.5201.
doi: 10.3934/dcdsb.2014.19.1311. |
[5] |
F. Camilli, C. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equation on networks, J. Differential Equations, 254 (2013), 4122-4143.
doi: 10.1016/j.jde.2013.02.013. |
[6] |
P. Cardaliaguet, Notes on Mean Field Games: from P.-L. Lions' lectures at Collège de France, Lecture Notes given at Tor Vergata, 2010. |
[7] |
G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783.
doi: 10.1137/090771417. |
[8] |
C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520.
doi: 10.1016/j.mcm.2010.06.012. |
[9] |
M. Freidlin and S. J. Sheu, Diffusion processes on graphs: Stochastic differential equations, large deviation principle, Probab. Theory Related Fields, 116 (2000), 181-220.
doi: 10.1007/PL00008726. |
[10] |
M. Freidlin and A. Wentzell, Diffusion processes on graphs and the averaging principle, Ann. Probab., 21 (1993), 2215-2245.
doi: 10.1214/aop/1176989018. |
[11] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, 2006. |
[12] |
D. Gomes and J. Saude, Mean field games - A brief survey, Dyn.Games Appl., 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
[13] |
M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240.
doi: 10.1007/s00245-006-0887-9. |
[14] |
O. Guéant, J-M. Lasry and P-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math. volume 2003, Springer, Berlin, (2011), 205-266.
doi: 10.1007/978-3-642-14660-2_3. |
[15] |
O. Guéant, Mean field games equations with quadratic Hamiltonian: A specific approach, M3AS Math. Models Methods Appl. Sci., 22 (2012), 1250022, 37pp.
doi: 10.1142/S0218202512500224. |
[16] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968. |
[17] |
J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[18] |
D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Het. Media, 2 (2007), 55-79.
doi: 10.3934/nhm.2007.2.55. |
[19] |
D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations, Math. Methods Appl. Sci., 30 (2007), 681-706.
doi: 10.1002/mma.805. |
[20] |
B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.
doi: 10.1007/s00205-010-0366-y. |
show all references
References:
[1] |
Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math. volume 2074, Springer, Berlin, (2013), 1-47.
doi: 10.1007/978-3-642-36433-4_1. |
[2] |
J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.
doi: 10.1016/0022-0396(88)90158-1. |
[3] |
J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion, Comm. Partial Differential Equations, 21 (1996), 255-279.
doi: 10.1080/03605309608821184. |
[4] |
M. Burger, M. Di Francesco, P. Markowich and M.-T. Wolfram, Mean field games with linear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333, arXiv:1304.5201.
doi: 10.3934/dcdsb.2014.19.1311. |
[5] |
F. Camilli, C. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equation on networks, J. Differential Equations, 254 (2013), 4122-4143.
doi: 10.1016/j.jde.2013.02.013. |
[6] |
P. Cardaliaguet, Notes on Mean Field Games: from P.-L. Lions' lectures at Collège de France, Lecture Notes given at Tor Vergata, 2010. |
[7] |
G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783.
doi: 10.1137/090771417. |
[8] |
C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520.
doi: 10.1016/j.mcm.2010.06.012. |
[9] |
M. Freidlin and S. J. Sheu, Diffusion processes on graphs: Stochastic differential equations, large deviation principle, Probab. Theory Related Fields, 116 (2000), 181-220.
doi: 10.1007/PL00008726. |
[10] |
M. Freidlin and A. Wentzell, Diffusion processes on graphs and the averaging principle, Ann. Probab., 21 (1993), 2215-2245.
doi: 10.1214/aop/1176989018. |
[11] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, 2006. |
[12] |
D. Gomes and J. Saude, Mean field games - A brief survey, Dyn.Games Appl., 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
[13] |
M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240.
doi: 10.1007/s00245-006-0887-9. |
[14] |
O. Guéant, J-M. Lasry and P-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math. volume 2003, Springer, Berlin, (2011), 205-266.
doi: 10.1007/978-3-642-14660-2_3. |
[15] |
O. Guéant, Mean field games equations with quadratic Hamiltonian: A specific approach, M3AS Math. Models Methods Appl. Sci., 22 (2012), 1250022, 37pp.
doi: 10.1142/S0218202512500224. |
[16] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968. |
[17] |
J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[18] |
D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Het. Media, 2 (2007), 55-79.
doi: 10.3934/nhm.2007.2.55. |
[19] |
D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations, Math. Methods Appl. Sci., 30 (2007), 681-706.
doi: 10.1002/mma.805. |
[20] |
B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.
doi: 10.1007/s00205-010-0366-y. |
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