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Regularity results for the solutions of a non-local model of traffic flow
| 1. | Laboratoire J. A. Dieudonné, UMR 7351 CNRS, Université Côte d'Azur, LJAD, CNRS, Inria, France |
| 2. | Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, France |
| 3. | Inria Sophia Antipolis - Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 2004 route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France |
We consider a non-local traffic model involving a convolution product. Unlike other studies, the considered kernel is discontinuous on $ \mathbb R $. We prove Sobolev estimates and prove the convergence of approximate solutions solving a viscous and regularized non-local equation. It leads to weak, $ {{\bf{C^{}}}}([0,T], {{\bf{L^2}}}( \mathbb R)) $, and smooth, $ {\bf{W}}^{2,2N}([0,T]\times \mathbb R) $, solutions for the non-local traffic model.
References:
| [1] |
A. Aggarwal, R. M. Colombo and P. Goatin,
Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.
doi: 10.1137/140975255. |
| [2] |
P. Amorim, R. Colombo and A. Teixeira,
On the numerical integration of scalar nonlocal conservation laws, ESAIM M2AN, 49 (2015), 19-37.
doi: 10.1051/m2an/2014023. |
| [3] |
F. Betancourt, R. Bürger, K. H. Karlsen and E. M. Tory,
On nonlocal conservation laws modelling sedimentation, Nonlinearity, 24 (2011), 855-885.
doi: 10.1088/0951-7715/24/3/008. |
| [4] |
S. Blandin and P. Goatin,
Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217-241.
doi: 10.1007/s00211-015-0717-6. |
| [5] |
M. Colombo, G. Crippa and L. V. Spinolo, On the singular local limit for conservation laws with nonlocal fluxes, arXiv e-prints, Oct. 2017. |
| [6] |
R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150023, 34p.
doi: 10.1142/S0218202511500230. |
| [7] |
J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of LWR traffic flow models with non-local flux, arXiv e-prints, Feb. 2018. |
| [8] |
P. Goatin and S. Scialanga,
Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.
doi: 10.3934/nhm.2016.11.107. |
| [9] |
S. Göttlich, S. Hoher, P. Schindler, V. Schleper and A. Verl,
Modeling, simulation and validation of material flow on conveyor belts, Applied Mathematical Modelling, 38 (2014), 3295-3313.
|
| [10] |
M. Gröschel, A. Keimer, G. Leugering and Z. Wang,
Regularity theory and adjoint-based optimality conditions for a nonlinear transport equation with nonlocal velocity, SIAM J. Control Optim., 52 (2014), 2141-2163.
doi: 10.1137/120873832. |
| [11] |
A. Keimer and L. Pflug,
Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations, 263 (2017), 4023-4069.
doi: 10.1016/j.jde.2017.05.015. |
| [12] |
A. Keimer, L. Pflug and M. Spinola,
Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping, J. Math. Anal. Appl., 466 (2018), 18-55.
doi: 10.1016/j.jmaa.2018.05.013. |
| [13] |
S. N. Kružkov,
First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.
|
| [14] |
D. Li and T. Li,
Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Networks and Heterogeneous Media, 6 (2011), 681-694.
doi: 10.3934/nhm.2011.6.681. |
| [15] |
E. Rossi and R. M. Colombo,
Nonlocal conservation laws in bounded domains, SIAM J. Math. Anal., 50 (2018), 4041-4065.
doi: 10.1137/18M1171783. |
| [16] |
A. Sopasakis and M. A. Katsoulakis,
Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), 921-944 (electronic).
doi: 10.1137/040617790. |
show all references
References:
| [1] |
A. Aggarwal, R. M. Colombo and P. Goatin,
Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.
doi: 10.1137/140975255. |
| [2] |
P. Amorim, R. Colombo and A. Teixeira,
On the numerical integration of scalar nonlocal conservation laws, ESAIM M2AN, 49 (2015), 19-37.
doi: 10.1051/m2an/2014023. |
| [3] |
F. Betancourt, R. Bürger, K. H. Karlsen and E. M. Tory,
On nonlocal conservation laws modelling sedimentation, Nonlinearity, 24 (2011), 855-885.
doi: 10.1088/0951-7715/24/3/008. |
| [4] |
S. Blandin and P. Goatin,
Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217-241.
doi: 10.1007/s00211-015-0717-6. |
| [5] |
M. Colombo, G. Crippa and L. V. Spinolo, On the singular local limit for conservation laws with nonlocal fluxes, arXiv e-prints, Oct. 2017. |
| [6] |
R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150023, 34p.
doi: 10.1142/S0218202511500230. |
| [7] |
J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of LWR traffic flow models with non-local flux, arXiv e-prints, Feb. 2018. |
| [8] |
P. Goatin and S. Scialanga,
Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.
doi: 10.3934/nhm.2016.11.107. |
| [9] |
S. Göttlich, S. Hoher, P. Schindler, V. Schleper and A. Verl,
Modeling, simulation and validation of material flow on conveyor belts, Applied Mathematical Modelling, 38 (2014), 3295-3313.
|
| [10] |
M. Gröschel, A. Keimer, G. Leugering and Z. Wang,
Regularity theory and adjoint-based optimality conditions for a nonlinear transport equation with nonlocal velocity, SIAM J. Control Optim., 52 (2014), 2141-2163.
doi: 10.1137/120873832. |
| [11] |
A. Keimer and L. Pflug,
Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations, 263 (2017), 4023-4069.
doi: 10.1016/j.jde.2017.05.015. |
| [12] |
A. Keimer, L. Pflug and M. Spinola,
Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping, J. Math. Anal. Appl., 466 (2018), 18-55.
doi: 10.1016/j.jmaa.2018.05.013. |
| [13] |
S. N. Kružkov,
First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.
|
| [14] |
D. Li and T. Li,
Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Networks and Heterogeneous Media, 6 (2011), 681-694.
doi: 10.3934/nhm.2011.6.681. |
| [15] |
E. Rossi and R. M. Colombo,
Nonlocal conservation laws in bounded domains, SIAM J. Math. Anal., 50 (2018), 4041-4065.
doi: 10.1137/18M1171783. |
| [16] |
A. Sopasakis and M. A. Katsoulakis,
Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), 921-944 (electronic).
doi: 10.1137/040617790. |
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