Regularity results for the solutions of a non-local model of traffic flow

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June 2019, 39(6): 3197-3213. doi: 10.3934/dcds.2019132

Regularity results for the solutions of a non-local model of traffic flow

Florent Berthelin ^1,2,, and Paola Goatin ^3,

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

* Corresponding author: Florent Berthelin

Received June 2018 Revised November 2018 Published February 2019

Fund Project: The first author was partially supported by Inria

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.

Keywords: Scalar conservation laws, anisotropic non-local flux, traffic flow models, viscous approximation, Sobolev estimates.

Mathematics Subject Classification: Primary: 35L65, 35B65; Secondary: 90B20.

Citation: Florent Berthelin, Paola Goatin. Regularity results for the solutions of a non-local model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3197-3213. doi: 10.3934/dcds.2019132

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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