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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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