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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

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.

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. AggarwalR. 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. AmorimR. 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

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F. BetancourtR. BürgerK. 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

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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. GöttlichS. HoherP. SchindlerV. 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. GröschelA. KeimerG. 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. KeimerL. 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. AggarwalR. 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. AmorimR. 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. BetancourtR. BürgerK. 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. GöttlichS. HoherP. SchindlerV. 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. GröschelA. KeimerG. 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. KeimerL. 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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