Consider a nonlocal conservation law where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from 0. We also exhibit a counter-example showing that, if the initial datum attains the value 0, then there are severe obstructions to a convergence proof.
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DOI : 10.1016/j.anihpc.2020.12.002
Mots-clés : Traffic model, Nonlocal conservation law, Anisotropic kernel, Nonlocal continuity equation, Local limit, Oleĭnik estimate
@article{AIHPC_2021__38_5_1653_0,
author = {Colombo, Maria and Crippa, Gianluca and Marconi, Elio and Spinolo, Laura V.},
title = {Local limit of nonlocal traffic models: {Convergence} results and total variation blow-up},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1653--1666},
publisher = {Elsevier},
volume = {38},
number = {5},
year = {2021},
doi = {10.1016/j.anihpc.2020.12.002},
mrnumber = {4300935},
zbl = {1473.35360},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.12.002/}
}
TY - JOUR AU - Colombo, Maria AU - Crippa, Gianluca AU - Marconi, Elio AU - Spinolo, Laura V. TI - Local limit of nonlocal traffic models: Convergence results and total variation blow-up JO - Annales de l'I.H.P. Analyse non linéaire PY - 2021 SP - 1653 EP - 1666 VL - 38 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2020.12.002/ DO - 10.1016/j.anihpc.2020.12.002 LA - en ID - AIHPC_2021__38_5_1653_0 ER -
%0 Journal Article %A Colombo, Maria %A Crippa, Gianluca %A Marconi, Elio %A Spinolo, Laura V. %T Local limit of nonlocal traffic models: Convergence results and total variation blow-up %J Annales de l'I.H.P. Analyse non linéaire %D 2021 %P 1653-1666 %V 38 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2020.12.002/ %R 10.1016/j.anihpc.2020.12.002 %G en %F AIHPC_2021__38_5_1653_0
Colombo, Maria; Crippa, Gianluca; Marconi, Elio; Spinolo, Laura V. Local limit of nonlocal traffic models: Convergence results and total variation blow-up. Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1653-1666. doi : 10.1016/j.anihpc.2020.12.002. http://www.numdam.org/articles/10.1016/j.anihpc.2020.12.002/
[1] On the numerical integration of scalar nonlocal conservation laws, ESAIM: Math. Model. Numer. Anal., Volume 49 (2015) no. 1, pp. 19-37 | DOI | Numdam | MR | Zbl
[2] Regularity results for the solutions of a non-local model of traffic flow, Discrete Contin. Dyn. Syst., Volume 39 (2019) no. 6, pp. 3197-3213 | DOI | MR | Zbl
[3] On nonlocal conservation laws modelling sedimentation, Nonlinearity, Volume 24 (2011) no. 3, pp. 855-885 | DOI | MR | Zbl
[4] Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., Volume 132 (2016) no. 2, pp. 217-241 | DOI | MR | Zbl
[5] On traffic flow with nonlocal flux: a relaxation representation, 2019 | arXiv | MR | Zbl
[6] Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM: M2AN, Volume 52 (2018) no. 1, pp. 163-180 | DOI | Numdam | MR | Zbl
[7] On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws, 2019 | arXiv | MR | Zbl
[8] On the singular local limit for conservation laws with nonlocal fluxes, Arch. Ration. Mech. Anal., Volume 233 (2019) no. 3, pp. 1131-1167 | DOI | MR | Zbl
[9] A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., Volume 22 (2012) no. 4 | DOI | MR | Zbl
[10] Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., Volume 17 (2011) no. 2, pp. 353-379 | DOI | Numdam | MR | Zbl
[11] Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, NoDEA Nonlinear Differ. Equ. Appl., Volume 20 (2013) no. 3, pp. 523-537 | DOI | MR | Zbl
[12] Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 325, Springer-Verlag, Berlin, 2016 | MR | Zbl
[13] Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, Volume 11 (2016) no. 1, pp. 107-121 | DOI | MR | Zbl
[14] Existence, uniqueness and regularity results on nonlocal balance laws, J. Differ. Equ., Volume 263 (2017) no. 7, pp. 4023-4069 | DOI | MR | Zbl
[15] On approximation of local conservation laws by nonlocal conservation laws, J. Math. Anal. Appl., Volume 475 (2019) no. 2, pp. 1927-1955 | DOI | MR | Zbl
[16] Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Netw. Heterog. Media, Volume 6 (2011) no. 4, pp. 681-694 | DOI | MR | Zbl
[17] Discontinuous solutions of non-linear differential equations, Am. Math. Soc. Transl., Volume 2 (1963) no. 26, pp. 95-172 | MR | Zbl
[18] On a nonlocal dispersive equation modeling particle suspensions, Q. Appl. Math., Volume 57 (1999) no. 3, pp. 573-600 | DOI | MR | Zbl
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