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
Keywords: Traffic model, Nonlocal conservation law, Anisotropic kernel, Nonlocal continuity equation, Local limit, Oleĭnik estimate
Colombo, Maria 1 ; Crippa, Gianluca 2 ; Marconi, Elio 2 ; Spinolo, Laura V. 3
@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},
year = {2021},
publisher = {Elsevier},
volume = {38},
number = {5},
doi = {10.1016/j.anihpc.2020.12.002},
mrnumber = {4300935},
zbl = {1473.35360},
language = {en},
url = {http://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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
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