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Local limit of nonlocal traffic models: convergence results and total variation blow-up

Colombo, Maria and Crippa, Gianluca and Marconi, Elio and Spinolo, Laura V.. (2021) Local limit of nonlocal traffic models: convergence results and total variation blow-up. Preprints Fachbereich Mathematik, 2021 (09).

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Abstract

Consider a nonlocal conservation 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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Crippa, Gianluca and Marconi, Elio
Item Type:Preprint
Publisher:Universität Basel
Language:English
edoc DOI:
Last Modified:08 Apr 2021 19:13
Deposited On:08 Apr 2021 19:13

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