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. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 38 (5). pp. 1653-1666.

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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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)
UniBasel Contributors:Crippa, Gianluca and Marconi, Elio
Item Type:Article, refereed
Article Subtype:Research Article
Note:Publication type according to Uni Basel Research Database: Journal article
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Last Modified:28 Sep 2021 09:41
Deposited On:28 Sep 2021 09:41

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