Calculus of Variations and Geometric Measure Theory

M. Colombo - G. Crippa - E. Marconi - L. V. Spinolo

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

created by marconi on 28 Jan 2021


Accepted Paper

Inserted: 28 jan 2021
Last Updated: 28 jan 2021

Journal: Ann. de l'institute H. Poincaré, Analyse non linéaire
Year: 2021


Consider a nonlocal conservation law where the ux 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 trac: 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 justi ed provided the initial datum satis es 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.

Keywords: traffic model, nonlocal conservation law, anisotropic kernel, nonlocal continuity equation, singular limit, local limit, Olenik estimate