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.
Cite this article
Gianluca Crippa, Elio Marconi, Laura V. Spinolo, Maria Colombo, Local limit of nonlocal traffic models: Convergence results and total variation blow-up. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 5, pp. 1653–1666DOI 10.1016/J.ANIHPC.2020.12.002