On the local limit of nonlocal conservation laws

Abstract

Consider a so-called nonlocal conservation law, that is a continuity equation where the velocity field depends on the solution through the convolution with a given kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a scalar conservation law. In this talk I will address the following question: can we rigorously justify this formal limit? In general, this is not possible, as shown by explicit counterexamples.
However, in the specific framework of traffic models (with anisotropic convolution kernels) convergence results have been recently established, under suitable assumptions. My presentation will be based on joint works with Maria Colombo, Gianluca Crippa and Elio Marconi.

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