Local limit of nonlocal traffic models: Convergence results and total variation blow-up
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.
- Publication:
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Annales de L'Institut Henri Poincare Section (C) Non Linear Analysis
- Pub Date:
- September 2021
- DOI:
- 10.1016/j.anihpc.2020.12.002
- arXiv:
- arXiv:1808.03529
- Bibcode:
- 2021AIHPC..38.1653C
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- doi:10.1016/j.anihpc.2020.12.002