Preprint
Inserted: 23 sep 2026
Last Updated: 24 sep 2026
Year: 2026
Abstract:
We consider a nonlocal conservation law for traffic flow in which drivers adjust their speed to a weighted average of the density ahead. We prove that, as the observation scale tends to zero, the nonlocal averages converge strongly in $\mathrm L^p_{\mathrm{loc}}$ to the entropy solution of the local Lighthill--Whitham--Richards model. The initial density is any bounded function, the velocity is only required to be Lipschitz and non-increasing, and the kernel is anisotropic, non-negative, normalized, monotone, and of bounded variation. The main point of the proof is that the drivers' speed does not oscillate on the observation scale. This is shown by a blow-up argument and a Liouville property of the Burgers equation, and it allows us to pass to the limit in the entropy inequalities.
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