Preprint

Inserted: 23 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 key step of the proof is a rescaling argument at the observation scale, combined with the rigidity of entire Lipschitz solutions of Burgers' equation with zero entropy production, which shows that velocity oscillations vanish at that scale. The remainder in the nonlocal entropy balance then disappears in the limit, and the limit is identified with the entropy solution by a comparison argument for measure-valued solutions.