Communications in Mathematical Sciences

Volume 19 (2021)

Number 5

Entropy admissibility of the limit solution for a nonlocal model of traffic flow

Pages: 1447 – 1450

(Fast Communication)

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n5.a12

Authors

Alberto Bressan (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Wen Shen (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Abstract

We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $\rho$ ahead. The averaging kernel is of exponential type: $w_\varepsilon (s)=\varepsilon^{-1} e^{-s / \varepsilon}$. For any decreasing velocity function $v$, we prove that, as $\varepsilon \to 0$, the limit of solutions to the nonlocal equation coincides with the unique entropy-admissible solution to the scalar conservation law $\rho_t + {(\rho v(\rho))}_x = 0$.

Keywords

traffic flow model, conservation law, nonlocal flux, singular limit

2010 Mathematics Subject Classification

35L65

This research was partially supported by the NSF with grant DMS-2006884, “Singularities and error bounds for hyperbolic equations”.

Received 10 November 2020

Accepted 8 February 2021

Published 7 July 2021