Communications in Mathematical Sciences

Volume 20 (2022)

Number 4

A sharp critical threshold for a traffic flow model with look-ahead dynamics

Pages: 1151 – 1172



Yongki Lee (Department of Mathematical Sciences, Georgia Southern University, Statesboro, Ga., U.S.A.)

Changhui Tan (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)


We study a Lighthill–Whitham–Richards (LWR) type traffic flow model, with a nonlocal look-ahead interaction that has a slow-down effect depending on the traffic ahead. We show a sharp critical threshold condition on the initial data that distinguishes global smooth solutions and finitetime wave breakdown. It is well-known that the LWR model leads to a finite-time shock formation, representing the creation of traffic jams, for generic smooth initial data with finite mass. Our result shows that the nonlocal slowdown effect can help to prevent shock formations, for a class of subcritical initial data.


nonlocal conservation law, traffic flow, critical threshold, global regularity, shock formation

2010 Mathematics Subject Classification

35B51, 35B65, 35L65, 35L67, 76Axx

Received 19 March 2021

Received revised 1 August 2021

Accepted 31 October 2021

Published 11 April 2022