Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics

TL;DR

This paper studies nonlocal traffic flow models with Arrhenius look-ahead dynamics, establishing conditions under which solutions remain smooth or develop shocks, with implications for understanding traffic congestion.

Contribution

It introduces sub-thresholds for shock formation in traffic models with Arrhenius look-ahead dynamics, extending the analysis of solution regularity and shock development.

Findings

Short-term regularity of solutions is guaranteed if the gradient remains bounded.

Identifies specific sub-threshold conditions preventing shock formation.

Provides insights into traffic flow behavior with nonlocal interactions.

Abstract

We investigate a class of nonlocal conservation laws with the nonlinear advection coupling both local and nonlocal mechanism, which arises in several applications such as the collective motion of cells and traffic flows. It is proved that the $C^1$ solution regularity of this class of conservation laws will persist at least for a short time. This persistency may continue as long as the solution gradient remains bounded. Based on this result, we further identify sub-thresholds for finite time shock formation in traffic flow models with Arrhenius look-ahead dynamics.