A nonlocal Aw-Rascle-Zhang system with linear pressure term

TL;DR

This work addresses a nonlocal extension of the Aw-Rascle-Zhang traffic model with a convolution-based linear pressure term, and proves well-posedness and an entropic selection principle via a sticky-particle Cucker–Smale approach. It derives a scalar balance law for the cumulative density $M$ and develops a full entropy framework, including Rankine–Hugoniot and Oleinik conditions, then constructs entropy weak solutions through discretized approximations and proves convergence. The scalar theory is lifted to the original system by setting $\rho = \partial_x M$ and $P = -\partial_t M$, yielding a unique, stable measure-valued ARZ solution and establishing convergence of atomic approximations to the continuum solution. By connecting nonlocal traffic dynamics to Euler–alignment-type behavior and providing robust stability bounds, the results offer a rigorous framework for anisotropic, nonlocal interactions in macroscopic traffic models with potential implications for long-time behavior and flocking phenomena.

Abstract

In this paper, we study a nonlocal extension of the Aw-Rascle-Zhang traffic model, where the pressure-like term is modeled as a convolution between vehicle density and a kernel function. This formulation captures nonlocal driver interactions and aligns structurally with the Euler-alignment system studied in [23]. Using a sticky particle approximation, we construct entropy solutions to the equation for the cumulative density and prove convergence of approximate solutions to weak solutions of the nonlocal system. The analysis includes well-posedness, stability estimates, and an entropic selection principle.

Theorems & Definitions (26)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3: The barycentric lemma
• proof
• Remark 3.1
• Theorem 3.2
• Definition 3.3
• Theorem 3.4