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

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.

💡 Research Summary

The paper investigates a non‑local extension of the Aw‑Rascle‑Zhang (ARZ) traffic model, in which the pressure term is replaced by a convolution of the vehicle density with a kernel ω. By fixing the pressure function to the linear case p(ρ)=ρ, the authors obtain the system

∂ₜρ + ∂ₓ(ρu) = 0,
∂ₜu + u∂ₓu = –(∂ₜ + u∂ₓ)(ω * ρ).

Introducing ψ = u + ω * ρ, they observe that ψ satisfies the transport equation ∂ₜψ + u∂ₓψ = 0, allowing the original equations to be rewritten in conservative form for (ρ, ρψ). Remarkably, after elementary manipulations the system becomes identical to the one‑dimensional Euler‑alignment model

∂ₜρ + ∂ₓ(ρu) = 0,
∂ₜ(ρu) + ∂ₓ(ρu²) = ρ(ϕ * (ρu)) – ρu(ϕ * ρ),

with ϕ = ω′. This connection enables the authors to import the extensive analytical machinery developed for the Cucker‑Smale flocking model.

The core analytical tool is a sticky‑particle approximation, originally devised for pressureless Euler equations. A finite set of particles {x_i(t), v_i(t)} with masses m_i ≥ 0 (∑ m_i = 1) evolves under the Cucker‑Smale interaction law. When particles collide they stick together and continue as a single cluster, preserving total mass and the quantity ψ_i = v_i + Σ_j m_j ω(x_i – x_j). Propositions 2.1 and 2.2 establish that ψ_i is constant along trajectories and remains bounded between its initial minimum and maximum; consequently the velocities v_i are also uniformly bounded. Lemma 2.3 (the barycentric lemma) shows that, at collision times, the mass‑weighted averages of ψ are monotone, guaranteeing that the sticky dynamics respects the underlying transport structure.

Using the particle system, the authors define cumulative distribution functions

M(x,t) = ∫{–∞}^x ρ(y,t) dy, Q(x,t) = ∫{–∞}^x ρψ(y,t) dy,

which satisfy the same transport equations ∂ₜM + u∂ₓM = 0 and ∂ₜQ + u∂ₓQ = 0. Section 3 treats this scalar balance law as a nonlinear conservation law with an entropy condition derived from the invariance of ψ. By passing to the limit in the sticky‑particle approximations, they obtain an entropy weak solution (M,Q) and consequently a weak solution (ρ, u) of the original non‑local ARZ system.

Section 4 contains the main existence‑uniqueness theorem (Theorem 4.3). For initial data consisting of a Radon measure ρ₀ (total mass = 1) and a bounded velocity u₀, there exists a global-in‑time weak solution (ρ,u) with ρ ∈ 𝔐⁺(ℝ) and u ∈ L^∞(ℝ×