Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Aw-Rascle-Zhang traffic models with nonlocal look-ahead interactions

Thomas Hamori, Changhui Tan

Abstract

We present a new family of second-order traffic flow models, extending the Aw-Rascle-Zhang (ARZ) model to incorporate nonlocal interactions. Our model includes a specific nonlocal Arrhenius-type look-ahead slowdown factor. We establish both local and global well-posedness theories for these nonlocal ARZ models. In contrast to the local ARZ model, where generic smooth initial data typically lead to finite-time shock formation, we show that our nonlocal ARZ model exhibits global regularity for a class of smooth subcritical initial data. Our result highlights the potential of nonlocal interactions to mitigate shock formations in second-order traffic flow models. Our analytical approach relies on investigating phase plane dynamics. We introduce a novel comparison principle based on a mediant inequality to effectively handle the nonlocal information inherent in our model.

On the Aw-Rascle-Zhang traffic models with nonlocal look-ahead interactions

Abstract

We present a new family of second-order traffic flow models, extending the Aw-Rascle-Zhang (ARZ) model to incorporate nonlocal interactions. Our model includes a specific nonlocal Arrhenius-type look-ahead slowdown factor. We establish both local and global well-posedness theories for these nonlocal ARZ models. In contrast to the local ARZ model, where generic smooth initial data typically lead to finite-time shock formation, we show that our nonlocal ARZ model exhibits global regularity for a class of smooth subcritical initial data. Our result highlights the potential of nonlocal interactions to mitigate shock formations in second-order traffic flow models. Our analytical approach relies on investigating phase plane dynamics. We introduce a novel comparison principle based on a mediant inequality to effectively handle the nonlocal information inherent in our model.
Paper Structure (17 sections, 18 theorems, 166 equations, 1 figure)

This paper contains 17 sections, 18 theorems, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Macroscopic traffic flow models
  3. Nonlocal traffic flow models
  4. An ARZ model with nonlocal look-ahead interactions
  5. Outline of the paper
  6. Statement of Main Results
  7. Local well-posedness and regularity criteria
  8. A priori bounds
  9. Energy estimates
  10. Regularity criteria
  11. Finite time blowup for local ARZ model
  12. Global well-posedness
  13. Phase plane dynamics on $(\rho, \partial_x\rho)$
  14. The threshold $\sigma$ for the first-order model
  15. A new threshold $\eta$
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $k>\frac{3}{2}$. Consider the Cauchy problem eq:main-eq:init where the initial data $(\rho_0,u_0)$ satisfy the assumptions A1-A4. Assume $U$ satisfies eq:Ucond and eq:Usmooth, and $w$ satisfies eq:w. Then

Figures (1)

  • Figure 1: Illustrations to the threshold function $\eta$ and the subcritical region in \ref{['eq:subcritical']}. The flux is chosen as $f(\rho)=\rho(1-\rho)^2$. Depending on different choices of $\psi_0$, $\eta$ can either be bounded in $[0,\rho_c]$ (left), or it may blow up to $-\infty$ at $\rho_*<\rho_c$ (right).

Theorems & Definitions (33)

  • Theorem 2.1: Local well-posedness
  • Remark 2.1
  • Theorem 2.2: Finite-time blowup for the local ARZ model with linear velocity
  • Theorem 2.3: Finite-time blowup for local ARZ model with general velocity
  • Remark 2.2
  • Theorem 2.4: Global well-posedness
  • Remark 2.3
  • Proposition 3.1: Maximum principles
  • proof
  • Remark 3.1
  • ...and 23 more