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Nonlocal Generalized Aw-Rascle-Zhang model: well-posedness and singular limit

Elio Marconi, Laura V. Spinolo

TL;DR

This work analyzes a nonlocal two-equation GARZ-type traffic model where the flux depends on a nonlocal density proxy $\xi$ computed with an anisotropic kernel. It first proves well-posedness (existence, uniqueness, stability) of the nonlocal Cauchy problem and derives key a priori bounds, including $0\le \xi\le 1$ and $0\le \rho\le \frac{1}{1- tC(V)\|z_0\|_{L^{\infty}}}$, with propagation of regularity. It then establishes a nonlocal-to-local limit for exponential kernels, showing that as the kernel concentrates to a Dirac delta the nonlocal system converges to the local GARZ model, with a novel Oleinik-type estimate playing a central role. The results include a global-in-time existence under additional structural assumptions and a stability framework with respect to initial data, marking the first rigorous limit for a coupled system of two non-decoupled equations with a nonlocal flux. Overall, the paper provides a comprehensive, mathematically rigorous foundation for nonlocal traffic models and their local limits, with implications for modeling and analysis of vehicular flows.

Abstract

We discuss a nonlocal version of the Generalized Aw-Rascle-Zhang model, a second-order vehicular traffic model where the empty road velocity is a Lagrangian marker governed by a transport equation. The evolution of the car density is described by a continuity equation where the drivers' velocity depends on both the empty road velocity and the convolution of the car density with an anisotropic kernel. We establish existence and uniqueness results. When the convolution kernel is replaced by a Dirac Delta, the nonlocal model formally boils down to the classical (local) Generalized Aw-Rascle-Zhang model, which consists of a conservation law coupled with a transport equation. In the case of exponential kernels, we establish convergence in the nonlocal-to-local limit by proving an Oleinik-type estimate for the convolution term. To the best of our knowledge, this is the first nonlocal-to-local limit result for a system of two non-decoupling equations with a nonlocal flux function.

Nonlocal Generalized Aw-Rascle-Zhang model: well-posedness and singular limit

TL;DR

This work analyzes a nonlocal two-equation GARZ-type traffic model where the flux depends on a nonlocal density proxy computed with an anisotropic kernel. It first proves well-posedness (existence, uniqueness, stability) of the nonlocal Cauchy problem and derives key a priori bounds, including and , with propagation of regularity. It then establishes a nonlocal-to-local limit for exponential kernels, showing that as the kernel concentrates to a Dirac delta the nonlocal system converges to the local GARZ model, with a novel Oleinik-type estimate playing a central role. The results include a global-in-time existence under additional structural assumptions and a stability framework with respect to initial data, marking the first rigorous limit for a coupled system of two non-decoupled equations with a nonlocal flux. Overall, the paper provides a comprehensive, mathematically rigorous foundation for nonlocal traffic models and their local limits, with implications for modeling and analysis of vehicular flows.

Abstract

We discuss a nonlocal version of the Generalized Aw-Rascle-Zhang model, a second-order vehicular traffic model where the empty road velocity is a Lagrangian marker governed by a transport equation. The evolution of the car density is described by a continuity equation where the drivers' velocity depends on both the empty road velocity and the convolution of the car density with an anisotropic kernel. We establish existence and uniqueness results. When the convolution kernel is replaced by a Dirac Delta, the nonlocal model formally boils down to the classical (local) Generalized Aw-Rascle-Zhang model, which consists of a conservation law coupled with a transport equation. In the case of exponential kernels, we establish convergence in the nonlocal-to-local limit by proving an Oleinik-type estimate for the convolution term. To the best of our knowledge, this is the first nonlocal-to-local limit result for a system of two non-decoupling equations with a nonlocal flux function.
Paper Structure (10 sections, 7 theorems, 162 equations)

This paper contains 10 sections, 7 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Well-posedness of the Cauchy problem \ref{['e:nlGARZ']},\ref{['e:idpose']}
  3. Approximation argument
  4. Passage to the limit
  5. Proof of \ref{['e:convu2']}
  6. Proof of Lemma \ref{['l:quenne']}
  7. Uniqueness
  8. Global-in-time existence
  9. Stability
  10. Oleinik estimate for a general function $V$

Key Result

Theorem 1.1

Assume e:V,e:eta, e:id and $\rho_0 \in L^1(\mathbb{R})$. Assume furthermore that Then there is $T= C(V,\| z_0 \|_{L^\infty})$ (that is, $T$ only depends on the velocity function $V$ and on $\| z_0 \|_{L^\infty}$) such that there is exactly one solution of e:nlGARZ,e:idpose in the class $(\rho, u) \in L^\infty_{\mathrm{loc}} ([0, T[; L^\infty(\mathbb{R}_+))\times L^\infty_{\mathr the two-sided est

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['l:gte']}
  • Corollary 2.5
  • ...and 5 more