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Well-posedness results for the Generalized Aw-Rascle-Zhang model

Elio Marconi, Laura V. Spinolo

TL;DR

The paper analyzes the well-posedness of the Generalized Aw-Rascle-Zhang (GARZ) traffic model, formulated as $\partial_t \rho + \partial_x [V(\rho,u)\rho] =0$ and $\partial_t u + V(\rho,u) \partial_x u =0$, and its entropy solution framework. It introduces an iterative approximation scheme, proves uniform BV/L^∞ bounds, and uses compactness arguments to obtain a global entropy solution with $\rho\in L^\infty_{loc}(\mathbb{R}_+;BV(\mathbb{R}))$, $u\in L^\infty(\mathbb{R}_+;W^{1\infty}(\mathbb{R}))$, and $\partial_x u = \rho z$ for some $z\in W^{1\infty}$. Uniqueness and stability are established via a stability theory for conservation laws with space–time dependent fluxes (MRS), yielding a Grönwall-type bound that implies continuous dependence on initial data and asymptotic states. The work provides a rigorous local-to-global well-posedness foundation for GARZ, connecting to the nonlocal GARZ program and supporting future nonlocal-to-local limit analyses in traffic-flow modeling.

Abstract

We establish existence, uniqueness and stability results for the so-called Generalized Aw-Rascle-Zhang model, a second order traffic model introduced by Fan, Herty and Seibold. Our analysis is motivated by the companion paper 'Nonlocal Generalized Aw-Rascle-Zhang model: well-posedness and singular limit'.

Well-posedness results for the Generalized Aw-Rascle-Zhang model

TL;DR

The paper analyzes the well-posedness of the Generalized Aw-Rascle-Zhang (GARZ) traffic model, formulated as and , and its entropy solution framework. It introduces an iterative approximation scheme, proves uniform BV/L^∞ bounds, and uses compactness arguments to obtain a global entropy solution with , , and for some . Uniqueness and stability are established via a stability theory for conservation laws with space–time dependent fluxes (MRS), yielding a Grönwall-type bound that implies continuous dependence on initial data and asymptotic states. The work provides a rigorous local-to-global well-posedness foundation for GARZ, connecting to the nonlocal GARZ program and supporting future nonlocal-to-local limit analyses in traffic-flow modeling.

Abstract

We establish existence, uniqueness and stability results for the so-called Generalized Aw-Rascle-Zhang model, a second order traffic model introduced by Fan, Herty and Seibold. Our analysis is motivated by the companion paper 'Nonlocal Generalized Aw-Rascle-Zhang model: well-posedness and singular limit'.
Paper Structure (6 sections, 2 theorems, 77 equations)

This paper contains 6 sections, 2 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Existence
  3. Construction of the approximation
  4. Passage to the limit
  5. Proof of Lemma \ref{['l:phienne']}
  6. Uniqueness and stability

Key Result

Theorem 1.2

Assume $V$ satisfies e:V and that the initial data $(\rho_0, u_0)$ satisfy e:id, $\rho_0 \in BV (\mathbb{R})$ and and Then there is a unique entropy admissible solution of e:GARZ,e:idpose in the class of functions $(\rho, u) \in L^\infty_{\mathrm{loc}} (\mathbb{R}_+;BV(\mathbb{R})) \times L^\infty (\mathbb{R}_+; W^{1 \infty}(\mathbb{R}))$ such that $\partial_x u = \rho z$ with $z \in W^{1 \inft

Theorems & Definitions (3)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1