Well-posedness for a monotone solver for traffic junctions

TL;DR

This paper proves the well-posedness of solutions for traffic junction models using vanishing viscosity limits, scalar conservation laws, and Kruzhkov-type entropies, ensuring consistent and unique solutions at junctions.

Contribution

It introduces a comprehensive framework for well-posedness of traffic junction solutions, including a detailed description of constant solutions and adapted entropy conditions.

Findings

Complete description of road-wise constant solutions

Introduction of Kruzhkov-type adapted entropies at junctions

Proof of well-posedness for the traffic junction problem

Abstract

In this paper we aim at proving well-posedness of solutions obtained as vanishing viscosity limits for the Cauchy problem on a traffic junction where $m$ incoming and $n$ outgoing roads meet. The traffic on each road is governed by a scalar conservation law $ \rho_{h,t} + f_h(\rho_h)_x = 0$, for $h\in \{1,\ldots, m+n\}$. Our proof relies upon the complete description of the set of road-wise constant solutions and its properties, which is of some interest on its own. Then we introduce a family of Kruzhkov-type adapted entropies at the junction and state a definition of admissible solution in the same spirit as in \cite{diehl, ColomboGoatinConstraint, scontrainte, AC_transmission, germes}.