An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic

TL;DR

This paper establishes the existence of solutions for a two-phase traffic flow model with a metastable phase, using wave-front tracking, under certain flux constraints and initial conditions.

Contribution

It introduces a novel existence result for a constrained two-phase traffic model with metastability, combining different models for free and congested traffic.

Findings

Global solutions exist when the flux constraint exceeds the minimal flux of the metastable phase.

Sufficient conditions are provided for solutions to exist even when the flux constraint is lower.

Wave-front tracking effectively proves the existence of solutions in this complex traffic model.

Abstract

In this paper we study a phase transition model for vehicular traffic flows. Two phases are taken into account, according to whether the traffic is light or heavy. We assume that the two phases have a non-empty intersection, the so called metastable phase. The model is given by the Lighthill-Whitham-Richards model in the free-flow phase and by the Aw-Rascle-Zhang model in the congested phase. In particular, we study the existence of solutions to Cauchy problems satisfying a local point constraint on the density flux. We prove that if the constraint F is higher than the minimal flux f -- c of the metastable phase, then constrained Cauchy problems with initial data of bounded total variation admit globally defined solutions. We also provide sufficient conditions on the initial data that guarantee the global existence of solutions also in the case F < f -- c. These results are obtained by applying the wave-front tracking technique.