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

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 \(f_\mathrm{c}^-\) 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 \(F < f_\mathrm{c}^-\). These results are obtained by applying the wave-front tracking technique.