In this paper, we present a general phase transition model that describes the evolution of vehicular traffic along a one-lane road. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the free-flow phase and by a system of 2 conservation laws in the congested phase. The free-flow phase is described by a one-dimensional fundamental diagram corresponding to a Newell-Daganzo type flux. The congestion phase is described by a two-dimensional fundamental diagram obtained by perturbing a general fundamental flux. In particular, we study the resulting Riemann problems in the case a local point constraint on the flow of the solutions is enforced. KEYWORDS conservation laws, phase transitions, Lighthill-Whitham-Richards model, Aw-Rascle-Zhang model, vehicular traffic, unilateral point constraint, Riemann problem INTRODUCTIONThis paper deals with phase transition models (PT models for short) of hyperbolic conservation laws for vehicular traffic. More precisely, we focus on macroscopic models that describe vehicular traffic along a unidirectional one-lane road, which has neither entrances nor exits and where overtaking is not allowed.In the specialized literature, vehicular traffic is shown to behave differently depending on whether it is free or congested. This leads to consider 2 different regimes corresponding to a free-flow phase (denoted by Ω f ) and a congested phase (denoted by Ω c ). The PT models analysed here are given by a scalar conservation law in the free-flow phase, coupled with a 2 × 2 system of conservation laws in the congested phase. The coupling is achieved via phase transitions, namely, discontinuities that separate 2 states belonging to different phases and that satisfy the Rankine-Hugoniot condition.This two-phase approach was introduced in Colombo 1 and is motivated by experimental observations; according to which for low densities, the flow of vehicles is free and approximable by a one-dimensional flux function, while at high densities, the flow is congested and covers a two-dimensional domain in the fundamental diagram, see Colombo 1, Figure 1.1 or Blandin et al. 2, Figure 3.1 Hence, it is reasonable to describe the dynamics in the free regime by a first-order model (a scalar PDE) and those in the congested regime by a second-order model (a 2 × 2 system of PDEs).In literature, 1 Colombo proposed to let the dynamics in free-flow phase Ω f be governed by the classical LWR model by Lighthill, Whitham, and Richards 3,4 ; on the other hand, the congested phase Ω c includes one more equation for the conservation of a linearized momentum. We recall that the LWR model expresses the conservation of the number of vehicles and assumes that the velocity is a function of the density alone. Furthermore, Colombo's model uses a Greenshields type (strictly parabolic) flux function in the free-flow regime and one consequence is that Ω f cannot intersect Ω c , see Colombo. 1, Remark 2Math Meth Appl Sci. 2017;40:6623-6641.wile...

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.2010 Mathematical Subject Classification. Primary: 35L65, 90B20, 35L45