Entropy solutions for a two-phase transition model for vehicular traffic with metastable phase and time depending point constraint on the density flow

We consider a macroscopic two-phase transition model for vehicular traffic flow subject to a point constraint on the density flux. The two phases correspond to light and heavy traffic and their dynamics are described respectively by the Lighthill-Whitham-Richards model and the Aw-Rascle-Zhang model. Their intersection, the so-called metastable phase, is assumed to be non empty. In order to guarantee the control of total variation of solutions, the t-dependent function prescribing the level of the constraint should be piecewise constant. It can be given a priori or it can be computed in a time-discrete way from the solution itself, thus giving rise to nonlocal phase transition models with point constraint in the spirit of LWR-based models described in Andreianov, Donadello, Razafison, Rosini, JMPA (2018). We introduce a new definition of admissible solutions for the Cauchy problem, for which we prove existence and we provide a characterization. In particular, these admissible solutions attain the maximal flux allowed by the constraint whenever it is enforced, which guarantees compatibility of the constructed solutions with the modeling assumption imposed at the level of the Riemann solver. These results rely on the wave front tracking method and on adaptation of the specific entropies and renormalization properties introduced in Andreainov, Donadello, Rosini, M3AS (2016) while dealing with the ARZ system. 2010 Mathematical Subject Classification. Primary: 35L65, 90B20, 35L4