The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow

We study the Cauchy problem for the Aw-Rascle-Zhang model for traffic flow with a flux constraint located at x = 0. The purpose of such a traffic model is to describe an obstruction in a road, such as a toll gate or a construction site. We consider here the concept of solutions introduced by Garavello and Goatin in 2011, which conserves the total number of cars flowing through the obstruction, but does not conserve the generalized momentum. We prove the existence of such solutions to the Cauchy problem by using the wavefront tracking method. We also compare our results with another concept of solutions introduced also by Garavello and Goatin in 2011 for the same model with a flux constraint. For this second formulation, Andreianov, Donadello, and Rosini had also proved the existence of solutions to the Cauchy problem