Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model

We study a Paveri-Fontana type model, which describes the evolution of the mesoscopic distribution of vehicles through a combined effect of adjustment of the velocity with respect to nearby vehicles, and slowing down and speeding up of the vehicles arising as a result of exchange of velocity with the vehicles on the same location on the road. We first prove the global-in-time existence of weak solutions. The proof is via energy, L^p, and compact support estimates together with a velocity averaging lemma. The combined effect of the alignment nature of Qr, which keeps the characteristic from spreading, and the dissipative nature of Q_i, which gives the uniform control on the size of the distribution function, is crucially used in the estimates. We also rigorously establish a hydrodynamic limit to the pressureless Euler equation by employing the relative entropy combined with the Monge-Kantorovich-Rubinstein distance.