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 velocity averaging lemma. The combined effect of alignment nature of $Q_r$, 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 presureless Euler equation by employing the relative entropy combined with the Monge-Kantorovich-Rubinstein distance.