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

TL;DR

This paper proves the existence of solutions and derives a hydrodynamic limit for a kinetic traffic model describing vehicle behavior, using advanced mathematical techniques to connect microscopic dynamics with macroscopic equations.

Contribution

It establishes the global existence of weak solutions for a Paveri-Fontana type model and rigorously derives a pressureless Euler equation as a hydrodynamic limit.

Findings

Proved global-in-time existence of weak solutions.

Established hydrodynamic limit to pressureless Euler equations.

Utilized energy, $L^p$, support estimates, and velocity averaging lemma.

Abstract

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.