Macroscopic limits of the Boltzmann equation: a review

Abstract

This document is concerned with the modeling of particle systems via kinetic equations. First, the hierarchy of available models for particle systems is reviewed, from particle dynamics to fluid models through kinetic equations. In particular the derivation of the gas dynamics Boltzmann equation is recalled and a few companion models are discussed. Then, the basic properties of kinetic models and particularly of the Boltzmann collision operator are reviewed. The core of this work is the derivation of macroscopic models (as e.g., the Euler or Navier-Stokes equations) from the Boltzmann equation by means of the Hilbert and ChapmanEnskog methods. This matter is first discussed in the context of the BGK equation, which is a simpler model than the full Boltzmann equation. The extension to the Boltzmann equation is summarized at the end of this discussion. Finally, a certain number of current research directions are reviewed. Our goal is to give a synthetic description of this subject, so as to allow the reader to acquire a rapid knowledge of the basic aspects of kinetic theory. The reader is referred to the bibliography for more details on the various items which are reviewed here.