Some recent results about the sixth problem of Hilbert: hydrodynamic limits of the Boltzmann equation

Abstract

The behaviour of a gas can be described by different models depending on the time and space scales to be considered. A natural question is therefore to know if there exist continuous transitions between these models: this is precisely the matter of the sixth problem asked by Hilbert on the occasion of the International Congress of Mathematicians held in Paris in 1900. Recent works allow to understand rigorously the passage from Boltzmann’s equation (which provides a statistical description of the microscopic state of the gas) to continuous models of fluid mechanics. The goal of the present paper is to present briefly the mathematical framework of the study, and to exhibit some difficulties of the asymptotic analysis related to the physics of the problem.