Hilbert’s sixth problem and the failure of the Boltzmann to Euler limit

Abstract

This paper addresses the main issue of Hilbert’s sixth problem, namely the rigorous passage of solutions to the mesoscopic Boltzmann equation to macroscopic solutions of the Euler equations of compressible gas dynamics. The results of the paper are that (i) in general Hilbert’s program will fail because of the appearance of van der Waals–Korteweg capillarity terms in a macroscopic description of motion of a gas, and (ii) the van der Waals–Korteweg theory itself might satisfy Hilbert’s quest for a map from the ‘atomistic view’ to the laws of motion of continua. This article is part of the theme issue ‘Hilbert’s sixth problem’.