The problem with Hilbert's 6th problem

Abstract

This paper reviews earlier results of the author regarding the hydrodynamic limit problem for the Boltzmann equation. In particular the key points are that the work of Gorban&Karlin suggests that Korteweg hydrodynamics is implied by the Boltzmann equation and this correspondence is validated by comparison with experimental, analytical, and numerical results. But if the correspondence is indeed valid then passage to limiting compressible Euler equations will not be generally possible after any time for which the the Euler system fails to have smooth solutions. Dedicated to my friend Alexander Gorban on the occasion of his sixtieth birthday