Boltzmann equation and hydrodynamics beyond Navier–Stokes

We consider in this paper the problem of derivation and regularization of higher (in Knudsen number) equations of hydrodynamics. The authors approach based on successive changes of hydrodynamic variables is presented in more detail for the Burnett level. The complete theory is briefly discussed for the linearized Boltzmann equation. It is shown that the best results in this case can be obtained by using the diagonal equations of hydrodynamics. Rigorous estimates of accuracy of the NavierStokes and Burnett approximations are also presented. This article is part of the theme issue Hilberts sixth problem.