Kinetic Equations and the Chapman–Enskog Projection Problem

It is well known that in the low-frequency cutoffs of the Chapman–Enskog projection of moment approximations of the Boltzmann kinetic equation, the so-called ultraviolet catastrophe occurs. For the first time, this phenomenon was pointed out by A. V. Bobylev in 1992 in the simplest mode (of one-dimensional linear deviation from global equilibrium). By an example of moment approximation of the Boltzmann–Peierls kinetic equation, we prove the existence of a Chapman–Enskog projection to the phase space of the conservative variable in the class of first-order hyperbolic pseudodifferential systems with relaxation. This result is used to explain the phenomenon of ultraviolet catastrophe.