On the mathematical structure of Eu's modified moment method

Any attempt to work with a kinetic equation via an infinite hierarchy of equations of transfer immediately faces the problem not only of determining the molecular density ƒ from its principal, conserved moments (p, u, e) and the Hermite coefficients bn but also of calculating the collision integrals Pn as exhibited functions of the conserved and Hermite variables. So as to obviate solutions ƒ of the kinetic equation of interest from the standpoint of irreversible thermodynamics, B.C. Eu [J. Chem. Phys. 73 (1980) 2958] chose to consider for quite general gases and for any spherically symmetrical model a class of molecular densities ƒ that can be written in an exponential form. In this paper we set up conditions under which some of the axioms of Eu's modified moment method are rigorously justified in case of a one-dimensional gas described by Ma's kinetic equation. Since many of the available analyses contain important gaps, our first purpose is to uncover these gaps and to illuminate them as challenges to future research by mathematicians.