Derivation and Analysis of a Class of Relaxation Operators in Kinetic Theory

Abstract

We aim to present a theory for the derivation of a class of relaxation operators approximating the Boltzmann collision operator. The construction is based on an approximation of the inverse Boltzmann linearized operator, on relaxation equations on the moments of the distribution function and finally on a variational problem to be solved. The theory comprises a characterization of the set of moments of non negative integrable functions, a study of those linear application whose range lies in this set and a generalization of the functional to be minimized under moment constraints. In particular we clarify but also modify some steps in the proof of Junk’s theorem on the characterization of moments of non negative functions (Junk in Math Models Methods Appl Sci 10:1001–1025, 2000). We also reestablish a theorem of Csiszar’s (Acta Math Hung 68:161–185, 1995) by different means on a class of functionals leading to well-posed variational problems. The present theory encompasses the derivation of known models and that of new models.