Compactness property of the linearized Boltzmann operator for a polyatomic gas undergoing resonant collisions

Abstract

In this paper, we investigate a compactness property of the linearized Boltzmann operator in the context of a polyatomic gas whose molecules undergo resonant collisions. The peculiar structure of resonant collision rules allows to tensorize the problem into a velocity-related one, neighbouring the monatomic case, and an internal energy-related one. Our analysis is based on a specific treatment of the internal energy contributions. We also propose a geometric variant of Grad's proof of the same compactness property in the monatomic case.

Compactness property of the linearized Boltzmann operator for a diatomic single gas model

“…We give now some models that satisfy assumptions (9), which have been recently studied in the literature (see [19,16,4]).…”

“…For a single diatomic gas, the operator K was proved to be a Hilbert-Schmidt operator in [13] under some assumption on the collision cross-section. In [4], L was proved to be Fredholm for a single polyatomic gas using a different approach, and in [9], the Fredholm property of L was proved for polyatomic gases undergoing resonance. In this work, we aim to generalize the work [13] for a single polyatomic gas using elementary arguments.…”