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

TL;DR

This paper studies the compactness of the linearized Boltzmann operator for polyatomic gases with resonant collisions, extending known results from monatomic gases by considering internal energy contributions.

Contribution

It introduces a tensorization approach for the operator and a geometric variant of Grad's proof tailored for polyatomic gases with internal energy effects.

Findings

Established compactness property for the linearized operator in polyatomic gases

Extended monatomic case techniques to include internal energy contributions

Proposed a geometric proof variant for the compactness property

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 contributions due to the internal energy of the molecules. We also propose a geometric variant of Grad's proof of the same compactness property in the monatomic case.