Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures

TL;DR

This paper investigates the mathematical properties of the linearized Boltzmann collision operator for polyatomic molecules modeled with a discrete internal energy variable, establishing compactness, self-adjointness, and Fredholm properties.

Contribution

It extends the analysis of the linearized collision operator to polyatomic molecules with discrete internal energy, proving compactness and related spectral properties.

Findings

Proved the compactness of the integral operator for polyatomic molecules.

Established self-adjointness of the linearized collision operator.

Derived bounds and coercivity for the collision frequency in a hard sphere model.

Abstract

The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding result for mixtures is more recently obtained. In this work the compactness of the operator for polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable, is studied. With a probabilistic formulation of the collision operator as a starting point, compactness is obtained by proving that the integral operator is a sum of Hilbert-Schmidt integral operators and approximately Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere model. Then it follows that the linearized collision operator is a Fredholm operator. The results can be extended to mixtures. For brevity, only the case of mixtures for monatomic species is accounted for.