Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

TL;DR

This paper proves the compactness and self-adjointness of the linearized Boltzmann collision operator for polyatomic molecules modeled with a continuous internal energy variable, extending classical results to more complex molecular models.

Contribution

It extends the analysis of the linearized Boltzmann collision operator to polyatomic molecules with continuous internal energy, proving compactness and self-adjointness.

Findings

Proved compactness of the integral operator for polyatomic molecules with continuous internal energy.

Established self-adjointness of the linearized collision operator.

Showed coercivity of the collision frequency for specific models.

Abstract

The linearized collision operator of the Boltzmann equation for single species can be written as a sum of a positive multiplication operator, the collision frequency, and a compact integral operator. This classical result has more recently, been extended to multi-component mixtures and polyatomic single species with the polyatomicity modeled by a discrete internal energy variable. In this work we prove compactness of the integral operator for polyatomic single species, with the polyatomicity modeled by a continuous internal energy variable, and the number of internal degrees of freedom greater or equal to two. The terms of the integral operator are shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators. Self-adjointness of the linearized collision operator follows. Coercivity of the collision frequency are shown for hard-sphere like and hard potential with cut-off like models, implying Fredholmness of the linearized collision operator.