Fredholm Property of the Linearized Boltzmann Operator for a Polyatomic Single Gas Model

TL;DR

This paper proves that the linearized Boltzmann operator for a polyatomic gas model is a Fredholm operator by demonstrating the compactness of a perturbation operator, under certain assumptions on the collision cross-section.

Contribution

It establishes the Fredholm property of the linearized Boltzmann operator for a polyatomic gas model, extending the mathematical understanding of such operators.

Findings

The linearized Boltzmann operator is a Fredholm operator.

The perturbation operator is shown to be compact.

The proof relies on kernel integrability and elementary arguments.

Abstract

In the following work, we consider the Boltzmann equation that models a polyatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section $\mathcal{B}$, we prove that the linearized Boltzmann operator $\mathcal{L}$ of this model is a Fredholm operator. For this, we write $\mathcal{L}$ as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator $\mathcal{K}$ is compact. The result is established after inspecting the kernel form of $\mathcal{K}$ and proving it to be $L^2$ integrable over its domain using elementary arguments.