Fredholm property of the linearized Boltzmann operator for a polyatomic single gas model

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 transition function $ \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.

Topics & Concepts