Compactness property of the linearized Boltzmann operator for a diatomic single gas model

Abstract

<p style='text-indent:20px;'>In the following work, we consider the Boltzmann equation that models a diatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, we prove that the linearized Boltzmann operator <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> of this model is a Fredholm operator. For this, we write <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{K} $\end{document}</tex-math></inline-formula> is compact. The result is established after inspecting the kernel form of <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{K} $\end{document}</tex-math></inline-formula> and proving it to be <inline-formula><tex-math id="M6">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> integrable over its domain using elementary arguments.This implies that <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{K} $\end{document}</tex-math></inline-formula> is a Hilbert-Schmidt operator.</p>