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 $\mathcal{B}$, 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.This implies that $\mathcal{K}$ is a Hilbert-Schmidt operator.

This paper is devoted to the study of the bitemperature Euler system in a polyatomic setting. Physically, this model describes a mixture of one species of ions and one species of electrons in the quasi-neutral regime. We firstly derive the model starting from a kinetic polyatomic model and performing next a fluid limit. This kinetic model is shown to satisfy fundamental properties. Some exact solutions are presented. Finally, a numerical scheme is derived and proved to coincide with an approximation designed in [3] and extended to second order and two space dimensions in [6]. Some numerical tests are presented.