Compactness Property of the Linearized Boltzmann Operator for a Mixture of Polyatomic Gases

In this paper we shall concern ourselves with a kinetic description of gas mixtures for polyatomic molecules. In particular, we will consider the Boltzmann equation that models a mixture of polyatomic gases of n species \((\mathcal {A}_i)_{i=1,\ldots ,n}\) . At the microscopic level, one additional argument of the distribution function is introduced which is the parameter I denoting the continuous internal energy. Under some convenient assumptions on the collision cross-section \(\mathcal {B}_{ij}\) , 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.