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 (A(i))(i=1, ... , 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 B-ij, we prove that the linearized Boltzmann operator L of this model is a Fredholm operator. For this, we write L as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator K is compact. The result is established after inspecting the kernel form of K and proving it to be L-2 integrable over its domain using elementary arguments.