Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures

The linearized collision operator of the Boltzmann equation can in a natural way be written as sum positive multiplication operator, frequency, and an integral operator. Compactness for monatomic single species is classical result, while corresponding result mixtures more recently obtained. In this work compactness polyatomic species, where polyatomicity modeled by discrete internal energy variable, studied. With probabilistic formulation starting point, obtained proving that Hilbert-Schmidt operators approximately operators, under some assumptions on kernel. Self-adjointness follows. Moreover, bounds - including coercivity frequency are hard sphere model. Then it follows Fredholm results extended to mixtures. For brevity, only case accounted for.