Compactness Property of the Linearized Boltzmann Collision Operator for a Mixture of Monatomic and Polyatomic Species

The linearized Boltzmann collision operator has a central role in many important applications of the Boltzmann equation. Recently some important classical properties of the linearized collision operator for monatomic single species were extended to multicomponent monatomic gases and polyatomic single species. For multicomponent polyatomic gases, the case where the polyatomicity is modelled by a discrete internal energy variable was considered lately. Here we consider the corresponding case for a continuous internal energy variable. Compactness results, stating that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a decomposition, such that the terms are, or at least are uniform limits of, Hilbert–Schmidt integral operators and therefore are compact operators. Moreover, bounds on—including coercivity of—the collision frequency are obtained for hard sphere like, as well as hard potentials with cutoff like, models, from which Fredholmness of the linearized collision operator follows, as well as its domain.