Linearized Boltzmann collision operator: II. Polyatomic molecules modeled by a continuous internal energy variable

Abstract

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 considers the corresponding case for a continuous internal energy variable. Compactness results, saying 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 a 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.

Linearized Boltzmann collision operator for a mixture of monatomic and polyatomic chemically reacting species

“…The decomposition follows by decomposition ( 9), ( 12) and Theorem 2, while the bounds (18) on the collision frequency will be proven in Sect. 5.…”

“…Compact properties of the integral operator K (for angular cut-off kernels) are extensively studied for monatomic single species, see, e.g., [13,[17][18][19]23], and more recently for monatomic multi-component mixtures [4,10]. Extensions to polyatomic gases, where the polyatomicity is modeled by either a discrete, or, a continuous internal energy variable for single species [4,5,12], or, mixtures [6,7] have very recently been conducted. Compactness results are also recently obtained for models of polyatomic single gases, with a continuous internal energy variable, where the molecules undergo resonant collisions (for which internal energy and kinetic energy, respectively, are conserved) [9].…”

“…Following the lines of a series of papers [4][5][6][7], motivated by an approach by Kogan in [22,Sect. 2.8] for the monatomic single species case, a probabilistic formulation of the collision operator is considered as the starting point.…”