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

Abstract

The linearized Boltzmann collision operator appears in many important applications of the Boltzmann equation. Therefore, knowing its main properties is of great interest. This work extends some classical results for the linearized Boltzmann collision operator for monatomic single species to the case of polyatomic single species, while also reviewing corresponding results for multicomponent mixtures of monatomic species. The polyatomicity is modeled by a discrete internal energy variable, that can take a finite number of (given) different values. Results concerning the linearized Boltzmann collision operator being a nonnegative symmetric operator with a finite-dimensional kernel are reviewed.A compactness result, 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 splitting, such that the terms can be shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators and as such being compact operators. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model, from which Fredholmness of the linearized collision operator follows, as well as its domain.

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

“…Lemma 4 in Section 4) [17], and so compactness of the integral operator K can be obtained. In this work, we extend the results of [4,5] for monatomic multicomponent mixtures and polyatomic single species, where the polyatomicity is modeled by a continuous internal energy variable [9,16], to the case of multicomponent mixtures of monatomic and/or polyatomic gases, where the polyatomicity is modeled by a continuous internal energy variable [13,1]. To consider mixtures of monatomic and polyatomic molecules are of highest relevance in, e.g., the upper atmosphere [1].…”

“…Compact properties of the integral operator K (for angular cut-off kernels) are extensively studied for monatomic single species, see, e.g., [18,14,12,21], and more recently for monatomic multicomponent mixtures [8,4]. Extensions to polyatomic single species, where the polyatomicity is modeled by either a discrete, or, a continuous internal energy variable [4,5] and polyatomic multicomponent mixtures [2], where the polyatomicity is modeled by discrete internal energy variables, have also been conducted. For models, assuming a continuous internal energy variable, see also [7] for the case of molecules undergoing resonant collisions (for which internal energy and kinetic energy, respectively, are conserved under collisions), and [10,11] for diatomic and polyatomic gases, respectively -with more restrictive assumptions on the collision kernels than in [5], but also a more direct approach.…”

“…Following the lines of [4,5,2], motivated by an approach by Kogan in [20,Sect. 2.8] for the monatomic single species case, a probabilistic formulation of the collision operator is considered as the starting point.…”

“…The linearized collision operator, obtained by considering deviations of an equilibrium, or Maxwellian, distribution, can in a natural way be written as a sum of a positive multiplication operator -the collision frequency -and an integral operator −K. Compact properties of the integral operator K (for angular cut-off kernels) are extensively studied for monatomic single species, see, e.g., [19,17,18,13,23], and more recently for monatomic multi-component mixtures [10,4]. 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.…”

“…Compact properties of the integral operator K (for angular cut-off kernels) are extensively studied for monatomic single species, see, e.g., [19,17,18,13,23], and more recently for monatomic multi-component mixtures [10,4]. 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].…”