Compactness property of the linearized Boltzmann collision operator for a multicomponent polyatomic gas

TL;DR

This paper proves the compactness of the linearized Boltzmann collision operator for a multicomponent polyatomic gas mixture, extending classical results to more complex models with internal energy variables.

Contribution

It introduces a probabilistic formulation and demonstrates the compactness and self-adjointness of the operator for polyatomic mixtures, generalizing previous monatomic results.

Findings

Proves the operator is a sum of Hilbert-Schmidt operators.

Establishes the operator's self-adjointness and Fredholm property.

Provides bounds and coercivity estimates for the collision frequency.

Abstract

The linearized Boltzmann collision operator is fundamental in many studies of the Boltzmann equation and its main properties are of substantial importance. The decomposition into a sum of a positive multiplication operator, the collision frequency, and an integral operator is trivial. Compactness of the integral operator for monatomic single species is a classical result, while corresponding results for monatomic mixtures and polyatomic single species are more recently obtained. This work concerns the compactness of the operator for a multicomponent mixture of polyatomic species, where the polyatomicity is modeled by a discrete internal energy variable. With a probabilistic formulation of the collision operator as a starting point, compactness is obtained by proving that the integral operator is a sum of Hilbert-Schmidt integral operators and operators, which are uniform limits of Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. The assumptions are essentially generalizations of the Grad's assumptions for monatomic single species. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model. Then it follows that the linearized collision operator is a Fredholm operator, and its domain is also obtained.