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

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.

Figures (3)

• Figure 1: Typical collision of $K_{\alpha \beta ,ij}^{(1)}$.
• Figure 2: Typical collision of $K_{\alpha \beta ,ij}^{(3)}$.
• Figure 3: Typical collision of $K_{\alpha \beta ,ij}^{(2)}$.

Theorems & Definitions (19)

• Remark 1
• Remark 2
• Lemma 1
• Proposition 1
• Definition 1
• Proposition 2
• Remark 3
• Lemma 2
• Lemma 3
• Proposition 3