Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures
Abstract
The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding result for mixtures is more recently obtained. In this work the compactness of the operator for polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable, is studied. 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 approximately Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere model. Then it follows that the linearized collision operator is a Fredholm operator. The results can be extended to mixtures. For brevity, only the case of mixtures for monatomic species is accounted for.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2022
- DOI:
- 10.48550/arXiv.2201.01365
- arXiv:
- arXiv:2201.01365
- Bibcode:
- 2022arXiv220101365B
- Keywords:
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- Mathematics - Analysis of PDEs;
- 82C40 (Primary);
- 35Q20;
- 35Q70;
- 76P05 (Secondary)
- E-Print:
- 45 pages, 5 figures