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 HilbertSchmidt integral operators and approximately HilbertSchmidt integral operators, under some assumptions on the collision kernel. Selfadjointness 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:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.01365
 Bibcode:
 2022arXiv220101365B
 Keywords:

 Mathematics  Analysis of PDEs;
 82C40 (Primary);
 35Q20;
 35Q70;
 76P05 (Secondary)
 EPrint:
 45 pages, 5 figures