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- Title
Compactness property of the linearized Boltzmann operator for a diatomic single gas model.
- Authors
Brull, Stéphane; Shahine, Marwa; Thieullen, Philippe
- Abstract
In the following work, we consider the Boltzmann equation that models a diatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section $ \mathcal{B} $, we prove that the linearized Boltzmann operator $ \mathcal{L} $ of this model is a Fredholm operator. For this, we write $ \mathcal{L} $ as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator $ \mathcal{K} $ is compact. The result is established after inspecting the kernel form of $ \mathcal{K} $ and proving it to be $ L^2 $ integrable over its domain using elementary arguments.This implies that $ \mathcal{K} $ is a Hilbert-Schmidt operator.
- Subjects
FREDHOLM operators; BOLTZMANN'S equation; GASES; MASS transfer coefficients
- Publication
Networks & Heterogeneous Media, 2022, Vol 17, Issue 6, p1
- ISSN
1556-1801
- Publication type
Article
- DOI
10.3934/nhm.2022029