Compactness property of the linearized Boltzmann operator for a diatomic single gas model

Stéphane Brull; Marwa Shahine; Philippe Thieullen; Stéphane Brull; Marwa Shahine; Philippe Thieullen

doi:10.3934/nhm.2022029

Networks and Heterogeneous Media

2022, Volume 17, Issue 6: 847-861. doi: 10.3934/nhm.2022029

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Compactness property of the linearized Boltzmann operator for a diatomic single gas model

Institute de Mathématiques de Bordeaux, Université de Bordeaux, France

Received: 01 February 2022 Revised: 01 May 2022 Published: 18 July 2022
Primary: 82-02; Secondary: 70-02

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.
- Diatomic gases,
- linearized Boltzmann operator,
- Fredholm alternative,
- continuous internal energy model
Citation: Stéphane Brull, Marwa Shahine, Philippe Thieullen. Compactness property of the linearized Boltzmann operator for a diatomic single gas model[J]. Networks and Heterogeneous Media, 2022, 17(6): 847-861. doi: 10.3934/nhm.2022029

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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.

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