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arXiv · 2208.14343

Fredholm Property of the Linearized Boltzmann Operator for a Polyatomic Single Gas Model

Stéphane Brull·Marwa Shahine·Philippe Thieullen
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Abstract

In the following work, we consider the Boltzmann equation that models a polyatomic 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.

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Stéphane Brull, Marwa Shahine, Philippe Thieullen. 2022-08-30. Fredholm Property of the Linearized Boltzmann Operator for a Polyatomic Single Gas Model. https://arxiv.org/abs/2208.14343

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