On the Cauchy problem for Boltzmann equation modelling a polyatomic gas

Gamba, Irene M.; Pavić-Čolić, Milana

doi:10.48550/arxiv.2005.01017

Cited by 5 publications

(13 citation statements)

References 30 publications

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“…results in a more familiar form of the Boltzmann collision operator for poly-atomic molecules modeled with a continuous energy variable [5,1,8]…”

Section: Collision Operatormentioning

confidence: 99%

“…More recently, compactness results were also obtained for monatomic multi-component mixtures [4], see also [2], and for polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable [2]. In this work, we consider polyatomic single species, where the polyatomicity is modeled by a continuous internal energy variable [5,1,8]. We restrict ourselves to the case when the number of internal degrees of freedom is greater or equal to two.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, the model considered is presented. The probabilistic formulation of the collision operator considered and its relation to a more classical formulation [5,1,8] is accounted for in Section 2.1 Some classical results for the collision operator in Section 2.2 and the linearized collision operator in Section 2.3 are reviewed. Section 3 is devoted to the main results of this paper.…”

Section: Introductionmentioning

confidence: 99%

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Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

Bernhoff¹

2022

Preprint

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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 integral operator for polyatomic single species, for which the number of internal degrees of freedom is greater or equal to two and the polyatomicity is modeled by a continuous internal energy variable, is studied. Compactness of the integral operator is obtained by proving that its terms are, or, at least, can be approximated by, 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 some particular collision kernels -corresponding to hard sphere models, but also hard potential with cut-off models. Then it follows that the linearized collision operator is a Fredholm operator.

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“…results in a more familiar form of the Boltzmann collision operator for poly-atomic molecules modeled with a continuous energy variable [5,1,8]…”

Section: Collision Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

Bernhoff¹

2022

Preprint

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“…Possible cross sections in the polyatomic case have been discussed in [18] for the Borgnakke-Larsen collisions. Since, for all γ ∈ (0, 2),…”

Section: Molecule Modelmentioning

confidence: 99%

Compactness property of the linearized Boltzmann operator for a polyatomic gas undergoing resonant collisions

Borsoni¹,

Boudin²,

Salvarani³

2022

Preprint

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In this paper, we investigate a compactness property of the linearized Boltzmann operator in the context of a polyatomic gas whose molecules undergo resonant collisions. The peculiar structure of resonant collision rules allows to tensorize the problem into a velocity-related one, neighbouring the monatomic case, and an internal energy-related one. Our analysis is based on a specific treatment of the internal energy contributions. We also propose a geometric variant of Grad's proof of the same compactness property in the monatomic case.

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“…The authors of [4] developed a generalized macroscopic 14 field theory for the polyatomic gases, based on the methods of extended thermodynamics [18]. In the full non-linear Boltzmann equation, Gamba and Pavić-Čolić [15] established existence and uniqueness theory in the space homogeneous setting.…”

mentioning

confidence: 99%

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

Brull

Shahine

Thieullen

2022

NHM

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<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, we prove that the linearized Boltzmann operator <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> of this model is a Fredholm operator. For this, we write <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{K} $\end{document}</tex-math></inline-formula> is compact. The result is established after inspecting the kernel form of <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{K} $\end{document}</tex-math></inline-formula> and proving it to be <inline-formula><tex-math id="M6">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> integrable over its domain using elementary arguments.This implies that <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{K} $\end{document}</tex-math></inline-formula> is a Hilbert-Schmidt operator.</p>

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On the Cauchy problem for Boltzmann equation modelling a polyatomic gas

Cited by 5 publications

References 30 publications

Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

Compactness property of the linearized Boltzmann operator for a polyatomic gas undergoing resonant collisions

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

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