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

Brull, Stéphane; Shahine, Marwa; Thieullen, Philippe

doi:10.3934/nhm.2022029

Cited by 11 publications

(10 citation statements)

References 30 publications

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“…The invariance under change of particles in a collision, which follows directly by the definition of the transition probability (9) and the symmetry relations (11) , (12) for the collision frequency, and the microreversibility of the collisions (10), implies that the transition probabilities (9) satisfy the relations Applying known properties of Dirac's delta function, the transition probabilities may be transformed to…”

Section: Collision Operatormentioning

confidence: 99%

“…Extensions to polyatomic single species, where the polyatomicity is modeled by either a discrete, or, a continuous internal energy variable [4,5] and polyatomic multicomponent mixtures [2], where the polyatomicity is modeled by discrete internal energy variables, have also been conducted. For models, assuming a continuous internal energy variable, see also [7] for the case of molecules undergoing resonant collisions (for which internal energy and kinetic energy, respectively, are conserved under collisions), and [10,11] for diatomic and polyatomic gases, respectively -with more restrictive assumptions on the collision kernels than in [5], but also a more direct approach. The integral operator can be written as the sum of a Hilbert-Schmidt integral operator and an approximately Hilbert-Schmidt integral operator -which is a uniform limit of Hilbert-Schmidt integral operators (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Compact properties of the integral operator K (for angular cut-off kernels) are extensively studied for monatomic single species, see, e.g., [19,17,18,13,23], and more recently for monatomic multi-component mixtures [10,4]. Extensions to polyatomic gases, where the polyatomicity is modeled by either a discrete, or, a continuous internal energy variable for single species [4,5,12], or, mixtures [6,7] have very recently been conducted. Compactness results are also recently obtained for models of polyatomic single gases, with a continuous internal energy variable, where the molecules undergo resonant collisions (for which internal energy and kinetic energy, respectively, are conserved) [9].…”

Section: Introductionmentioning

confidence: 99%

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Linearized Boltzmann collision operator: II. Polyatomic molecules modeled by a continuous internal energy variable

Bernhoff¹

2023

KRM

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The linearized Boltzmann collision operator has a central role in many important applications of the Boltzmann equation. Recently some important classical properties of the linearized collision operator for monatomic single species were extended to multicomponent monatomic gases and polyatomic single species. For multicomponent polyatomic gases, the case where the polyatomicity is modelled by a discrete internal energy variable was considered lately. Here we considers the corresponding case for a continuous internal energy variable. Compactness results, saying that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a decomposition, such that the terms are, or at least are uniform limits of, Hilbert-Schmidt integral operators and therefore are compact operators. Moreover, bounds on -including coercivity of -the collision frequency are obtained for a hard sphere like, as well as hard potentials with cutoff like, models, from which Fredholmness of the linearized collision operator follows, as well as its domain.

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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

2023

KRM

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“…3/2 e −mα |ξ | 2 /2 , of the form (13). Insertion in the Boltzmann equation ( 1) results in a system (14), where the components of the linearized collision operator L = (L 1 , . .…”

Section: Linearized Collision Operatormentioning

confidence: 99%

“…Related studies have attracted recent attention. The case of polyatomic single species, where the polyatomicity is modeled by a continuous internal energy variable is considered in [5], see also [9] for the case of molecules undergoing resonant collisions (for which internal energy and kinetic energy, respectively, are conserved under collisions), and [13,14] for diatomic and polyatomic gases, respectively -with more restrictive assumptions on the collision kernels than in [5], but also a more direct approach.…”

Section: Introductionmentioning

confidence: 99%

Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures

Bernhoff

2022

Acta Appl Math

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The linearized Boltzmann collision operator appears in many important applications of the Boltzmann equation. Therefore, knowing its main properties is of great interest. This work extends some classical results for the linearized Boltzmann collision operator for monatomic single species to the case of polyatomic single species, while also reviewing corresponding results for multicomponent mixtures of monatomic species. The polyatomicity is modeled by a discrete internal energy variable, that can take a finite number of (given) different values. Results concerning the linearized Boltzmann collision operator being a nonnegative symmetric operator with a finite-dimensional kernel are reviewed.A compactness result, saying that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a splitting, such that the terms can be shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators and as such being compact operators. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model, from which Fredholmness of the linearized collision operator follows, as well as its domain.

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Linearized Boltzmann collision operator for a mixture of monatomic and polyatomic chemically reacting species

Bernhoff

2024

J Math Chem

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At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of the considered reactions. Polyatomicity is here modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In the Chapman-Enskog process—and related half-space problems—the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized operator to the considered model with chemical reactions. A compactness result, that the linearized operator can be decomposed into a sum of a positive multiplication operator—the collision frequency—and a compact integral operator, is obtained. The terms of the integral operator are shown to be (at least) uniform limits of Hilbert-Schmidt integral operators and, thereby, compact operators. Self-adjointness of the linearized operator follows as a direct consequence. Also, bounds on—including coercivity of—the collision frequency is obtained for hard sphere, as well as hard potentials with cutoff, like models. As consequence, Fredholmness as well as the domain of the linearized operator are obtained.

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

Cited by 11 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

Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures

Linearized Boltzmann collision operator for a mixture of monatomic and polyatomic chemically reacting species

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