Scattering Kernels for Gas-Surface Interaction

Cercignani, Carlo

doi:10.1007/978-3-642-84580-2_5

Cited by 7 publications

(7 citation statements)

References 30 publications

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“…Then we introduce the boundary conditions for the distribution functions. As in Cercignani's work [14][15][16] (see also [6,53]), we define the sets of outgoing (Σ x + ) and incoming (Σ x − ) velocities at point x ∈ ∂Ω such that Σ x ± := {v ∈ R d : ±v · n(x) > 0} and denote the boundary sets…”

Section: Preliminaries and Main Results 21 Boundary And Initial Condimentioning

confidence: 99%

“…We refer to [6] for possible minimal assumptions on R such that (2.1) is well-defined, i.e., R is nonnegative and it verifies the normalization condition as well as the reciprocity principle. Detailed discussions on the boundary conditions can be found in [14][15][16].…”

Section: Preliminaries and Main Results 21 Boundary And Initial Condimentioning

confidence: 99%

“…Let γh be the trace of function h and γ ± h = 1 (0,+∞)×Σ ± γh. Reflection boundary conditions for the kinetic equations take the form of integral (balance) relations between the densities of the particles on the outgoing and incoming velocity subsets of the boundary ∂Ω at a given time [14][15][16]. For instance, given x ∈ ∂Ω and t > 0, we have (cf.…”

Section: Preliminaries and Main Results 21 Boundary And Initial Condi...mentioning

confidence: 99%

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Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles

Lin

Chun

2014

Arch Rational Mech Anal

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In ionic solutions, there are multi-species charged particles (ions) with different properties like mass, charge etc. Macroscopic continuum models like the Poisson-Nernst-Planck (PNP) systems have been extensively used to describe the transport and distribution of ionic species in the solvent. Starting from the kinetic theory for the ion transport, we study a Vlasov-Poisson-Fokker-Planck (VPFP) system in a bounded domain with reflection boundary conditions for charge distributions and prove that the global renormalized solutions of the VPFP system converge to the global weak solutions of the PNP system, as the small parameter related to the scaled thermal velocity and mean free path tends to zero. Our results may justify the PNP system as a macroscopic model for the transport of multi-species ions in dilute solutions.

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Section: Preliminaries and Main Results 21 Boundary And Initial Condimentioning

confidence: 99%

Section: Preliminaries and Main Results 21 Boundary And Initial Condimentioning

confidence: 99%

Section: Preliminaries and Main Results 21 Boundary And Initial Condi...mentioning

confidence: 99%

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Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles

Lin

Chun

2014

Arch Rational Mech Anal

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“…The reflection law (1.2) was the only model for the gas/surface interaction that appeared in the literature before the late 1960s. In order to describe with more accuracy the interaction between molecules and wall, other models have been proposed in [26], [27], [51] where the reflection operator R is a general integral operator satisfying the so-called non-negative, normalization and reciprocity conditions, see [30] and Remark 6.4. We do not know whether our analysis can be adapted to such a general kernel.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Kinetic equations with Maxwell boundary conditions

Mischler¹

2010

Ann. Sci. École Norm. Sup.

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-We prove global stability results of DiPerna-Lions renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weakweak convergence (the renormalized convergence and the biting L 1 -weak convergence), as well as the Darrozès-Guiraud information in a crucial way.R. -Nous montrons la stabilité des solutions renormalisées au sens de DiPerna-Lions pour des équations cinétiques avec conditions initiale et aux limites. La condition aux limites (qui peut être non linéaire) est partiellement diffuse et est réalisée (c'est-à-dire qu'elle n'est pas relaxée). Les techniques que nous introduisons sont illustrées sur l'équation de Fokker-Planck-Boltzmann et le système de Vlasov-Poisson-Fokker-Planck ainsi que pour des conditions aux limites linéaires sur l'équation de Boltzmann et le système de Vlasov-Poisson. Les démonstrations utilisent des théorèmes de trace du type de ceux introduits par l'auteur pour les équations de Vlasov, des résultats d'analyse fonctionnelle sur les convergences faible-faible (la convergence renormalisée et la convergence au sens du biting lemma), ainsi que l'information de Darrozès-Guiraud d'une manière essentielle.

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“…where all the physics is embodied in the kernel k and the coefficient 0 ≤ α ≤ 1. We refer for the derivation of boundary kernels and comments to [14,15,21,36,37,42]. It is natural to require:…”

Section: Remark 34 the Discussion Of Knudsen Layers And Their Numermentioning

confidence: 99%

Hydrodynamic Regimes, Knudsen Layer, Numerical Schemes: Definition of Boundary Fluxes

et al. 2011

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We propose a numerical solution to incorporate in the simulation of a system of conservation laws boundary conditions that come from a microscopic modeling in the small mean free path regime. The typical example we discuss is the derivation of the Euler system from the BGK equation. The boundary condition relies on the analysis of boundary layers formation that accounts from the fact that the incoming kinetic flux might be far from the thermodynamic equilibrium.

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Scattering Kernels for Gas-Surface Interaction

Cited by 7 publications

References 30 publications

Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles

Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles

Kinetic equations with Maxwell boundary conditions

Hydrodynamic Regimes, Knudsen Layer, Numerical Schemes: Definition of Boundary Fluxes

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