Asymptotic Stability of Boundary Layers to the Euler–Poisson Equations Arising in Plasma Physics

Nishibata, Shinya; Ohnawa, Masashi; Suzuki, Masahiro

doi:10.1137/110835657

Cited by 30 publications

(36 citation statements)

References 10 publications

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“…Moreover, the stability of the stationary solution was shown under a condition slightly stronger than (4) in [13]. Recently, Nishibata, Ohnawa, and Suzuki [9] obtained the stability theorem under (4). These results rigorously clarify that a sheath is regarded as a stationary solution.…”

Section: Masahiro Suzukimentioning

confidence: 99%

“…It is reasonable to expect that the asymptotic state is given by a stationary solution, since a sheath is observed as a stationary boundary layer. The papers [1,2,9,13] studied this expectation for initial-boundary value problems of (1) with k = 1. Ambroso, Méhats, and Raviart [2] showed the unique existence of a monotone stationary solution over a one-dimensional bounded domain under assumption (4).…”

Section: Masahiro Suzukimentioning

confidence: 99%

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Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma

Suzuki¹

2016

KRM

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The main concern of this paper is to analyze a boundary layer called a sheath that occurs on the surface of materials when in contact with a multicomponent plasma. For the formation of a sheath, the generalized Bohm criterion demands that ions enter the sheath region with a high velocity. The motion of a multicomponent plasma is governed by the Euler-Poisson equations, and a sheath is understood as a monotone stationary solution to those equations. In this paper, we prove the unique existence of the monotone stationary solution by assuming the generalized Bohm criterion. Moreover, it is shown that the stationary solution is time asymptotically stable provided that an initial perturbation is sufficiently small in weighted Sobolev space. We also obtain the convergence rate, which is subject to the decay rate of the initial perturbation, of the time global solution toward the stationary solution.

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Section: Masahiro Suzukimentioning

confidence: 99%

Section: Masahiro Suzukimentioning

confidence: 99%

Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma

Suzuki¹

2016

KRM

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“…We shall combine many energy estimates for the proof of Proposition 2. As already pointed out in the introduction, our approach shares features with the analysis led in [8]. The starting point of all the energy estimates will be the following lemma:…”

Section: Linear Stabilitymentioning

confidence: 99%

“…The idea is to use the stabilizing effect of convection. We shall also borrow some ideas from a recent paper of Nishibata, Ohnawa and Suzuki [8] on a related problem. Namely, this problem is the stability of a special solution of the unscaled Euler-Poisson system, that is when ε = 1.…”

Section: Introductionmentioning

confidence: 99%

Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II

Gérard-Varet¹,

Han-Kwan²,

Rousset³

2014

Journal De L’École Polytechnique — Mathématiques

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“…In order to figure out the most technical part of the analysis, for brevity we only consider in the paper the motion of one-species VPB system (1.1) for ions under the generalized Boltzmann relation satisfying the assumption (A). Here, we remark that the Boltzmann relation ρ e = ρ e (φ) has been extensively used in the mathematical study of both the fluid dynamic equations, for instance, Guo-Pausader [28], Suzuki [54], Nishibata-Ohnawa-Suzuki [48], and the kinetic Vlasov-type equations, for instance, Han-Kwan [29], Charles-Després-Perthame-Sentis [6].…”

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confidence: 99%

Stability of the Rarefaction Wave of the Vlasov--Poisson--Boltzmann System

Duan¹,

Liu²

2015

SIAM J. Math. Anal.

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This paper is devoted to the study of the nonlinear stability of the rarefaction waves of the Vlasov-Poisson-Boltzmann system with slab symmetry in the case where the electron background density satisfies an analogue of the Boltzmann relation. We allows that the electric potential may take distinct constant states at both far-fields. The rarefaction wave whose strength is not necessarily small is constructed through the quasineutral Euler equations coming from the zero-order fluid dynamic approximation of the kinetic system. We prove that the local Maxwellian with macroscopic quantities determined by the quasineutral rarefaction wave is time-asymptotically stable under small perturbations for the corresponding Cauchy problem on the Vlasov-Poisson-Boltzmann system. The main analytical tool is the combination of techniques we developed in [10] for the viscous compressible fluid with the self-consistent electric field and the reciprocal energy method based on the macro-micro decomposition of the Boltzmann equation around a local Maxwellian. Both the time decay property of the rarefaction waves and the structure of the Poisson equation play a key role in the analysis.

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Asymptotic Stability of Boundary Layers to the Euler–Poisson Equations Arising in Plasma Physics

Cited by 30 publications

References 10 publications

Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma

Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma

Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II

Stability of the Rarefaction Wave of the Vlasov--Poisson--Boltzmann System

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