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

Duan, Renjun; Liu, Shuangqian

doi:10.1137/140995179

Cited by 34 publications

(27 citation statements)

References 68 publications

(121 reference statements)

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“…x φ = n i − n e , t > 0, x ∈ R. Finally we conclude the note with discussions on several closely relative problems arising from the current work [7]. First of all, it is of course an interesting problem to justify the fluid dynamic limit of the VPB system to the one-fluid Euler-Poisson system for ions, cf.…”

Section: Rarefaction Waves Of Vlasov-poisson-boltzmann Systemmentioning

confidence: 89%

“…In particular, the potential function may take the distinct states at both far fields x = ±∞. In what follows, we will present the main result of [7].…”

Section: Rarefaction Waves Of Vlasov-poisson-boltzmann Systemmentioning

confidence: 99%

“…However, for the Boltzmann equation with forces, to the best of our knowledge, there are few results on the same issue. In [7], we have showed the time-asymptotic stability of the rarefaction wave for the model mentioned above. In particular, the potential function may take the distinct states at both far fields x = ±∞.…”

Section: Rarefaction Waves Of Vlasov-poisson-boltzmann Systemmentioning

confidence: 99%

“…We may start from the simple case when only the potential force is present, and mainly focus on the interesting situation where the potential function is non-trivial, cf. Duan-Liu [6,7].…”

Section: Motivationsmentioning

confidence: 99%

“…Readers may refer to [7] for the complete proof of the above theorem. The main idea is based on the study of the same problem on the Navier-StokesPoisson system; see [6].…”

Section: Rarefaction Waves Of Vlasov-poisson-boltzmann Systemmentioning

confidence: 99%

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Large-time behavior for fluid and kinetic plasmas with collisions

Duan

2016

Bull Braz Math Soc, New Series

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The motion of collisional plasmas can be governed either by the EulerMaxwell system with damping at the fluid level or by the Vlasov-Maxwell-Boltzmann system at the kinetic level. In the note, we present some recent results in [8] and [7] for the study of the non-trivial large-time behavior of solutions to the Cauchy problem on the related models in perturbation framework.[42] S.-H. Yu. Nonlinear wave propagations over a Boltzmann shock profile.

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Section: Rarefaction Waves Of Vlasov-poisson-boltzmann Systemmentioning

confidence: 89%

“…In particular, the potential function may take the distinct states at both far fields x = ±∞. In what follows, we will present the main result of [7].…”

Section: Rarefaction Waves Of Vlasov-poisson-boltzmann Systemmentioning

confidence: 99%

Section: Rarefaction Waves Of Vlasov-poisson-boltzmann Systemmentioning

confidence: 99%

“…We may start from the simple case when only the potential force is present, and mainly focus on the interesting situation where the potential function is non-trivial, cf. Duan-Liu [6,7].…”

Section: Motivationsmentioning

confidence: 99%

“…Readers may refer to [7] for the complete proof of the above theorem. The main idea is based on the study of the same problem on the Navier-StokesPoisson system; see [6].…”

Section: Rarefaction Waves Of Vlasov-poisson-boltzmann Systemmentioning

confidence: 99%

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Large-time behavior for fluid and kinetic plasmas with collisions

Duan

2016

Bull Braz Math Soc, New Series

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The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential

Liu¹,

Yang²

2016

Arch Rational Mech Anal

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Abstract. Boundary effects are central to the dynamics of the dilute particles governed by Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for Boltzmann equation with soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate L ∞ analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in L ∞ space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori L ∞ estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the L 2 − L ∞ theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted L ∞ theory so that we could deal with the greater difficulties stemmed from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in L ∞ space with the aid of the L 2 theory and a bootstrap argument. These methods in the latter case can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.

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Stability of Nonlinear Wave Patterns to the Bipolar Vlasov–Poisson–Boltzmann System

Wang

Yang

et al. 2017

Arch Rational Mech Anal

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Abstract. The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction wave for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in [21,23], we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as the applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction wave, are proved for the 1D bipolar Vlasov-PoissonBoltzmann system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of rarefaction wave fan to compressible Euler equations is proved to 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in [18] sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.

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

Cited by 34 publications

References 68 publications

Large-time behavior for fluid and kinetic plasmas with collisions

Large-time behavior for fluid and kinetic plasmas with collisions

The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential

Stability of Nonlinear Wave Patterns to the Bipolar Vlasov–Poisson–Boltzmann System

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