KdV Limit of the Euler–Poisson System

Guo, Yan; Pu, Xueke

doi:10.1007/s00205-013-0683-z

Cited by 42 publications

(47 citation statements)

References 18 publications

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“…where compared to the two-component VPB system (1.1), the dynamical equation of electrons and the ions-electrons collisions have been omitted, and the number density n e = R 3 f e dξ has been replaced by an analogue of the classical Boltzmann relation n e = exp{φ/T e }, or a general function depending on the potential function φ. We remark that the Boltzmann relation has been recently extensively visited in a lot of studies of kinetic and related fluid dynamic equations, for instance, [16,47,48,49,50,72]. Inspired by our previous works [25,26,28], we expect in the paper to further consider the much more physical two-component VPB system, particularly extending the results in [42,44] to the case of perturbations of the non-constant equilibrium state.…”

Section: Literature and Backgroundmentioning

confidence: 85%

The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

Duan

Liu²

2017

J Stat Phys

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The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in [42]. It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-trivial long-term dynamics of the binary system. In the paper, we consider the problem in the setting of rarefaction waves. The analytical tool is based on the macro-micro decomposition introduced in [59] that we can be able to develop into the case for the two-component Boltzmann equations around local bi-Maxwellians. Our focus is to explore how the disparate masses and charges of particles play a role in the analysis of the approach of the complex coupling system time-asymptotically toward a nonconstant equilibrium state whose macroscopic quantities satisfy the quasineutral nonisentropic Euler system. 10 2.3. Diffusion and heat-conductivity 15 3. Quasineutral Euler equations and rarefaction waves 20 4. Preliminary estimates on two-component collision operator 23 5. Proof of the main result 27 6. A priori estimates on the fluid part 32 6.1. Estimate on zero-order energy 32 6.2. Estimate on first-order dissipation 37 6.3. Estimate on first-order energy 44 7. A priori estimates on the non-fluid part 51 7.1. Estimate on zero-order dissipation 51 7.2. Estimate on high-order energy 55 7.3. Estimate on energy with mixed derivatives 60 References 61 2010 Mathematics Subject Classification. Primary: 35Q20, 76P05; Secondary: 35B35, 35B40.

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Section: Literature and Backgroundmentioning

confidence: 85%

The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

Duan

Liu²

2017

J Stat Phys

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“…then each component of (n (1) , u (1) , φ (1) ) satisfies (KdV). A mathematical validity of this result has not been proved until the work of [5], in which it is proved that the solutions to (GMEP) with the prepared initial data converge as ε → 0 to that of (KdV) on any fixed time interval. This result deals also with the case σ > 0 with the choice of V = √ 1 + σ.…”

Section: 2mentioning

confidence: 98%

Small amplitude limit of solitary waves for the Euler–Poisson system

Bae

Kwon

2019

Journal of Differential Equations

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The one-dimensional Euler-Poisson system arises in the study of phenomena of plasma such as plasma solitons, plasma sheaths, and double layers. When the system is rescaled by the Gardner-Morikawa transformation, the rescaled system is known to be formally approximated by the Korteweg-de Vries (KdV) equation. In light of this, we show existence of solitary wave solutions of the Euler-Poisson system in the stretched moving frame given by the transformation, and prove that they converge to the solitary wave solution of the associated KdV equation as the small amplitude parameter tends to zero. Our results assert that the formal expansion for the rescaled system is mathematically valid in the presence of solitary waves and justify Sagdeev's formal approximation for the solitary wave solutions of the pressureless Euler-Poisson system. Our work extends to the isothermal case.

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“…For details, see Section 2. To this aspect, one may refer to the recent papers [10,21,27]. In particular, Guo and Pu established rigorously the KdV limit for the ion Euler-Poisson system in 1D for both the cold and hot plasma case, where the electron density satisfies the classical Maxwell-Boltzmann law.…”

Section: Tomentioning

confidence: 99%

Long Wavelength Limit for the Quantum Euler--Poisson Equation

Liu¹,

Pu²

2016

SIAM J. Math. Anal.

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Abstract. In this paper, we consider the long wavelength limit for the quantum EulerPoisson equation. Under the Gardner-Morikawa transform, we derive the quantum Korteweg-de Vries (KdV) equation by a singular perturbation method. We show that the KdV dynamics can be seen at time interval of order O(ǫ −3/2 ). When the nondimensional quantum parameter H = 2, it reduces to the inviscid Burgers equation.

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KdV Limit of the Euler–Poisson System

Cited by 42 publications

References 18 publications

The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

Small amplitude limit of solitary waves for the Euler–Poisson system

Long Wavelength Limit for the Quantum Euler--Poisson Equation

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