Ion-acoustic shock in a collisional plasma

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Duan, Renjun¹  Liu, Shuangqian^2,3  Zhang, Zhu⁴ 

[1] Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China

[2] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China

[3] Jinan Univ, Dept Math, Guangzhou 510632, Peoples R China

[4] City Univ Hong Kong, Hong Kong Inst Adv Study, Kowloon, Hong Kong, Peoples R China

关键词: Collisional plasma;    Ion-acoustic shock;    Navier-Stokes-Poisson equations;    KdV-Burgers equations;    Nonlinear stability;   

DOI  :  10.1016/j.jde.2020.03.012

来源: Elsevier

【 摘 要 】

The paper is concerned with the propagation of ion-acoustic shock waves in a collision dominated plasma whose equations of motion are described by the one-dimensional isothermal Navier-Stokes-Poisson system for ions with the electron density determined by the Boltzmann relation. The main results include three parts: (a) We establish the existence and uniqueness of a small-amplitude smooth traveling wave by solving a 3-D ODE in terms of the center manifold theorem. (b) We study the shock structure in a specific asymptotic regime where the viscosity coefficient and the shock strength are proportional to epsilon and the Debye length is proportional to (delta epsilon)(1/2) with two parameters epsilon and delta, and show that in the limit epsilon -> 0, shock profiles obtained in (a) can be approximated by the profiles of KdV-Burgers uniformly for 0 < delta <= delta(0) with some delta(0)> 0. The proof is based on the suitable construction of the KdV-Burgers shock profiles together with the delicate analysis of a linearized variable coefficient system in exponentially weighted Sobolev spaces involving parameters epsilon and delta. (c) We also prove the large time asymptotic stability of traveling waves under suitably small smooth zero-mass perturbations. Note that the ions' temperature is allowed to be zero in parts (a) and (b), but necessarily required to be strictly positive in the proof of part (c). (C) 2020 Elsevier Inc. All rights reserved.

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