Ion-acoustic shock in a collisional plasma

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)3721-3768
Journal / PublicationJournal of Differential Equations
Issue number4
Online published18 Mar 2020
Publication statusPublished - 5 Aug 2020


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 ε and the Debye length is proportional to (δε)1/2 with two parameters ε and δ, and show that in the limit ε → 0, shock profiles obtained in (a) can be approximated by the profiles of KdV-Burgers uniformly for 0 < δδ0 with some δ> 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 ε and δ. (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).

Research Area(s)

  • Collisional plasma, Ion-acoustic shock, KdV-Burgers equations, Navier-Stokes-Poisson equations, Nonlinear stability

Citation Format(s)

Ion-acoustic shock in a collisional plasma. / Duan, Renjun; Liu, Shuangqian; Zhang, Zhu.
In: Journal of Differential Equations, Vol. 269, No. 4, 05.08.2020, p. 3721-3768.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review