TY - JOUR
T1 - Ion-acoustic shock in a collisional plasma
AU - Duan, Renjun
AU - Liu, Shuangqian
AU - Zhang, Zhu
PY - 2020/8/5
Y1 - 2020/8/5
N2 - 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 < δ ≤ δ0 with some δ0 > 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).
AB - 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 < δ ≤ δ0 with some δ0 > 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).
KW - Collisional plasma
KW - Ion-acoustic shock
KW - KdV-Burgers equations
KW - Navier-Stokes-Poisson equations
KW - Nonlinear stability
UR - http://www.scopus.com/inward/record.url?scp=85081967291&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85081967291&origin=recordpage
U2 - 10.1016/j.jde.2020.03.012
DO - 10.1016/j.jde.2020.03.012
M3 - 21_Publication in refereed journal
VL - 269
SP - 3721
EP - 3768
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 4
ER -