Ion-acoustic shock in a collisional plasma

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).