Stability of small-amplitude shock profiles for the navier-stokes-poisson system

Abstract

We study stability of small-amplitude ion-acoustic shock profiles for the one-dimensional compressible Navier-Stokes-Poisson system which describes dynamics of positive ions in a collision-dominated plasma. The first main difficulty with stability analysis for shock profile solutions is to handle unstable (stationary) mode which is from translational invariance of the waves. Duan-Liu-Zhang [8] have shown asymptotic stability for zero mass initial perturbations. The main goal of this thesis is to establish nonlinear orbital stability without any assumptions and obtain sharp decay rate of considered perturbations. We first show that the shock profiles are strongly spectrally stable. Also, we prove simplicity of the zero eigenvalue by direct computation of the Evans function. From the spectral informations, we can find that there is no spectral gap between essential spectrum and the zero eigenvalue, of the associated linearized operator. This is the second difficulty in our analysis. To resolve this, we adopt pointwise semigroup methods introduced in [43]. Once we define a resolvent kernel, it can be extended on a small neighborhood of the origin, i.e., to regions of essential spectrum. This, together with spectral resolution formula, allows us to obtain sharp pointwise bounds on the Green’s function of the linearized operator. Then, we establish orbital stability by using the bounds of Green’s function and modulation techniques.