A half-space problem on the full Euler-Poisson system

TL;DR

This paper investigates the existence and stability of stationary solutions to the full Euler-Poisson system for ions on a half line, demonstrating convergence and stability under small perturbations using energy methods.

Contribution

It extends the analysis of the Euler-Poisson system by establishing stationary solutions and their large-time stability with explicit convergence rates, considering boundary effects.

Findings

Existence of stationary solutions under the Bohm criterion.

Large-time asymptotic stability of small-amplitude solutions.

Explicit convergence rates toward stationary solutions.

Abstract

This paper is concerned with the initial-boundary value problem on the full Euler-Poisson system for ions over a half line. We establish the existence of stationary solutions under the Bohm criterion similar to the isentropic case and further obtain the large time asymptotic stability of small-amplitude stationary solutions provided that the initial perturbation is sufficiently small in some weighted Sobolev spaces. Moreover, the convergence rate of the solution toward the stationary solution is obtained. The proof is based on the energy method. A key point is to capture the positivity of the temporal energy dissipation functional and boundary terms with suitable space weight functions either algebraic or exponential depending on whether or not the incoming far-field velocity is critical.