Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations

TL;DR

This paper proves the asymptotic stability of stationary solutions to the three-dimensional compressible Euler-Maxwell equations with a small perturbation in background density, providing convergence rates through advanced mathematical estimates.

Contribution

It establishes the stability and convergence rates of solutions near stationary states for the Euler-Maxwell system with variable background density.

Findings

Proved asymptotic stability under small initial perturbations.

Derived explicit convergence rates using $L^p$-$L^q$ estimates.

Extended stability analysis to nonconstant background densities.

Abstract

In this paper, we are concerned with the compressible Euler-Maxwell equations with a nonconstant background density (e.g. of ions) in three dimensional space. There exist stationary solutions when the background density is a small perturbation of a positive constant state. We first show the asymptotic stability of solutions to the Cauchy problem near the stationary state provided that the initial perturbation is sufficiently small. Moreover the convergence rates are obtained by combining the $L^p$-$L^q$ estimates for the linearized equations with time-weighted estimate.