Convergence to Steady-States of Compressible Navier–Stokes–Maxwell Equations

In this paper, we consider the compressible Navier–Stokes–Maxwell equations with a non-constant background density in ℝ^3 . We first show the existence and uniqueness of the non-trivial equilibrium (steady-state) of the system when the background density is a small variation of certain constant state, then we prove the asymptotic stability of the steady-state once the initial perturbation around the steady-state is small. Furthermore, by establishing the time-decay estimates for the corresponding linearized homogeneous equations, we artfully derive the time-algebraic convergence rates. The proof is based on the time-weighted energy method but with some new developments on the weight settings.