Perturbation theory of the compressible Navier-Stokes equations and its application

Abstract

In this article, a perturbation theory of the compressible Navier-Stokes equations in ${\mathbb{R}}^n$ $(n\ge 3)$ is studied to investigate decay estimate of solutions around a non-constant state. As a concrete problem, stability is considered for a perturbation system from a stationary solution $u_{\omega }$ belonging to the weak $L^n$ space. Decay rates of the perturbation including $L^{\infty }$ norm are obtained which coincide with those of the heat kernel except a bit loss. The proof is based on deriving suitable resolvent estimates with perturbation terms in the low frequency part having a parabolic spectral curve. Our method can be applicable to dispersive hyperbolic systems like wave equations with strong damping. Indeed, a parabolic type decay rate of a solution is obtained for a damped wave equation including variable coefficients which satisfy spatial decay conditions.

Revised a proof of Proposition 4.15