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

In this article, a perturbation theory of the compressible Navier-Stokes equations in $\mathbb{R}^n$ $(n \geq 3)$ 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 $u_ω$ 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.

💡 Research Summary

The paper develops a perturbation framework for the compressible Navier‑Stokes (CNS) system in $\mathbb R^{n}$ with $n\ge3$, focusing on the long‑time decay of solutions that are small perturbations of a non‑constant stationary state $u_{\omega}$. Unlike most previous works that treat perturbations around a constant equilibrium, the authors allow $u_{\omega}$ to decay only like $|x|^{-1}$ at infinity and to belong merely to weak Lebesgue spaces $L^{n,\infty}$ (for the state) and $L^{n/2,\infty}$ (for its gradient). This setting is motivated by physically relevant stationary solutions such as the PR‑type solutions and by recent results on exterior domain flows.

The main result (Theorem 1.5) states that if the initial perturbation $u_{0}$ lies in $L^{p}\cap H^{s}$ with $1<p<2$, $s\ge n/2+2$, and is sufficiently small, then a global solution $u(t)$ exists and satisfies \