Asymptotic Stability of 3D Out-flowing Compressible Viscous Fluid under Non-Spherical Perturbation

Abstract

We study an outflow problem for the $3$-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of the unit ball $\Omega=\{x\in\mathbb{R}^3\,\vert\, |x|\ge 1\}$ and it is flowing out from the unit ball $\Omega$ at a constant speed $|u_b|$, in the normal direction to the boundary surface $\partial\Omega$. The existence of a unique spherically symmetric stationary solution $(\tilde{\rho},\tilde{u})$ is obtained by I.~Hashimoto and A.~Matsumura in 2021, provided that the fluid velocity at the far-field is assumed to be zero, and $|u_b|$ is sufficiently small. Subsequently, authors of the present article prove in 2024 that $(\tilde{\rho},\tilde{u})$ is time-asymptotically stable under large spherically symmetric initial perturbations in the suitable Sobolev norm. The main purpose of the present paper is to investigate the case when the initial perturbations are possibly non-spherically symmetric. We show that $(\tilde{\rho},\tilde{u})$ remains asymptotically stable in time, under general small initial perturbations in the $H^3$-norm.