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

TL;DR

This paper proves that a specific 3D outflowing compressible viscous fluid solution remains stable over time even when initial disturbances are not spherically symmetric, extending previous spherical symmetry results.

Contribution

It extends the stability analysis of outflow solutions to include non-spherical initial perturbations in the 3D compressible Navier-Stokes equations.

Findings

Asymptotic stability under non-spherical perturbations in 3D.

Stability holds for small initial disturbances in the H^3 norm.

Generalization of previous spherical symmetry stability results.

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.