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

TL;DR

The paper investigates the time-asymptotic stability of a 3D outflowing, isentropic compressible Navier–Stokes flow in the exterior domain $\Omega=\{ |x|\, oeq 1 ight\\\\}$ under non-spherical perturbations. Building on the known spherically symmetric stationary state $( ilde ho, ilde u)$, the authors develop an energy-method framework around perturbations $( ho- ilde ho,u- ilde u)$, denoted by $(\,\phi,\\psi)$, and derive a full suite of a priori estimates: L^2 relative-energy bounds, time-derivative controls, angular-derivative bounds via a tangential-calculus decomposition, radial-damping for $\partial_r\phi$, and elliptic/Stokes regularity to close high-order norms. A smallness condition on the boundary outflow speed $|u_b|$ and the initial perturbation in $H^3$ ensures global existence and convergence to the stationary state as $t\to\infty$, i.e., $( ho,u)\to(\tilde ho,\tilde u)$ in an appropriate Sobolev topology. The approach extends prior results from spherically symmetric perturbations to general non-symmetric disturbances, establishing robust stability in the exterior-domain setting with precise energy-dissipation mechanisms and cancellations. This yields a rigorous foundation for the persistence of stationary outflow states under realistic perturbations, with potential implications for compressible flow control in exterior-domain configurations.

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 $Ω=\{x\in\mathbb{R}^3\,\vert\, |x|\ge 1\}$ and it is flowing out from the unit ball $Ω$ at a constant speed $|u_b|$, in the normal direction to the boundary surface $\partialΩ$. The existence of a unique spherically symmetric stationary solution $(\tildeρ,\tilde{u})$ 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 $(\tildeρ,\tilde{u})$ 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 $(\tildeρ,\tilde{u})$ remains asymptotically stable in time, under general small initial perturbations in the $H^3$-norm.

Theorems & Definitions (30)

• Lemma 1.1
• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6