On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluid

TL;DR

This paper studies the stability of a spherically symmetric solution to an inflow problem for the isentropic compressible Navier-Stokes equations in an exterior domain, establishing decay rates and stability under small perturbations.

Contribution

It classifies the density profile of the stationary solution and proves its time-asymptotic stability using energy methods in Lagrangian coordinates.

Findings

Density is either monotone increasing or has a unique minimum.

Derived decay rates for the stationary solution.

Proved stability under small initial perturbations.

Abstract

We investigate an inflow problem for the multi-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of unit ball, $\Omega=\{x\in\mathbb{R}^n\,\vert\, |x|\ge 1\}$, and a constant stream of mass is flowing into the domain from the boundary $\partial\Omega=\{|x|=1\}$. It is shown in Hashimoto-Matsumura(2021) that if the fluid velocity at the far-field is assumed to be zero, then there exists a unique spherically symmetric stationary solution, denoted as $(\tilde{\rho},\tilde{u})(r)$ with $r\equiv |x|$. In this paper, we show that either $\tilde{\rho}$ is monotone increasing or $\tilde{\rho}$ attains a unique global minimum, and this is classified by the boundary condition of density. In addition, we also derive a set of spatial decay rates for $(\tilde{\rho},\tilde{u})$ which allows us to prove the time-asymptotic stability of $(\tilde{\rho},\tilde{u})$ using the energy method. More specifically, we prove this under small initial perturbation on $(\tilde{\rho},\tilde{u})$, provided that the density at the far-field is supposed to be strictly positive but suitably small, in other words, the far-field state of the fluid is not vacuum but suitably rarefied. The main difficulty for the proof is the boundary terms that appears in the a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.