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

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.