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

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, =\xⁿ\, \, |x| 1\, and a constant stream of mass is flowing into the domain from the boundary =\|x|=1\. The existence and uniqueness of a spherically symmetric stationary solution, denoted as (, u), is first proved by I. Hashimoto and A. Matsumura in 2021. In this paper, we show that either is monotone increasing or attains a unique global minimum, and this is classified by the boundary condition of density. Moreover, we also derive a set of decay rates for (, u) which allows us to prove the long time stability of (, u) under small initial perturbations using the energy method. The main difficulty for this is the boundary terms that appears in the a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.