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 multidimensional isentropic compressible NavierStokes 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 $\partial\Omega=\{x=1\}$. The existence and uniqueness of a spherically symmetric stationary solution, denoted as $(\tilde{\rho},\tilde{u})$, is first proved by I. Hashimoto and A. Matsumura in 2021. 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. Moreover, we also derive a set of decay rates for $(\tilde{\rho},\tilde{u})$ which allows us to prove the long time stability of $(\tilde{\rho},\tilde{u})$ under small initial perturbations using the energy method. The main difficulty for this is the boundary terms that appears in the apriori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.07469
 arXiv:
 arXiv:2404.07469
 Bibcode:
 2024arXiv240407469H
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics
 EPrint:
 31 pages