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 $\partial\Omega=\{|x|=1\}$. 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 a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2024
- DOI:
- 10.48550/arXiv.2404.07469
- arXiv:
- arXiv:2404.07469
- Bibcode:
- 2024arXiv240407469H
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematical Physics
- E-Print:
- 31 pages