On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluid
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917170720538624 |
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| author | Huang, Yucong Hashimoto, Itsuko Nishibata, Shinya |
| author_facet | Huang, Yucong Hashimoto, Itsuko Nishibata, Shinya |
| contents | 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, $Ω=\{x\in\mathbb{R}^n\,\vert\, |x|\ge 1\}$, and a constant stream of mass is flowing into the domain from the boundary $\partialΩ=\{|x|=1\}$. 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 $(\tildeρ,\tilde{u})(r)$ with $r\equiv |x|$. In this paper, we show that either $\tildeρ$ is monotone increasing or $\tildeρ$ 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 $(\tildeρ,\tilde{u})$ which allows us to prove the time-asymptotic stability of $(\tildeρ,\tilde{u})$ using the energy method. More specifically, we prove this under small initial perturbation on $(\tildeρ,\tilde{u})$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07469 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluid Huang, Yucong Hashimoto, Itsuko Nishibata, Shinya Analysis of PDEs Mathematical Physics 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, $Ω=\{x\in\mathbb{R}^n\,\vert\, |x|\ge 1\}$, and a constant stream of mass is flowing into the domain from the boundary $\partialΩ=\{|x|=1\}$. 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 $(\tildeρ,\tilde{u})(r)$ with $r\equiv |x|$. In this paper, we show that either $\tildeρ$ is monotone increasing or $\tildeρ$ 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 $(\tildeρ,\tilde{u})$ which allows us to prove the time-asymptotic stability of $(\tildeρ,\tilde{u})$ using the energy method. More specifically, we prove this under small initial perturbation on $(\tildeρ,\tilde{u})$, 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. |
| title | On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluid |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2404.07469 |