Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909548768395264 |
|---|---|
| author | Li, Zhile |
| author_facet | Li, Zhile |
| contents | We consider solutions to the Navier-Stokes equations on $\mathbb{R}^2$ close to the Poiseuille flow with viscosity $0< ν< 1$. For the linearized problem, we prove that when the $x$-frequency satisfy $|k| \ge ν^{-\frac{1}{3}}$, the perturbation decays on a time-scale proportional to $ν^{-\frac{1}{2}}|k|^{-\frac{1}{2}}$. Since it decays faster than the heat equation, this phenomenon is referred to as enhanced dissipation. Then we concern the non-linear equations. We show that if the initial perturbation $ω_{in}$ is at most of size $ν^\frac{7}{3}$ in an anisotropic Sobolev space, then the size of the perturbation remains no more than twice the size of its initial value. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19716 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$ Li, Zhile Analysis of PDEs We consider solutions to the Navier-Stokes equations on $\mathbb{R}^2$ close to the Poiseuille flow with viscosity $0< ν< 1$. For the linearized problem, we prove that when the $x$-frequency satisfy $|k| \ge ν^{-\frac{1}{3}}$, the perturbation decays on a time-scale proportional to $ν^{-\frac{1}{2}}|k|^{-\frac{1}{2}}$. Since it decays faster than the heat equation, this phenomenon is referred to as enhanced dissipation. Then we concern the non-linear equations. We show that if the initial perturbation $ω_{in}$ is at most of size $ν^\frac{7}{3}$ in an anisotropic Sobolev space, then the size of the perturbation remains no more than twice the size of its initial value. |
| title | Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.19716 |