Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.06807 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909785236963328 |
|---|---|
| author | Wang, Fatao Wang, Guodong Zuo, Bijun |
| author_facet | Wang, Fatao Wang, Guodong Zuo, Bijun |
| contents | For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $ψ$ and its vorticity $ω$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if $0<\nablaω/\nablaψ<C_{ar}$ for some $C_{ar}>0$. In this paper, we show that, for a bounded domain, $C_{ar}$ can be taken as the first eigenvalue $\bmΛ_1$ of a certain Laplacian eigenvalue problem. When $\nablaω/\nablaψ$ reaches $\bmΛ_1$, instability may occur, as illustrated by a non-circular steady flow in a disk; however, a certain form of structural stability still holds. Based on these results, we establish a theorem on the rigidity and orbital stability of steady Euler flows in a disk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06807 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the positive constant in Arnold's second stability theorem for a bounded domain Wang, Fatao Wang, Guodong Zuo, Bijun Analysis of PDEs For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $ψ$ and its vorticity $ω$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if $0<\nablaω/\nablaψ<C_{ar}$ for some $C_{ar}>0$. In this paper, we show that, for a bounded domain, $C_{ar}$ can be taken as the first eigenvalue $\bmΛ_1$ of a certain Laplacian eigenvalue problem. When $\nablaω/\nablaψ$ reaches $\bmΛ_1$, instability may occur, as illustrated by a non-circular steady flow in a disk; however, a certain form of structural stability still holds. Based on these results, we establish a theorem on the rigidity and orbital stability of steady Euler flows in a disk. |
| title | On the positive constant in Arnold's second stability theorem for a bounded domain |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.06807 |