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arXiv · 2505.06807

On the positive constant in Arnold's second stability theorem for a bounded domain

Fatao Wang·Guodong Wang·Bijun Zuo
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Abstract

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 $0<\nablaω/\nablaψ 0$. 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.

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Fatao Wang, Guodong Wang, Bijun Zuo. 2025-09-14. On the positive constant in Arnold's second stability theorem for a bounded domain. https://arxiv.org/abs/2505.06807

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