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

TL;DR

This paper refines Arnold's second stability theorem for 2D ideal fluid flows in bounded domains by identifying the positive constant as the first Laplacian eigenvalue, linking stability bounds to spectral properties.

Contribution

It demonstrates that the constant in Arnold's stability criterion equals the first eigenvalue of a Laplacian problem for bounded domains, extending the theorem's applicability.

Findings

The constant $C_{ar}$ can be taken as the first eigenvalue $m\Lambda_1$ in bounded domains.

Instability can occur when $ abla\omega/ abla\psi$ reaches $m\\Lambda_1$.

Structural stability persists under certain conditions despite reaching the eigenvalue.

Abstract

For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $\psi$ and its vorticity $\omega$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if $0<\nabla\omega/\nabla\psi<C_{ar}$ for some $C_{ar}>0$. In this paper, we show that, for a bounded domain, $C_{ar}$ can be taken as the first eigenvalue $\bm\Lambda_1$ of a certain Laplacian eigenvalue problem. When $\nabla\omega/\nabla\psi$ reaches $\bm\Lambda_1$, 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.