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

Abstract

For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $\ddot{I}ˆ$ and its vorticity $\ddot{I}‰$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if $0<\nabla \ddot{I}‰/\nabla \ddot{I}ˆ<C_{ar}$ for some $C_{ar}>0$. In this paper, we show that, for a bounded domain, $C_{ar}$ can be taken as the first eigenvalue $\hat{\mathbf{I}}{›}_1$ of a certain Laplacian eigenvalue problem. When $\nabla \ddot{I}‰/\nabla \ddot{I}ˆ$ reaches $\hat{\mathbf{I}}{›}_1$, 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.

25 pages; Some writing improvements are provided in this version