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

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.