On Arnold's second stability theorem for two-dimensional steady ideal flows in a bounded domain

Abstract: For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $\psi$ and of 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 $C_{ar}$ can be chosen as $\bm\Lambda_1,$ the first eigenvalue of $-\Delta$ in the space of mean-zero functions that are piecewise constant on the boundary. When $\nabla\omega/\nabla\psi$ is allowed to reach $\bm\Lambda_1$, instability may occur, as illustrated by a non-circular steady flow in a disk; however, we show that certain structural stability still holds. As an application, we establish a general stability criterion for steady flows in a disk.