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 $ψ$ and its vorticity $ω$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if 0<∇ω/∇ψ<Cₐᵣ for some Cₐᵣ>0. In this paper, we show that, for a bounded domain, Cₐᵣ can be taken as the first eigenvalue Λ₁ of a certain Laplacian eigenvalue problem. When ∇ω/∇ψ reaches Λ₁, 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.