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

2025/05/11 by Fatao Wang, Guodong Wang, Wang, Fatao +3
Engineering · Computer Science · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Modeling in Engineering #Algebraic and Geometric Analysis

paper · pdf · doi:10.48550/arxiv.2505.06807

Abstract

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/∇ψ0. In this paper, we show that, for a bounded domain, C_ar can be taken as the first eigenvalue \bmΛ₁ of a certain Laplacian eigenvalue problem. When ∇ω/∇ψ reaches \bmΛ₁, 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.

Citations

• Stability of a class of exact solutions of the incompressible Euler equation in a disk
• Stability of degree-2 Rossby-Haurwitz waves
• Nonlinear stability of sinusoidal Euler flows on a flat two-torus
• An extension of Arnold's second stability theorem in a multiply-connected domain
• Stability of two-dimensional steady Euler flows with concentrated vorticity
• Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems

Related

• An extension of Arnold's second stability theorem in a multiply-connected domain
• Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems
• Quantitative stability of a class of explicit steady Euler flows in a disk
• Nonlinear stability of plane ideal flows in a periodic channel
• On Arnold-type stability theorems for the Euler equation on a sphere