An extension of Arnold's second stability theorem in a multiply-connected domain

TL;DR

This paper extends Arnold's second stability theorem to multiply-connected domains, providing a new sufficient condition for the nonlinear stability of steady flows in such complex geometries.

Contribution

It generalizes Arnold's stability criterion to multiply-connected domains using a variational approach and energy-Casimir methods, broadening the applicability of stability analysis.

Findings

Established a variational characterization for steady flows in multiply-connected domains.

Proved a sufficient condition for nonlinear stability in these domains.

Extended Arnold's second stability theorem to more complex geometries.

Abstract

We give a sufficient condition for the nonlinear stability of steady flows of a two-dimensional ideal fluid in a bounded multiply-connected domain, which generalizes a stability criterion proved by Arnold in the 1960s. The most important ingredient of the proof is to establish a variational characterization for the steady flow under consideration, which is achieved based on the energy-Casimir method proposed by Arnold, and the supporting functional method introduced by Wolansky and Ghil. Nonlinear stability then follows from a compactness argument related to the variational characterization and proper use of conserved quantities of the two-dimensional Euler equations.