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

Guodong Wang; Bijun Zuo

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arXiv · 2208.10697

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

Guodong Wang·Bijun Zuo

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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.

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Guodong Wang, Bijun Zuo. 2022-08-23. An extension of Arnold's second stability theorem in a multiply-connected domain. https://arxiv.org/abs/2208.10697

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