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<p>We prove a nonlinear stability result for steady flows of a two-dimensional ideal fluid in a bounded multiply-connected domain, which generalizes a stability theorem 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 by using 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Guodong Wang,
  • Bijun Zuo

摘要

We prove a nonlinear stability result for steady flows of a two-dimensional ideal fluid in a bounded multiply-connected domain, which generalizes a stability theorem 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 by using 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.