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<p>This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets following the formation of curvature singularities due to the Kelvin-Helmholtz instability. Furthermore, they constitute plausible candidates for demonstrating non-uniqueness within the class of Delort’s weak solutions. The most challenging part of this paper is handling the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.</p>

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Self-Similar Algebraic Spiral Vortex Sheets of 2-D Incompressible Euler Equations

  • 期刊论文

  • 发表日期：

    2026年3月6日

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  • Feng Shao,
  • Dongyi Wei,
  • Zhifei Zhang

摘要

This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets following the formation of curvature singularities due to the Kelvin-Helmholtz instability. Furthermore, they constitute plausible candidates for demonstrating non-uniqueness within the class of Delort’s weak solutions. The most challenging part of this paper is handling the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.