Self-similar algebraic spiral vortex sheets of 2-D incompressible Euler equations

2025/05/06 by Feng Shao, Dongyi Wei, Shao, Feng +3 · 1 citation
Earth and Planetary Sciences · Engineering · Mathematics · #Analysis of PDEs (math.AP) #Aquatic and Environmental Studies #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2505.03309

openalex publication_date 2025/05/06 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

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 after the formation of curvature singularities. The most challenging part of this paper is to handle the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.

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• Finite-time self-similar implosion of hollow vortices

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