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

TL;DR

This paper rigorously constructs self-similar algebraic spiral vortex sheet solutions for 2-D incompressible Euler equations, offering insights into vortex sheet roll-up patterns post-singularity formation.

Contribution

It introduces the first rigorous construction of algebraic spiral vortex sheets, addressing complex singular integral challenges beyond classical theory.

Findings

First rigorous construction of algebraic spiral vortex sheets

Provides mathematical insight into vortex sheet roll-up patterns

Handles complex singular integral operators in the construction

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.