Self-similar algebraic spiral solution of 2-D incompressible Euler equations

TL;DR

This paper establishes the existence of self-similar algebraic spiral solutions for 2-D incompressible Euler equations with specific initial vorticity profiles, advancing understanding of vortex structures and weak solutions.

Contribution

It proves the existence of such spiral solutions for a broad class of initial vorticities, including Radon measures, and connects to vortex sheet solutions.

Findings

Existence of self-similar algebraic spiral solutions for specified initial vorticity.

Construction of weak solutions with Radon measure initial vorticity.

Application to vortex sheet solutions in 2-D Euler flows.

Abstract

In this paper, we prove the existence of self-similar algebraic spiral solutions for 2-D incompressible Euler equations for the initial vorticity of the form $|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)$ with $\mu>\frac12$ and $\mathring{\omega}\in L^1(\mathbb T)$ satisfying $m$-fold symmetry ($m\geq 2$) and a dominant condition. As an important application, we prove the existence of weak solution when $\mathring{\omega}$ is a Radon measure on $\mathbb T$ with $m$-fold symmetry, which is related to the vortex sheet solution.