Self-Similar Algebraic Spiral Solution of 2-D Incompressible Euler Equations

Abstract

In this paper, we prove the existence of self-similar algebraic spiral solutions of the 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 (\(m\ge 2\)) 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.