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

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 $\mathrm{\mid }y{\mathrm{\mid }}^{-\frac{1}{\mu }}\mathring{\omega }(\theta )$ with $\mu >\frac{1}{2}$ 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.