A solution of 2D incompressible Euler equation with algebraic spiral roll-up in the presence of Wiener type perturbation

Extending the results of Elling \cite{Elling-2013, Elling-2016}, we construct a weak solution of 2D incompressible Euler equation with initial vorticity of the form $w_0(x)={\mid x\mid }^{-1\mathrm{/}\mu }g(\theta )$, where $g\in L^p(\mathbb{T})$ satisfies ${\sum }_{\mathbb{Z}}{\mid n\mid }^{-0.5}\mid \widehat{g}(n)\mid <\mathrm{\infty }$. In particular, the solution is self-similar and shows algebraic spiral roll-up.