Sharp Global Well-Posedness for the Logarithmically Regularized 2D Euler Equations

We consider solutions to the two-dimensional Euler equations with a logarithmically regularized velocity given by $u={\mathrm{\nabla }}^{\perp }(-\mathrm{\Delta }{)}^{-1}{\ln }^{-\gamma }(\mathrm{\mid }\mathrm{\nabla }\mathrm{\mid }+e)\omega$ and initial vorticity in ${\dot{H}}^1\cap {\dot{H}}^{-1}({\mathbb{R}}^2)$. We show that if $\gamma >1\mathrm{/}2$, then global well-posedness holds.