Exactly self-similar blow-up of the generalized De Gregorio equation

We study exactly self-similar blow-up profiles fot the generalized De Gregorio model for the three-dimensional Euler equation: $w_t+au{w}_x=u_{x}w,u_x=Hw$ We show that for any $\alpha \in (0,1)$ such that $\mathrm{\mid }a\alpha \mathrm{\mid }$ is sufficiently small, there is an exactly self-similar $C^{\alpha }$ solution that blows up in finite time. This simultaneously improves on the result in \cite{ElJe} by removing the restriction $1\mathrm{/}\alpha \in \mathbb{Z}$ and \cite{El-GhMa,ChHoHu}, which only deals with asymptotically self-similar blow-ups.