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: $w_t + auw_x = u_xw, \quad u_x = Hw$ We show that for any $\alpha \in (0, 1)$ such that $|a\alpha|$ 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 $1/\alpha \in \mathbb Z$ and \cite{El-GhMa,ChHoHu}, which only deals with asymptotically self-similar blow-ups.