On the Stability of Self-similar Blow-up for $C^{1,\alpha }$ Solutions to the Incompressible Euler Equations on ${\mathbb{R}}^3$

Abstract

We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,\alpha }$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale-Kato-Majda criterion cannot be improved in the class of $C^{1,\alpha }$ solutions.