Finite time singularities to the 3D incompressible Euler equations for solutions in $C^{\infty }({\mathbb{R}}^3\setminus \{0\})\cap C^{1,\alpha }\cap L^2$

Abstract

We introduce a novel mechanism that reveals finite time singularities within the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably, we do not construct our blow up using self-similar coordinates, but build it from infinitely many regions with vorticity, separated by vortex-free regions in between. It yields solutions of the 3D incompressible Euler equations in ${\mathbb{R}}^3\times [-T,0]$ such that the velocity is in the space $C^{\infty }({\mathbb{R}}^3\setminus \{0\})\cap C^{1,\alpha }\cap L^2$ for times $t\in (-T,0)$ and is not $C^1$ at time 0.

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