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 \(C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,\alpha}\cap L^2\) where \(0 < \alpha \ll 1\) for times \(t\in (-T,0)\) and is not \(C^1\) at time 0.