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

TL;DR

This paper introduces a new mechanism demonstrating finite time singularities in the 3D incompressible Euler equations, constructing solutions with specific regularity properties that blow up at a finite time without using self-similar coordinates.

Contribution

It presents a novel construction of finite time singularities for the 3D Euler equations using multiple vorticity regions separated by vortex-free zones, avoiding self-similar blow-up methods.

Findings

Constructed solutions blow up at finite time with specific regularity.

Solutions are smooth away from the origin and in certain function spaces.

Blow-up occurs without self-similar coordinate techniques.

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.