Finite-Time Singularity Formation for $C^{1,\alpha}$ Solutions to the Incompressible Euler Equations on $\mathbb{R}^3$

TL;DR

This paper demonstrates that smooth solutions to the 3D incompressible Euler equations with Hölder continuous gradients can develop finite-time singularities, challenging previous notions of well-posedness in this class.

Contribution

It proves finite-time singularity formation for $C^{1,eta}$ solutions of the 3D Euler equations, extending understanding of solution behavior in classical function spaces.

Findings

Finite-time singularities can occur for $C^{1,eta}$ solutions.

Singularities arise even in simple three-dimensional flows.

Challenges previous assumptions of well-posedness in Hölder spaces.

Abstract

It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with H\"older continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.