Finite-time singularity formation for $C^{1,\alpha }$ solutions to the incompressible Euler equations on $\mathbb{R}^3$

Abstract

It has been known since work of Lichtenstein and Gunther in the 1920s that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder 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.