Geometric properties and non-blowup of 3-D incompressible Euler flow

TL;DR

This paper investigates the geometric structure of vorticity in 3D Euler flows, establishing a relationship with vortex stretching that improves understanding of conditions for global existence.

Contribution

It introduces a new geometric perspective linking vorticity properties to vortex stretching, leading to improved global existence results for the 3D Euler equations.

Findings

Established a sharp relationship between vorticity geometry and vortex stretching.

Provided an improved global existence criterion for 3D Euler flows.

Supported findings with numerical observations.

Abstract

By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum vortex stretching. This new understanding leads to an improved result of the global existence of the 3-D Euler equation under mild assumptions that are consistent with the observations from recent numerical computations.