Open problems concerning the Holder continuity of the direction of vorticity for the Navier-Stokes equations

TL;DR

This paper discusses open problems related to the Holder continuity of vorticity direction in 3D Navier-Stokes solutions and explores how this affects regularity and integrability properties of vorticity.

Contribution

It highlights unresolved issues and proposes an approach to analyze the impact of Holder continuity assumptions on vorticity regularity in Navier-Stokes equations.

Findings

Identifies key open problems in vorticity regularity

Connects Holder continuity with space-time integrability

Provides a framework for future investigations

Abstract

In these notes we consider solutions u to the evolution three dimensional Navier-Stokes equations in the whole space. Set w= curl u, the vorticity of u. Our study concerns mainly the relation between Holdercontinuity assumptions on the direction of the vorticity and induced space-time integrability regularity results for w. We discuss some related open problems, consistent with our calculations. We hope that our approach should help readers interesting in carrying on the investigation.