A geometric measure-type regularity criterion for solutions to the 3D Navier-Stokes equations

TL;DR

This paper introduces a new geometric regularity criterion based on local anisotropic measure conditions that prevent finite-time singularities in solutions to the 3D Navier-Stokes equations, emphasizing weak one-dimensional sparseness of intense fluid regions.

Contribution

It proposes a novel geometric measure-type regularity condition involving local anisotropic sparseness to control solution regularity in 3D NSE.

Findings

Prevents finite-time singularities under the proposed condition

Establishes a link between geometric sparseness and solution regularity

Provides a new criterion for analyzing 3D Navier-Stokes solutions

Abstract

A local anisotropic geometric measure-type condition on the super-level sets of solutions to the 3D NSE preventing the formation of finite-time singularity is presented; essentially, local one-dimensional sparseness of the regions of intense fluid activity in a very weak sense.