A spatially localized $L \log L$ estimate on the vorticity in the 3D NSE

TL;DR

This paper establishes a spatially localized $L ext{log} L$ bound on vorticity in 3D Navier-Stokes equations under a mild geometric condition, leading to insights on vorticity distribution decay and blow-up scenarios.

Contribution

It introduces a new localized $L ext{log} L$ estimate for vorticity in 3D NSE based on geometric assumptions, advancing understanding of vorticity behavior.

Findings

Extra-log decay of vorticity distribution function.

Breaks criticality in blow-up scenarios.

Provides a geometric condition for vorticity bounds.

Abstract

The purpose of this note is to present a spatially localized $L \log L$ bound on the vorticity in the 3D Navier-Stokes equations, assuming a very mild, \emph{purely geometric} condition. This yields an extra-log decay of the distribution function of the vorticity, which in turn implies \emph{breaking the criticality} in a physically, numerically, and mathematical analysis-motivated blow-up scenario based on vortex stretching and anisotropic diffusion.