# About some possible blow-up conditions for the 3-D Navier-Stokes equations

## TL;DR

This paper investigates conditions under which solutions to the 3D Navier-Stokes equations may blow up, showing that smallness in certain critical spaces can ensure global well-posedness, with implications for understanding turbulence.

## Contribution

It refines existing blow-up criteria by demonstrating global well-posedness under smallness conditions in specific critical Sobolev and Besov spaces for one velocity component or vorticity.

## Key findings

- Smallness in a subspace of ot;H^{1/2} ensures global solutions.
- Conditions on vorticity and velocity derivatives in Besov spaces prevent blow-up.
- Modified proof of recent results extends understanding of Navier-Stokes regularity.

## Abstract

In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from \cite{Isab}, we prove that if one component of the velocity remains small enough in a sub-space of $\dot{H}^{\frac{1}{2}}$ "almost" scaling invariant, then the 3D Navier Stokes is globally wellposed. In a second time, we investigate the same question under some conditions on one component of the vorticity and unidirectional derivative of one component of the velocity in some critical Besov spaces of the form $L^p_T(\dot{B}_{2,\infty}^{\alpha, \frac{2}{p}-\frac{1}{2}-\alpha})$ or $L^p_T(\dot{B}_{q,\infty}^{ \frac{2}{p}+\frac{3}{q}-2})$.