Some remarks about the possible blow-up for the Navier-Stokes equations

TL;DR

This paper explores conditions under which controlling a single velocity component in the Navier-Stokes equations can prevent singularities, highlighting limitations and proposing a near-invariant space for blow-up prevention.

Contribution

It introduces a new space nearly invariant under scaling and demonstrates that controlling one velocity component in this space can prevent blow-up, advancing understanding of Navier-Stokes regularity.

Findings

One component cannot tend to zero too fast near blow-up.

A new nearly invariant space under scaling is proposed.

Controlling one component in this space can prevent singularities.

Abstract

In this work we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant norms. In particular we prove that it is not possible for one component of the velocity field to tend to~$0$ too fast near blow up. We also introduce a space "almost" invariant under the action of the scaling such that if one component of the velocity field measured in this space remains small enough, then there is no blow up.