Small $\dot B^{-1}_{\infty,\infty}$ implies regularity

TL;DR

This paper proves that if the solution to the Navier-Stokes equations has a sufficiently small $ abla^{-1}_{ ext{infty}, ext{infty}}$ norm, then the solution remains regular and does not blow up in finite time.

Contribution

The paper establishes a new regularity criterion for Navier-Stokes solutions based on the smallness of the $ abla^{-1}_{ ext{infty}, ext{infty}}$ norm, extending understanding of blowup prevention.

Findings

Small $ abla^{-1}_{ ext{infty}, ext{infty}}$ norm prevents blowups in Navier-Stokes solutions.

Provides a new criterion for regularity based on Besov space norms.

Applicable to $d ext{--} 3$ dimensional incompressible flows.

Abstract

We show that smallness of $\dot B^{-1}_{\infty,\infty}$ norm of solution to $d$-dimensional ($d\ge 3$) incompressible Navier-Stokes prevents blowups.