A smallness regularity criterion for the 3D Navier-Stokes equations in the largest class

TL;DR

This paper establishes that smallness in the $ ext{B}^{-1}_{ ext{infty}, ext{infty}}$ norm of the velocity field guarantees classical solutions for the 3D Navier-Stokes equations, providing a new regularity criterion.

Contribution

It introduces a new regularity criterion based on the smallness of the $ ext{B}^{-1}_{ ext{infty}, ext{infty}}$ norm, expanding the class of initial data ensuring smooth solutions.

Findings

Small $ ext{B}^{-1}_{ ext{infty}, ext{infty}}$-norm implies classical solutions.

Regularity criterion in the largest known class.

Extension of previous regularity results.

Abstract

In this paper, we consider the three-dimensional Navier-Stokes equations, and show that if the $\dot B^{-1}_{\infty,\infty}$-norm of the velocity field is sufficiently small, then the solution is in fact classical.