The space $B^{-1}_{\infty, \infty}$, volumetric sparseness, and 3D NSE

TL;DR

This paper demonstrates that small solutions in the Besov space $B^{-1}_{ infty, infty}$ can prevent blow-up in 3D Navier-Stokes equations, advancing understanding of solution regularity.

Contribution

It establishes a new regularity criterion based on volumetric sparseness in the Besov space $B^{-1}_{ infty, infty}$ for the 3D Navier-Stokes equations.

Findings

Smallness in $B^{-1}_{ infty, infty}$ prevents blow-up

Provides a new regularity criterion for 3D NSE

Enhances understanding of solution behavior in critical spaces

Abstract

In the context of the $L^\infty$-theory of the 3D NSE, it is shown that suitable smallness of a solution in Besov space $B^{-1}_{\infty, \infty}$ suffices to prevent a possible blow-up.