Regularity of Solutions to the Navier-Stokes equations in $\dot{B}_{\infty,\infty}^{-1}$

TL;DR

This paper proves that suitable weak solutions to the 3D Navier-Stokes equations in the space ;_{;}^(-1) have bounded energy quantities, and axially symmetric solutions in this space are smooth, advancing understanding of solution regularity.

Contribution

It establishes boundedness of energy quantities for solutions in ;_{;}^(-1) and proves smoothness for axially symmetric solutions in this space.

Findings

Bounded energy quantities for solutions in ;_{;}^(-1).

Axially symmetric solutions in this space are smooth.

Advances regularity criteria for Navier-Stokes solutions.

Abstract

We prove that if $u$ is a suitable weak solution to the three dimensional Navier-Stokes equations from the space $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, then all scaled energy quantities of $u$ are bounded. As a consequence, it is shown that any axially symmetric suitable weak solution $u$, belonging to $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, is smooth.