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

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.