On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$

Abstract

We show that if a Leray-Hopf solution $u$ to the 3D Navier-Stokes equation belongs to $C((0,T]; B^{-1}_{\infty,\infty})$ or its jumps in the $B^{-1}_{\infty,\infty}$-norm do not exceed a constant multiple of viscosity, then $u$ is regular on $(0,T]$. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya-Prodi-Serrin criterion.