Blow-up of critical norms for the 3-D Navier-Stokes equations

Abstract

Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in $\R^3\times [0,T)$. It was proved that if $u_3\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$ for $3＜p,q＜\infty$ and $u_h\in L^{\infty}(0,T; {\rm BMO}^{-1}(\R^3))$ with $u_h(T)\in {\rm VMO}^{-1}(\R^3)$, then $u$ can be extended beyond $T$. This result generalizes the recent result proved by Gallagher et al. (2016), which requires $u\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$. Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.