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

Let u = (u _h, u ₃) be a smooth solution of the 3-D Navier-Stokes equations in ?³ × 0, T). It was proved that if u ₃ ∈ L ^∞(0, T; ? _p,q ^?1+3/p (?³)) for 3 < p,q < ∞ and u _h ∈ L ^∞(0, T; BMO^?1(?³)) with u _h(T) ∈ VMO^?1(?³), then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al. (2016), which requires u ∈ L ^∞(0, T; ? _p,q ^?1+3/p (?₃)). Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.