Sharp One Component Regularity for Navier–Stokes

We consider the conditional regularity of mild solution ({nu}) to the incompressible Navier–Stokes equations three dimensions. Let ({e in mathbb{S}^{2}}) and ({0 u003c {T}^{*} u003c infty}). Chemin and Zhang (Ann Sci Ec Norm Super 49:131–167, 2016) proved the regularity of ({nu}) on (0, T*] if there exists ({p in (4, 6)}) such that $$int_{0}^{T^ast}|vcdot e|^p_{dot{H}^{frac{1}{2}+frac{2}{p}}} {rm d}t u003c infty. $$Chemin et al. (Arch Ration Mech Anal 224(3):871–905, 2017) extended the range of p to ({(4,infty)}). In this article we settle the case ({p in [2, 4]}). Our proof also works for the case ({p in (4,infty)}).