Sharp One Component Regularity for Navier–Stokes

Abstract

We consider the conditional regularity of mild solution ν to the incompressible Navier–Stokes equations in three dimensions. Let e∈ S2 and 0 < T∗< ∞. Chemin and Zhang (Ann Sci Éc Norm Supér 49:131–167, 2016) proved the regularity of ν on (0, T*] if there exists p∈ (4 , 6) such that∫0T*‖v·e‖H˙12+2ppdt<∞.Chemin et al. (Arch Ration Mech Anal 224(3):871–905, 2017) extended the range of p to (4 , ∞). In this article we settle the case p∈ [2 , 4]. Our proof also works for the case p∈ (4 , ∞).