Remarks on the Possible Blow-Up Conditions via One Velocity Component for the 3D Navier–Stokes Equations

In this paper, we study some blow-up conditions via one velocity component for the 3D incompressible Navier–Stokes equations in the framework of scaling invariant anisotropic Besov spaces. In particular, we prove that if one component of the velocity remains small enough in the space \(\dot{H}^{\frac{1}{2}}\) ${\dot{H}}^{\frac{1}{2}}$ , then there is no blow-up. This result improves the previous ones by Chemin et al. (Commun Partial Differ Equ 44:1387-1405, 2019) and Houamed (J Differ Equ 275:116–138, 2021).