An optimal regularity criterion for 3D Navier-Stokes equations involving the gradient of one velocity component

In this paper, we provide an optimal regularity criterion for 3D Navier-Stokes equations involving the gradient of one velocity component in the framework of anisotropic Lebesgue spaces. More precisely, employing the anisotropic Littlewood-Paley theory, we prove that a weak solution u is regular if ∇ u 3 belongs to scaling invariant space L 2 ( 0 , T ; L v ∞ L h 2 ) , where h and v denote the horizontal and vertical components, respectively. This result verifies the limiting case of a previous result established by Guo, Caggio and Skalák (2017). • Regularity criterion via one velocity component in anisotropic Lebesgue space. • Improvement of criterion using the anisotropic Littlewood-Paley theory. • Regularity criterion in scaling invariant Lebesgue space.