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

Abstract

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 ∇u3 belongs to scaling invariant space L2(0,T;Lv∞Lh2), 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).