Remarks on the regularity criterion to the Navier–Stokes equations via the gradient of one velocity component

In this paper, we consider the classical three-dimensional (3D) incompressible Navier–Stokes equations. By making use of a new anisotropic Sobolev inequality, we improve a regularity criterion for the weak solution to the 3D Navier–Stokes equations involving the gradient of one velocity component. This result improves the previous work of Zhou and Pokorný (2010).