Gradient estimation on Navier-Stokes equations

In this note, we present a prior uniform gradient estimates on solutions to the S-dimensional Navier-Stokes equations. It is shown that the gradient of the velocity field is locally uniformly bounded in L-infinity-norm provided that either the scaled local L-2-norm of the vorticity or the scaled local total energy is small. In particular, our results imply that the smooth solutions to 3-dimensional Navier-Stokes equations cannot develop finite time singularity and suitable weak solutions are in fact regular if either the scaled local L-2-norm of the vorticity or the scaled local energy is small.