Gradient estimation on Navier-Stokes equations (Q1293486)

A priori uniform gradient estimates on solutions to the 3-dimensional Navier-Stokes equations are presented. It is shown that the gradient of the velocity field is locally uniformly bounded in \(L^{\infty}\)-norm, provided that either the scaled local \(L^2\)-norm of the vorticity or the scaled local total energy is small. Smooth solutions to 3-dimensional Navier-Stokes equations cannot develop a 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.