Gradient estimation on Navier–Stokes equations
Gang Tian, Zhouping Xin · Communications in Analysis and Geometry · 1999
In this note, we present a prior uniform gradient estimates on solutions to the 3-dimensional Navier-Stokes equations.It is shown that the gradient of the velocity field is locally uniformly bounded in L^-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.