A localized criterion for the regularity of solutions to Navier-Stokes equations

The Ladyzhenskaya-Prodi-Serrin type $L^{s,r}$ criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global $L^{s,r}$ norm is usually large and hence hard to control. Replacing the global $L^{s,r}$ norm with some kind of local norm is interesting. In this article, we introduce a local $L^{s,r}$ space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local $L^{s,r}$ type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local $L^{s,r}$ type norms is necessary and sufficient to affirmatively answer the millennium problem.