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

The Serrin-Prodi-Ladyzhenskaya type $L^{p,q}$ 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. In this article, we establish some localized $L^{p,q}$ 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 $L^{p,q}$ type norms. These local $L^{p,q}$ type 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 $L^{p,q}$ type norms is necessary and sufficient to affirmatively answer the millennium problem. Our work provides an interesting and plausible approach to study the millennium problem.