Smooth Solutions to the 3d Navier-Stokes Equations Exist

In 1999, J.C. Mattingly and Ya. G. Sinai used elementary methods to prove the existence and uniqueness of smooth solutions to the 2D Navier-Stokes equations with periodic boundary conditions. And they were almost successful in proving the existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations with periodic boundary conditions using the same strategy. In this paper, we modify their technique slightly to prove the existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations with periodic boundary conditions