Trapping Regions for the Navier-Stokes Equations
  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 using the same strategy. In this paper, we modify their technique to obtain a simpler proof of one of their results. We also argue that there is no logical reason why the 3D Navier-Stokes equations must always have solutions, even when the initial velocity vector field is smooth; if they do always have solutions, it is due to probability and not logic. 
Craig Alan Feinstein