  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.
