A bounded numerical solution with a small mesh size implies existence of   a smooth solution to the Navier-Stokes equations
  We prove that for a given smooth initial value, if the finite element solution of the three-dimensional Navier-Stokes equations is bounded in a certain norm with a relatively small mesh size, then the solution of the Navier-Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical solution. 
Buyang Li
math.AP cs.NA math.NA