text: (C) Existence and smoothness of the Navier–Stokes solutions in 
  Let . For any initial condition  satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector  and a pressure  satisfying conditions 3 and 4 above.
  (D) Breakdown of the Navier–Stokes solutions in 
  There exists an initial condition  and an external force  such that there exists no solutions  and  satisfying conditions 3 and 4 above.