# Finite difference method proved to be convergent for the Navier–Stokes equations and the equations are numerically solved by the 1960s. It is proved that there are smooth and globally defined solutions to the Navier–Stokes equations in 2 dimensions. 
# If the initial velocity  is sufficiently small then the statement is true: there are smooth and globally defined solutions to the Navier–Stokes equations.
# Given an initial velocity  there exists a finite time T, depending on   such that the Navier–Stokes equations on  have smooth solutions  and . It is not known if the solutions exist beyond that "blowup time" T. 
#Jean Leray in 1934 proved the existence of so-called weak solutions to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise.
#Terence Tao in 2016 published a finite time blowup result for an averaged version of the 3-dimensional Navier–Stokes equation.  He writes that the result formalizes a "supercriticality barrier" for the global regularity problem for the true Navier–Stokes equations, and claims that the method of proof hints at a possible route to establishing blowup for the true equations.