The only problem I encountered with Navier-Stokes equations, while studying them at university, was the fact that you could plug the same problem, with the same mesh size into three different CFD packages and get three different answers.
As far as I recall the equations have no cast in stone solution, but a series of opposing variables which are balanced via successive iterations, until they seem to work.
We then had several discussions as to whether they can be used to describe or approximate the lift/flow theories, and the debates ended pretty much the same as this forum. A Newtonian camp, Bernoulli camp, a few abstentions, and a couple of perplexed by-standers.
The only thing we agreed on is that this has been debated for 100 years, and will probably still be debated on in 100 years!