Question

Why the Navier-Stokes equations are so difficult to solve?

Answer 1

yeah they are impossible to solve analytically sometimes...specially when we work on cylinderical and and spherial coordinates

Answer 2

unless we reduce the variables to one or 2 sometimes with some assumptions...

Answer 3

The Navier-Stokes equations describe the behaviour of fluids, which is, even iteratively, difficult to solve. Now, to give a closed-form solution (if at all possible), requires a ridiculous amount of general work. But it's not difficult to see why they are so hard to solve by observing the behaviour of water with any action upon it.