"Navier–Stokes Equation: This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding."
There are already a large number of solutions to Navier-Stokes that are known. The millennium problem asks to prove existence or non-existence of smooth (infinitely differentiable) solutions that are valid for all time See pg 2, here: http://www.claymath.org/sites/default/files/navierstokes.pdf
http://www.claymath.org/millennium-problems
Relevant summary:
"Navier–Stokes Equation: This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding."