Can any one explain to me (without using mathematical equations) that what is Navier-Stokes existence smoothness problem. I read a lot about Navier Stokes existence smoothness problem, but I still can not understand what the Navier-Stokes problem actually is!
Is it a kind of pure mathematical problem?
What are, its physical applications?
What really Navier stokes existence smoothness is?
See http://www.claymath.org/millennium/Navier-Stokes_Equations/navierstokes.pdf for the detailed description of the (purely) mathematical problem. You can also watch http://claymath.msri.org/navierstokes.mov if you like movies. I am not an expert, but from what I understand, physical significance of this problem is that NS-equation describes basic fluid dynamics. Since smoothness is unknown, all the current numerical approximation schemes for NS equation are highly questionable, as it is unclear what (if anything) do they actually approximate.