The current state of Navier-Stokes existence and smoothness

[AVFistula]

Can anyone point me to some publications or archives which feature developments in solving the N-S existence and smoothness problem? Basically, I'd like to read up about how far people have gone towards solving the problem, e.g. a new method to analyze the equations.

Also, what fields of mathematics and physics do the Navier-Stokes equations employ? I've taken differential equations (the class had barely any content on partial diffeq) and just physics I+II. Can I even begin to understand the equations in-depth without any other math or fluid mechanics?

 

[The_Duck]

I don't know anything about status of the millenium problem. But the Navier-Stokes equations are just a partial differential equation. If and when you have a good handle on PDE's the Navier-Stokes equations are not too hard to understand. The physics they encode is quite simple--it's just the equivalent of F = ma except it describes a continuous medium instead of a point particle.

 

[torquil]

Also, what fields of mathematics and physics do the Navier-Stokes equations employ?

Regarding the math, I would guess that the theory of Sobolev spaces would be very relevant.

 

[AVFistula]

Okay, thanks for the heads up. I've since read a lot of explanations and derivations of the N-S equations, so I think the math isn't too bad. I'll read up about Sobolev spaces, though.

If anyone knows where I can read general science/math publications so that I can search for developments in this particular subject, I am still looking and would greatly appreciate a heads-up.