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?