I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to "understand" but some approaches (as published papers) in solving it seem to require high knowledge of Mathematics, e.g., advanced Topology.
By the way, I have some elementary knowledge in Maths and Physics. Here is a list of books that I have studied which may be relevant to the problem: Topology (Munkres), Introduction to ODE (Rabenstein), Real Analysis (Royden & Fitzpatrick), Abstract Algebra (Dummit & Foote) and Classical Dynamics (Thornton & Marion).
I would highly appreciate it if someone please guide me a comprehensive list of areas (e.g. Differential Topology (?)) and related levels of studies of each, needed to do a real approach on the problem of Navier–Stokes Existence and Smoothness.
Thank you.
Best Answer
You need to know advanced real analysis and also PDE and introductory fluid mechanics. Look at the references on the mentioned article and you find your pathway. Also the book G.P. Galdi - An Introduction to the Mathematical Theory of the Navier-Stokes Equations (sec.Ed) could be helpful.
"H L Royden & P Fitzpatrick - Real Analysis" for example, provides you enough knowledge to start. You need to know at least an undergraduate topology as well.