What courses should I take in order to understand Navier-Stokes equations?

Question

I've recently been interested in the Navier-Stokes equations. I would like to understand them deeply from a mathematical point of view.

What courses in advanced mathematics (and physics?), and in what order, should I take in order to understand the Navier-Stokes equations and eventually understand scientific papers on Navier-Stokes existence and smoothness problem?

I've found this Math.SE question but it asks about references in general, whereas I'm asking about specific undergraduate and graduate level courses. Let's assume for the purpose of this question that I am an undergraduate mathematics student who can focus 100% of his studies only on understanding the Navier-Stokes equations. Note that I am asking for a typical academic courses and that I'm especially interested in the sequence/order of the courses.

EDIT: As for me specifically, I've taken courses in Mathematical analysis, as well as Geometry and linear algebra so far (as well as some introductory courses in computational mathematics and computer science).

Answer 1

Okay, I'm no expert but somewhat recently I read, while an undergraduate, Leray's paper on the existence of weak solution in the whole space $\mathbb{R}^3$. So maybe I can be of some help.

1. First I would say to get a good real analysis, graduate level, course. Mostly the theory of the $L^p$ spaces, the basic inequalities in measure theory and Arzela-Ascoli is what's needed. Some knowledge of Hausdorff measure is requiered for the more sophisticated estimate on the size of the set of singular times (which is good to know before going to more modern partial regularity results).

2. A course in functional analysis is advisable since you're going to need some weak-* convergence familiarity and several arguments with weak compactness in $L^2$ and $H^1$ are used.

3. A course in PDE is obvious, the arguments are based first on a linearization of the equation which of course assumes some knowledge of the theory for these equations (for instance the heat equation). Another thing from this area that you'll need is the theory of Sobolev spaces, and the course on which this'll be seen varies from program to program (I learned the basics from a real analysis course, for instance) so you should look for a course that'll cover this.

4. The rest of the background is covered by topics courses (if at all), like "singular" integral equations and differential inequalities (think variants of Gronwall's inequalities) and you'll have to look for courses that cover them.

I'm sure, as you approach the modern results on the regularity of N-S the prerrequisites will grow, but I think these are a good place to start, since it's background for any serious study of PDE.