For understanding the Navier-Stokes equations, are there any references which may include one or more of the followings:
- mathematical rigorousness
As this is one of the Clay Institute of Mathematics Millennium Problems, I refer you to their webpage on the problem and to their write-up for the prize. It contains several references about the problem and some of the results on the problem. Their webpage also has a video lecture by Luis Caffarelli on the problem, which I think is really cool.