…but to get some intuition that could provide useful heuristics – hacking lattices gets old – it helps to get the heart of the matter, so to speak. these are two people have thought deeply about matter, C. Fefferman and L Caffarelli. and here are truly beautiful paper and video by them: http://www.claymath.org/prizeproblems/navierstokes.htm
pre-reqs, after linear algebra (the functional analysis can be thought of as the grown up version of lin alg, but powerful enough to deal with applications to analysis, (infinite dimensional function spaces) etc):
the basic canonical models to understand well: wave eqn, heat eqn the trouble is, there are so many bad books, and i don't know what to recommend as a good book except maybe those above. the only i can say from bitter experience is skip the engineers books, they say a huge amount to confuse. i bet some physics books would be good. the most common ones are the landau-lifshitz series and feynman's series. (i would take the russians since they generally write better for learning.)
what we did in the cat group for simulations and games was to write a numerics class framework that could integrate de's – you specified the eqn, region and boundary condition, and the framework automatically did the lattice discretization and integration. (like josh's jit ) this is not easy – lots of numerical instabilites. of course blow-ups are fun, but not all blow-ups are usable – the study of singularities has a language which is quite useful – that's where (geometric) measure theory comes into its own.
Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System > http://arxiv.org/abs/math/0609740