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You don't. Unless you think about the magnetohydrodynamics of stars, or are a biophysicist, you probably will never touch those equations. Engineers tend to be more interested in them, and even then they just plug in parameters into COMSOL or some shit.
If you really want to work with them at a deep, theoretical level, you go into math. There's a reason this is a problem of mathematics, not physics. But I would highly not recommend going into math for 1 equation, especially one as notorious as NS. That said, PDEs is a fairly wide field and certainly an interesting research area, since you can trick funding agencies into thinking that your proofs of existence and uniqueness of some obscure nonlinear PDE have real world, practical value.
