That's awesome and clicking through the links I learned that the proof for Navier-Stokes has a million dollar prize on it. I didn't even know that it's not proven, back in engineering school we basically took that as a given and starting point for so much useful stuff. I guess that's the difference between science and engineering.
The PDE that most analyses start with does not need to be proven. This has been rigorously derived since forever. The thing that actually needs to be proven is whether or not smooth solutions exist to the PDE for all cases of parameters. In practice, people tend to stick to working with simple cases that can be solved analytically, or use computer simulations to compute approximate solutions (to very high precision).
To add on that, for all cases really means a lot more than bends in pipe or air off a wing or smoke which we have difficulty with today. You'll need to hit everything, plasma on the Sun, exotic matter, it leads to quasiparticles for the many-body problem.
My interest in this subject is I hope new developments lead to cardiac therapies. I send some buzzing acoustic signal with a pacemaker, I can treat blockages in the brain or tumors in hard to reach organs. A lot better to me than building faster missiles to blow more people up.
Physicists normally care little whether their mathematical tools are formally proven correct. See: the Feynman path integral. It makes sense and gives correct answers, but as far as I know, it has no rigorous mathematical basis.
Indeed, natural science deals in models, not 'the truth'.
In most cases 'correct' isn't an option, rather a degree of accuracy. If it's consistent with the measurements, or even better if it has predictive power, then it's useful.
I wouldn't say they don't deal in truth. Scientific theories clearly capture a lot of true elements of how the universe works. But proving that the mathematical methods are correct, starting from basic axioms, is something for mathematicians. Most physicists will use a mathematical tool if it's useful, even if it's not been formally proven correct.
