[btrettel]

Fluid dynamicist here. At the risk of looking like a heretic, I am willing to say that I don't think the NS Millennium Prize problem and related things will end up being very useful practically.

I can't see how a definitive answer to the question will result in better turbulence modeling, which is what matters from a practical point of view. If it turns out that the solutions are not unique then we could probably find an additional condition to add (e.g., the entropy condition) to make the solutions unique. If the solutions are unique, bounded, etc. then that's great and it would have no impact practically speaking aside from perhaps helping the reputation NS has for accuracy. Some people seem to think that solving the NS Millennium Prize problem would likely lead to a solution for the turbulence problem, but as I said, I can't see how. I'd be interested if anyone could explain this belief better.

There may be other benefits. I've found papers that find bounds on different fluid dynamics quantities to be interesting, and the motivation for these studies are the NS problem from what I understand. Unfortunately the results from these papers tend to be less useful than bounds I can derive specifically for applications myself.

(In a nutshell the turbulence problem is that NS has far too high a computational cost/complexity to be used in practical simulations. So cheaper approximations to NS are used, which you can cladsify as "turbulence models". How steep the drop-off in accuracy is as you reduce complexity is an open question. My opinion is that fluids probably require high computational cost for accuracy a-priori. Things like correlations from experiments can get around this as you are using pre-computed results, and that may be what we should go for in my philosophy.)