That is fascinating. Is there any sort of immediate real-world impact (like to weather forecasting)?
It's possible that this can explain some of the differences between models or ensemble runs... but you have to realize that most of the error comes from incomplete data in the initial and boundary conditions. Looking for weather model effects is like looking for relativistic effects in automobiles.
Edit: typos
> I realize that some of us have worked a lot more on numerical methods, but other hackers never made it past Calculus 2.
While that didn't occur to me, I do appreciate trying to keep things accessible.
The right way to do this is through uncertainty quantification techniques, and I don't know a lot about those at the moment. Until then, all I can say is that there are multiple sources of error.
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.)
I’m an applied mathematician and a macroeconomist that studied turbulence in financial markets and crashes thereof. I can assure you that the dynamics are pretty distinct. In economics wealth is not a conserved quantity whereas in physics energy and momentum are.
That's what makes it unsuitable for being a currency.
> a uniquely closed & conserved system
Nope. Bitcoin may or may not be (lost wallets, as you point out, is one way in which it is not). But the ‘system’ is the economy, because money is moving in and out of bitcoin because it can be exchanged for other assets (goods, services, or other currencies when doing conversions).
So no. I'm not trying to be condescending, I'm just trying to nip this apparently valid but flawed analogy in the bud. The only commonality is the word ‘turbulence’ which is being used as a label for two entirely different phenomena that have some similitude and points of contact but are largely distinct and unrelatable.