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It's a rather general equation governing fluids.
Describe the fluid by its velocity u(x,t) with x space and t time, mass ρ, then it's

ρ (∂u/∂t + (u • ∇)u) = -∇p + μ ∇2u + f
and
∂•u = 0

By analogy, if r(t) in R^3 is a trajectory with velocity v:=r', then it determines a collection of paths

ρ Acc = F
where
Acc := (∂/∂t + (v • ∇)) v
and a differences of field- and pressure forces
F = f - ∇p

In this way it's like Newton's equation, except it's not really for a single path r(t) but for the whole flow field u(x,t).

The initial conjecture was that no matter how/were you start and what the force field f and pressure p is, it would always give you bounded velocity flow, everywhere smooth.
Over the last decades, that was more and more doubted. People have conjectured, contra the NS conjecture, that you can find initial conditions and locations and force fields f in which the solution doesn't smoothly spread to all of R^3, but instead "v blows up in finite time", meaning u gets somewhere bumped to infinity by f and p (and then you can probably choose t=1 as "blow-up time"), and thus the equation likely describes unphysical scenarios (for contrieved boundary conditions).

What people did for a while, was to try and find scenarios where u goes to infinity, look at what f is needed to get this, and just check if this evil f still fits to what the Navier-Stokes official problem statements allows f to do (some mathematical regularity assumption on f, else it's too easy to blow up u).

Now OpenAI did just that, pay >8$ million in compute and let the machines tweak the scenario to indeed reject the NS conjecture, and do the work to always check the remaining f terms stay allowed w.r.t. the problem. That's the fastering but flattening vortext scneario in the pic.

Btw. the problem is still open in the harder variant where f(x,t)=0, i.e. you can't use rest terms. I.e. it's still not know if just with a force of form -∇p you maybe still get u=inf

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As for your question - well just from more fundamental physics of the last 130 years, it's clear that this classical fluid equation doesn't literally describe the physical world - just because we have more small scale quantum mechanics, and genral relativitstic invariances that it doesn't respect.
It producing infinities in principle means it's more effectively "removed from reality", which is a learning. But then again, finding a place and an f/p where it explicitly breaks with reality is done in a contreived way.
I'd say it's always good to learn that yes, it has this and that unniceness, or this failure is possible. Means there's very very likely more of those scnearios, maybe some less weird.
You could also try and mimic the scneario and force in reality and see what happens instead (u is not expected to go to infinity I assume, but then again the vortex in the scenario is also vanishing, so mabye that's legal and just difficile). I'm not an expert, obviously.

The f here is adverserialy constructed. So the result is rather negative, logically speaking, so I don't know if it will help with anything.

I think it's a learning in principle either way. You can always demand more insight and shittalk it, but in reality learnings are almost always incremental.

The equation is also not "solved" per se, it's a hard thing that has many other use equations as special cases, and many open problems, e.g. with the f(x,t)=0 situation, which as I said is still open. (And much harder since less stuff to wiggle)
One can only hope interested doesn't fade just due to this counter-example to the initial conjecture as written in the Millenium price formulation.

PS I don't post here much anymore more now. I'm traveling Japan now. Anybody in Japan?

>>17050893
Thanks!

>>17049829
>Are there direct applications for solving the problem
No, numerical methods are used for differential equations for a reason.

>>17050985
what reason?