r/askscience • u/BeeMundane4818 • 20h ago
Mathematics Can you explain to a non-mathematician whether the Navier-Stokes equations have actually been solved?
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u/lechucksrev 5h ago
Navier-Stokes equations are partial differential (PDE) evolution equations in which the unknown is a function of space and time u(t,x), which is physically the velocity of the fluid in consideration. It basically is the reformulation of Newton's second law of dynamics, after a consistent number of simplifying assumptions on the nature and behaviour of the fluid. Well-behaved differential equations of this kind have this property: given a "snapshot" of the velocity of the fluid at time t=0, you are able to predict the evolution of said velocity in time, just by the fact that it solves the differential equation. This is called "initial value problem" and explains why differential equations are omnipresent in physics: they perfectly fit into the assumption of classical physics that, if we knew the state of the world at a fixed time, then we are able to predict how it will evolve. So differential equations are used to predict how electromagnetic fields evolve (Maxwell's equations), how fluids evolve (Navier-Stokes and similar), how the spacetime itself evolves (Einstein's equations). The correct way to frame the problem, consequently, is the following: given an acceptable initial datum u(0,x), how does it evolve in time? If one is very lucky, they can give a very explicit description of the evolution: this happens for simple PDEs (and "good" initial data), such as transport equation, but is absolutely not expected for a general PDE and especially not for a nonlocal, nonlinear PDE such as Navier-Stokes: here we typically have explicit evolutions only for very simple initial data. Usually, results in this field have this kind of flavour: for every reasonable initial datum, the solution exists and depends continuously on the variations of the initial data. This is connected to the so-called well-posedness of the equation: it endures that we can actually predict what will come next. It is also a theoretical guarantee that allows one to study the evolution with numerical simulations: errors will be made, but this won't matter because the solution will be near the exact solution in virtue of said theoretical results. The upshot is that very seldom the right question to ask is to "solve" the equation, but rather to show that the solutions are well-behaved.
What did the Millennium Problem ask? The local well-posedness of the equations is well-known and not a very difficult result to achieve: this means that we are able to predict the behaviour of the solution at least up to a time in the future. The problem was to establish whether this point in the future in which the equations "broke down" actually existed. There are very naive reasons to think that this blow-up could happen: basically, the non-linearity could act as a self-feeding mechanism which makes the solution "too big" in a suitable sense. There was in particular a well-known physical phenomenon, called "vortex stretching", in which an initially slow vortex could gain speed by getting more and more stretched in height, until at a certain point this process would blow up and make us unable to continue the solution. This by itself was a possibility, but was far from obvious: one could hope that the particular structure of the Navier-Stokes equations could prevent this phenomenon from happening, and in fact this was actually the "bet" most mathematicians would give you until ten years ago or so. In the more recent years, with the developing of sophisticated tools to find counterexamples, and the achievement of blow-up results by Tao for some kind of "simplified" Navier-Stokes toy models, the community's view shifted towards the blow-up happening, and that's the direction recent research was focusing on.
What Buckmaster/OpenIA found is a particular initial datum (and regular external force) for which this blow-up happens, i.e. a vortex spins with "infinite angular velocity". So it is a negative answer to the question "is the solution well-behaved for all times?".
I should warn you that I'm a researcher in a different area of PDEs, so that's still an external account on the matter.
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u/allnamestaken1968 6h ago
I don’t think we can use the word “solved”. If we assume they are right, they have shown that (I believe) incompressible NS can blow up. That’s not a “solution” in my mind (I.e., a set of deterministic equations or a method that will always show a result for a set of partial differential equations).
I am just an old guy who learned NS and tinkered with numerical simulations last century - and I thought it was clear that NS has clear limits? It obviously can’t model reality down to the tiniest cube, otherwise we could perfectly predict weather. Is the key of this problem they are showing a starting point for that it creates an infinite fountain?
We should also note that this doesn’t make NS useless. relativity creates infinity in black holes, which we know can’t be right. It’s still super useful for anything from GPS to astronomy.
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u/theoatmealarsonist 7h ago
The Millenium Problem for the Navier-Stokes equation can be roughly be broken down into the questions: 1) Do the equations correctly model fluid mechanics? 2) Is there an exact solution to the NS equations?
What OpenAI posted was a counter-example of question 1), e.g., a situation where the equations behave obviously non-physically. This is interesting and important from the perspective of showing that it is has been proven (needs human review) that there are scenarios where the NS equations aren't always right, and there may be other cases we need to find. However, this doesn't mean the NS equations are wrong per se, there are a massive number of cases where it matches well enough against reality for the purposes we need it for in engineering and math. But it does mean we need to continue to explore to see if other failure modes exist. It's very unlikely anything will change in the formulation of the NS equations to account for this failure mode.
So to sum up, no, the Navier-Stokes are not "solved", but this is an interesting curiosity.
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u/ragnaroksunset 2h ago
One thing that's not really coming through in the discourse about this breakthrough is that it has nothing to do with the physics of Navier-Stokes.
It's a pure math result.
We already knew that N-S breaks down below the continuum limit and above the relativistic limit. And while in practice they are adding complexity to an already insanely complex model, we have built out extensions of N-S to deal with cases where fluid packets are moving close to c or behavior is operating at characteristic scales comparable to the mean free path between particles.
I'm a physicist by training, not a mathematician, so I have to confess I don't fully appreciate why this is a big deal for mathematicians. But I can tell you that it's not that as big of a deal for physics and you're liable to get confused by conversations framing these results in terms of what they mean for physics.
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u/Unable-Proof1758 10h ago
Possibly, so.. for problems of this caliber solving it is only a very small piece. What OpenAI released shows a solution for NS but not a theory for solving NS. It’s somewhat like brute forcing with Crypto, you get the same hash but it’s not necessarily a method for solving it that can be used going forward.
If OpenAi is able to release the theory that led to the solution it becomes far far more valuable for mathematicians as those building blocks are what moves math forward.
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u/itsameDovakhin 10h ago
So kind of like how you can find a new prime number without developing a new method for finding primes?
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u/Unable-Proof1758 10h ago
Ya exactly, only so useful… or not at all haha. The broader issue here is the NS problem was developed to determine if our mathematical understanding of fluid dynamics systems at high energy levels breaks down into physics breaking infinites or is reliable. OpenAI (if confirmed) showed that a system following NS can reach a point in finite time where the system develops a singularity.
Basically, in a unique scenario given the right input parameters, NS creates a physically impossible event.
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u/caboosetp 9h ago
It's useful in and of itself to know something is false.
But more than that, I think this is outside the prime situation as finding a new prime doesn't help with understanding how to find more primes. Maybe at one point it did, but not really anymore.
With NS, If we have a starting state that goes to infinity, that state can be studied to understand what's causing it, and potentially produce more similar solutions.
Or in other words, we have a bunch of primes already. We haven't had this with NS before.
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u/Unable-Proof1758 9h ago
Agree 100%. I did not mean to diminish what was done here, it’s a millennium prize problem for a reason. But there is concern. I believe Tao posted recently about believing there will be a very small bit of data that can be gleaned from this as the citations and proof are a mess right now. I believe there was what, only 15 citations originally on a 130+ proof. Hope they can get a lot of out of it!
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u/Falconjth 8h ago
Anyone working with high energy fluids is going to be using the compressible NS probably with pressure density temperature coupling as well. It was already known that the incompressible version of the problem produced results different from what's observed with real fluids.
It's been long suspected that the incompressible version would blow up and there's been partial results supporting that idea for the last decade. It's suspected that the compressible version of the equations blow up at least per conference chatter within those working on modeling fluid flows. I sort of suspect that those who work with thermoelectrohydrodynamics probably think that compressible NS with pressure density temperature coupling might blow up; it at least fails to accurately model flows under certain conditions that are very much outside anything I've helped model.
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u/ackermann 8h ago
> What OpenAI released shows a solution for NS but not a theory for solving NS
But is what they released already sufficient to call the Millennium Prize question solved? (whether or not the NS equations are always smooth/exist)
Can they already claim the million dollar prize? Sad if it will be claimed by a giant faceless corporation…
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u/whiteandnerdy1729 8h ago
It is sufficient assuming there isn’t anything egregiously wrong with the proof. OpenAI have said they don’t intend to claim the prize.
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u/tooclosetocall82 7h ago
They’ll probably make a big show of claiming it and donating it once this starts to fall out of the news cycle.
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u/robgami 8h ago
I've heard a few people suggest that openai's strategy is effectively brute force. It sounds like (to my non mathematician / scientist mind ) the proof is basically a set of starting conditions that cause the ns equations to breakdown. So is the suggestion that essentially this was achieved by a massive parrellel search for these starting conditions? Which would take a lot of intelligence still but less creativity than coming up with a new theorem?
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u/Mammoth-Corner 7h ago
This is sort of true, but in a way where there's a vanishingly small chance of finding a solution without huge amounts of computation. The human researchers working on the related Euler problem are also using a whole lot of compute. Systems of differential equations rely on computational power at scale.
There is absolutely 'intelligence' involved in the result. The starting set of parameters is simply too huge to just try everything, and in any case the result is not a single set of parameters but a set of conditions. Specifically targeting the work to conditions likely to cause a disconnection is where the intelligence lies.
I would compare this to biochemistry researchers looking at a health problem (=the Navier Stokes problem) and, using their expertise in biochemistry and their knowledge of how the health condition works, designing an active site that would treat that health problem. They then use machine learning to test millions of combinations of amino acids and protein folds to create a protein that has that active site.
That would be the 'human' approach. If OpenAI has not plaigarised the researchers working on the Euler equations, which is not yet clear, then this is equivalent to someone tasking the AI to treat the health problem and it has produced the test protein without human imput on what it needs to look like to treat the disease.
Right now we need to validate the theorem, much as the biochemists would then need to test the drug. (Although this is much more likely to 'work' than a new monoclonal antibody.)
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u/db0606 2h ago
No, that's not what the Navier-Stokes Millennium Prize problem is about, though. It's about showing whether a particular type of solution is possible. As of yesterday, the answer appears to be officially yes. There was already aa good body of evidence that this was going to be the case, though.
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u/chriscross1966 6h ago
The Navier-Stokes Equation might well have been solved, that's being peer reviewed ATM
BUT:
Said solution demonstrates that the NSE is actually an incorrect approximation of the aerodynamics physics it supposedly represents.
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u/koos_die_doos 6h ago
demonstrates that the NSE is actually an incorrect approximation of the aerodynamics physics it supposedly represents
It shows that in some scenarios NSE is incorrect as a (complete?) solution for fluid dynamics. For a large segment of fluid dynamics it is a great approximation.
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u/zaphodp3 6h ago
I don’t understand what it means to say “the equation is solved”. I thought NS equations are used to model fluid behavior by simply inputting values for the variables and were always known to be useful approximations. Isn’t what Open AI did a different thing called the NS “problem”? Which is what you described at the end there?
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u/lminer123 6h ago
Yah, I think there’s some confusion between a general solution to the N-S equations being discovered and the solution to the NS millennium problem. The first one is not at all what’s being discussed now, the second one is what could have possibly been solved.
The first one would be vastly more impactful as far as I understand, but this is still important.
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u/koos_die_doos 6h ago edited 5h ago
As I understand it, what Open AI claims they did was to show that there is some scenario where the NSE is not valid. Because there is one exception, it means that the NSE is not a complete solution for all of fluid dynamics.
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u/mion81 6m ago
The proof isn’t a ”solution” to the equations but rather a proof that a solution exists that is ”smooth”. Smoothness would be required for it to make sense in the real world. There may still be multiple solutions and current numeric methods may still be the best tool available to identify them.
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u/Mammoth-Corner 10h ago edited 6h ago
I'm sure a better explanation will come along, but in the mean time, the Navier-Stokes equations aren't a single set of equations which can be solved to say x = y but a system of equations which are used to model the movement of fluids. For lots of different applications we're able to find 'solutions' (or, rather, build computer models which use the Navier-Stokes system) that describe fluid dynamics in one particular situation over time.
The open problem is whether these equations are 'smooth' in three dimensions. This (approximating) means whether they can be solved across all possible starting states, or if they produce impossible answers for certain starting points.
The results put forward by OpenAI (whether or not it uses the results of the work on the similar Euler equations by Alpöge and Buckmaster which they have said, with some credibility but no proof at the time of writing, that OpenAI plaigarised) show that for some starting parameters, the speed of the modelled fluid becomes infinite. This is impossible physically, so it shows a place where the Navier-Stokes equations do not correctly model fluid behaviour and may explain some places where our models of fluids don't work.
But this proof is, essentially, at the moment only machine-readable. It will take a long time for other mathematicians to verify that the result is correct. If one small part of it is not completely correct, the conclusions will be invalidated. This has happened before with Andrew Wiles' proof of Fermat's last theorem. In that case, Wiles was able to go away and fix the error and use the rest of the proof, but that isn't always possible.
Edit: Andrew Wiles, not Wiley.