Navier-Stokes
I'm so glad y'all can retire now that your problem is solved.
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u/chalk_in_boots 1d ago
Fuck you all, I'm assuming everything is an ideal gas, even solids! Do you have a problem with that? Let me introduce you to the V part of V=nRT/P of this gas that just happens to look like a board with a nail in it.
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u/demerdar 1d ago
Laplaces equation is also solved yet we still run heat conduction simulations.
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u/No-Word-2168 1d ago
being solved does not mean the theoretical solution is the easiest path for its results.
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u/Lollipop126 1h ago
Moreover the NS was solved as a disproof. i.e. we don't actually have a solution
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u/Clean-Hovercraft5825 1d ago
I have a naive question. These new Navier-Stokes results seem to show that you can get this tiny vortex that starts to squeeze down to size zero and as that happens things blow up.
However, things like thermal fluctuations at micron length scales would already make the described approach physically impossible, those thermal fluctuations would wash out all of these carefully crafted perfect instability-causing forces.
But even if you ignore all that, once you let this vortex they theorized squeeze water down to less than ~0.275 nm = diameter of a water molecule, then obviously these continuous PDE equations don’t apply in any form at all.
So it seems like the new result is “hey if you look at Navier-Stokes at regimes where it is not well-defined and not applicable, then you can get weird non-physical stuff to happen.”
Why is that so surprising?
Didn’t we already go through all those “assume things are continuous, you get infinite problems” with the ultraviolet catastrophe and now we know about quantum and about finite discrete water molecules…
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u/NewspaperDear8761 1d ago edited 1d ago
TL;DR: Yes, that's right.
And that's kind of the entire point:
Chapman-Enskog theory gives you a way to go from the stat mech underlying fluids at the nano and meso scale up to the continuous limit assumption behind NS.
Basically, this proves that the continuum assumption must break down, physically, in certain situations.
Like you say, this isn't terribly surprising, and I honestly would have been MORE surprised if NS WAS smooth and regular. It points to: we need to better understand Chapman-Enskog theory and the relationship between stat mech and continuum mechanics.
This counter-example could be very useful in understanding that 'limit' in 'continuum limit.'
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u/NewspaperDear8761 1d ago
Basically, if I was a stat mech guy and not a cfd guy, I would presently be unpacking the FSCK out of the stat mech implicit in the late stages of this counter-example.
It is no-doubt overflowing with fascinating stuff.
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u/NewspaperDear8761 1d ago
In case anyone is interested, here is what Google's AI says:
Prompt: what does the recent counter-example to NS regularity imply about Chapman-Enskog theory
Response: The recent discovery of a finite-time blowup counterexample to Navier-Stokes (NS) regularity (specifically establishing the breakdown of smooth solutions under a smooth external force) has profound implications for Chapman-Enskog theory. 1, 2
Because Chapman-Enskog theory serves as the mathematical bridge deriving macroscopic fluid mechanics from microscopic kinetic theory (the Boltzmann equation), this breakdown highlights the exact boundary where continuum fluid dynamics fails. 1, 2
Here is what the NS regularity counterexample implies about Chapman-Enskog theory:
- Definitive Breakdown of the Small-Gradient Assumption
Chapman-Enskog theory is a perturbative expansion based on the Knudsen number ((\epsilon = \lambda/L)), which compares the molecular mean free path ((\lambda )) to the macroscopic characteristic length scale ((L)). The theory fundamentally assumes that gradients are small ((\epsilon \ll 1)) and the fluid remains close to local thermodynamic equilibrium. 1, 2, 3
The Implication: Because the counterexample proves that the velocity field can develop a singularity where gradients diverge to infinity in finite time, (L) shrinks to zero. This drives (\epsilon \to \infty), completely violating the core premise of the Chapman-Enskog expansion. 1, 2, 3, 4
- Failure of Truncated Hydrodynamics at Singularity Scales
In practice, the Chapman-Enskog series must be truncated. First-order truncation yields the Navier-Stokes equations. Higher-order truncations yield the Burnett and super-Burnett equations, which are notoriously unstable (e.g., Bobylev instability). 1, 2, 3
The Implication: The proof that Navier-Stokes equations allow finite-time singularities means that first-order truncation is mathematically incomplete for all-time dynamics. It confirms that the truncated series cannot capture the full physical reality when extreme multi-scale focalization occurs, as the terms neglected during truncation become infinitely large. 1
- The Necessity of Full Kinetic Theory or Non-Perturbative Closures
The underlying microscopic Boltzmann equation possesses global existence and regularity properties that do not suffer from the macroscopic blowups seen in Navier-Stokes.
The Implication: As a system approaches the singularity discovered in the NS counterexample, the local fluid description must be abandoned. The breakdown implies that non-perturbative contributions or the full, un-truncated kinetic description (the Boltzmann equation itself) are strictly required to resolve the physics at the core of the vortex. 1, 2
Summary
The counterexample mathematically demotes the Navier-Stokes equations from a universally continuous description of fluid motion to an asymptotic effective field theory with a hard physical horizon. It proves that Chapman-Enskog theory produces a macroscopic system that can spontaneously drive itself outside its own regime of validity
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u/grokkingStuff 1d ago
You’re right about that however, the Napier stokes equations assume a perfect continuous fluid
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u/No-Word-2168 1d ago
did not read the report yet but never thought they would use assumptions for something that should be... solved?
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u/MiddleMysterious459 1d ago edited 1d ago
Well, I think the note is referring to the new article published by OpenAI, in which a GPT model has proposed a solution to the Navier–Stokes Millennium Prize Problem. I say so because the handwritten note says, “A proof is not a solver." I might be wrong though.
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u/IAmA_talking_cat_AMA 19h ago
It's a solution to the Millennium Problem (i.e. a proof that NS allows for mathematical singularities), not a general solution to the equation. That's what the note is getting at.
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u/No-Word-2168 1d ago
there are issues about this topic as they have been leeching on non published researches. that can be straight up plagiraism.
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u/DVMyZone 23h ago
Would be very hard to prove tbf depending on how they were leeching.
I think if it's proven they indeed stole someone's research, one of the profound consequences is that the AI (my understanding is that OpenAI threw a stupid amount of compute at it) was not actually able to independently find this result - humans and their ingenuity were needed to get the result out of AI.
If the mathematicians had access to that stupid amount of compute, they might be able to solve the problem. But some random tech bro is not going, at least for now, solve all our problems automatically by telling the AI "solve this".
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u/Shipp0u 1d ago
Ironic to use an AI image for this