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- Hacker News
- Interesting to see that they are not using the coordinate-free representation (exterior calculus, differential forms) that mathematical physicists prefer to use today.by amelius
- Those notations are used when writing down the models, because they make clear the intrinsic geometry, the basic symmetries, etc. But they're not used so much in the study of solutions to the equations. Solutions tend to have peculiar features, tend to break underlying symmetries, etc. and there only needs to be one nasty particular solution to prove the NS conjecture wrong.by auntienomen
- It’s pretty crazy what dynamics you get from NS.. until one realise they emerge from a tiny part of the solution space of Einstein equations.. which themselves emerge at the low energy limit of sth much bigger.by immmmmm
- Could you expand? Curious.by cacio-e-pepe
- Can you actually credibly find NS or even Euler’s equations as an effective theory from GR?
Euler’s equations and NS have this pesky velocity field, which requires the fluid’s state to be well described by a velocity at each point in space (and a density and a pressure, but I think GR has no problem with those). This means that you need some kind of interaction between particles to get them to exchange energy so that they thermalize instead of staying in the collisionless regime. (In other words, if you have two blobs of fluid collide, you need them to not go right through each other.) And I don’t think that GR is dissipating on the relevant scales.
As a real-world example, the universe contains neat structures (the horsehead nebula is a somewhat famous example) that are consistent with dark matter distributions that don’t really resemble fluids.
by amluto - I don’t think this problem is very relevant. The Millennium Problem only asks about the smoothness of solutions to the incompressible Navier-Stokes equations. Real fluids are compressible and also conduct heat. In other words, if you solve the problem, you only learn how fluids behave under unrealistic assumptions. Physically speaking, one learns nothing from such mathematical exercises. Furthermore, singularities in the velocity field essentially imply velocities greater than the speed of light. That, too, is physically nonsensical, as we know from the theory of relativity. In other words, the solution only tells us that the incompressible Navier-Stokes equations are not realistic. But physicists have known that for a long time.by amai
- In classical Newtonian physics, singularities exist, but only in frictionless systems. One example is the driven oscillator. Another more complex one the https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture . However, as soon as friction is taken into account, the singularities (e.g., infinitely high amplitudes in the oscillator) are damped.
Fluids without internal friction or viscosity are described by the Euler equations. Therefore, singularities (finite time blowups) occur there, which has recently been shown with a computer assisted proof:
https://www.quantamagazine.org/computer-helps-prove-long-sou...
The Navier-Stokes equations are the Euler equations plus friction/viscosity: As a physicist, I do not expect singularities there, since energy is always lost due to friction.
by amai - This is essentially irrelevant to the content of the post, but it's amusing to me that he casually mentions submitting to JAMS as if its acceptance were a mere formality.
- It basically is a formality at this level. Many top math researchers now hardly even submit to journals at all and just put up a preprint.
At this scale, peer review happens by the audience. They don't need a journal to get people reviewing their work.
by hodgehog11 - He has rejected papers too https://mathstodon.xyz/@tao/113721192051328193by gus_massa
- See https://www.quantamagazine.org/theory-of-fluids-enters-the-2... if you want to know what the Navier Stokes equations are about.by MeteorMarc
- @dang please can we add a [2014] to the title ?by goldenarm
- Yes please. For a moment I thought Terrence Tao had scooped Anthropic.by thomasahle
- This is going around due to rumors and baseless speculation on Twitter [1] right now that Anthropic has solved the Millennium problem related to Navier-Stokes [2]
[1] https://x.com/AndrewCurran_/status/2096062392442724805 for example
[2] https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...
by v64 - if there's anything that would convince that LLMS are one of the biggest innovations ever, it would be this :-D
- If it is solved what are the applications of that? What changes?by bilsbie
- For a second I thought they were aiming the scary proof machine at us mortals doing PDE stuff. Fortunately the speculation is just that they happen to be aiming it at a nearby mathematician type problem. Phew.by bee_rider
- Why would they send it out for "expert review"? Every time, they have just made the AI generate a Lean proof. In fact, it seems like the most plausible direction to NS is computationally assisted detection of a blowup solution, which has fantastic automatic validation.by hodgehog11
- Lol as far as I know that post was the origin of that claim and it's clearly just a guy predicting something that will happen in the future with no information about it.by CSMastermind
- This is a step beyond baseless predictions. Tao also had a "weird" "hypothetical" comment about LLMs solving complex proofs with impossible to human verify Lean.
- by cacio-e-pepe