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Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014)
- goldenarm 1mo ago@dang please can we add a [2014] to the title ?
- thomasahle 1mo agoYes please. For a moment I thought Terrence Tao had scooped Anthropic.
- tacomonstrous 1mo agoThis 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.
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- gus_massa 1mo agoHe has rejected papers too https://mathstodon.xyz/@tao/113721192051328193 https://mathstodon.xyz/@tao/113721192051328193
- Sharlin 1mo ago> With hindsight, some of my past rejections have become amusing. With a coauthor, I once almost solved a conjecture, establishing the result with an "epsilon loss" in a key parameter. We submitted to a highly reputable journal, but it was rejected on the grounds that it did not resolve the full conjecture. So we submitted elsewhere, and the paper was accepted. > The following year, we managed to finally prove the full conjecture without the epsilon loss, and decided to try submitting to the highly reputable journal again. This time, the paper was rejected for only being an epsilon improvement over the previous literature!
- hodgehog11 1mo agoIt 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.
- tacomonstrous 1mo agoNone of this is true.
- hodgehog11 1mo agoUh, care to explain? I have several colleagues that stopped submitting to journals once they reached full professor. They only submit papers from their students for the benefit of their careers. First-author papers, not so much.
- tacomonstrous 1mo agoThe fact that you talk about 'first author' papers in math only reinforces my point. It's not the same culture as other STEM fields.
- hodgehog11 1mo agoTouche, I used to work in pure probability where that ridiculous Hardy-Littlewood rule used to cause all sorts of problems, but now work in statistics, where it is no longer an issue. To be clear, the colleagues I am referring to mostly work in math phys. They are sole author papers, but I refer to them as first author, since that is the language I am now accustomed to. And no, the culture in maths vs. theoretical stats is not really that different at the end of the day, and the latter is most assuredly like other STEM fields. I still collaborate on pure math papers (geometric analysis and PDEs mostly) from time to time, and it isn't really a different head space. My name is typically later in the alphabet, so I never even think about the alphabetical ordering.
- v64 1mo agoThis 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 https://x.com/AndrewCurran_/status/2096062392442724805 for example [2] https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...
- CSMastermind 1mo agoLol 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.
- krackers 1mo agoElliot Glazer (FrontierMath lead) traces how it snowballed over time https://x.com/ElliotGlazer/status/2096298696438906934 https://x.com/ElliotGlazer/status/2096298696438906934
- Arodex 1mo agoNot reading any x.com content until xcancel and nitter are back.
- sawjet 1mo agoDear diary...
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- perching_aix 1mo agohere you go: https://nitter.kareem.one https://nitter.kareem.one
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- amelius 1mo agoInteresting to see that they are not using the coordinate-free representation (exterior calculus, differential forms) that mathematical physicists prefer to use today.
- auntienomen 1mo agoThose 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.
- immmmmm 1mo agoIt’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.
- cacio-e-pepe 1mo agoCould you expand? Curious.
- immmmmm 1mo agosorry for the late reply, see answer above
- amluto 1mo agoCan 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.
- immmmmm 1mo agosorry for the late answer.. yes you can : https://arxiv.org/abs/1211.1983 https://arxiv.org/abs/1211.1983 but you might need to understand it in the context of holography in some frame you can recover non-relativistic symmetries, we got a paper on this back then, but so hardcore i only understand parts of it https://arxiv.org/abs/1205.5777 https://arxiv.org/abs/1205.5777
- immmmmm 1mo ago
- MeteorMarc 1mo agoSee https://www.quantamagazine.org/theory-of-fluids-enters-the-21st-century-20260817/ https://www.quantamagazine.org/theory-of-fluids-enters-the-2... if you want to know what the Navier Stokes equations are about.
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- amai 1mo agoI 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.
- amai 1mo agoIn 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 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-sought-fluid-equation-singularity-20221116/ 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.
- michalsustr 1mo agoIf you'd like to explore what the Navier–Stokes blow-up construction looks like visually, I vibe-coded an interactive 3D visualization based on the published result for fun :) Demo: https://minfx.ai/navier-stokes/ https://minfx.ai/navier-stokes/ Source code: https://github.com/minfx-ai/navier-stokes-blowup https://github.com/minfx-ai/navier-stokes-blowup