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  • 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
  • If 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/ Source code: https://github.com/minfx-ai/navier-stokes-blowup
    by michalsustr
  • 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.
    by tacomonstrous
  • 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
  • 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

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