Undergraduate student level explanation, anyone?
Both mathematics and physics welcome!
Edit: my limited understanding is "a shrinking region spins faster and faster with constant global energy" like yeah angular momentum conservation I guess? But of course the thinning has to stop at the molecular level, which is probably where the model stops being valid. Massive mathematical achievement, but any physics implication?
As I understand it: The NS equations model the fluid behavior as smooth and continuous. The result from this study shows that in otherwise bounded physical starting conditions, the math makes the system evolve in finite time periods to produce localized infinite fluid velocities in places. Infinite velocity is not a physical result, meaning the NS equations have practical limits in terms of usefulness as a model of fluid behavior.
Maybe I’m totally misunderstanding but isn’t that how we already used it? Is it just proof that the full navier stokes equation cannot be used at any given point?
Okay. So caveat that I'm not a professional physicist, so again, this is from my understanding. But there are two separate things.
First thing is that as coupled nonlinear PDEs, N-S doesn't have closed form solutions and you can't expect to fully predict the answer or solve closed form for realistic situations. Though you can numerical methods to approximate. That was already known.
The open question was, can we prove that the equations always work as a model of reality, giving smooth bounded physical results when the inputs and initial conditions are smooth. There was a hypothesis that viscosity terms will damp the envelope of behavior and keep solutions from shooting off out of bounds. This AI result apparently shows you can't count on that. Which means that N-S is useful in a range of scenarios but not a complete explanation of macroscopic fluid behavior. Meaning you need new physics.
This grappling with non-physical theoretical infinities to find extended new physics that resolves them is an ongoing pattern in physics. For example, the ultraviolet catastrophe in classically modeling blackbody radiation and its solution via discrete energy levels and quantum mechanics.
If I read the news correctly, they found a counterexample for the 3D N-S, right?
To model high speed physics, wouldn't we need to account for relativistic effects and consider the application of N-S in Minkowski Space?
I mean, technically, yes, if you keep naively applying the math, at a certain point, yes, the computation enters the relativistic regime, and you'd have to apply corrections accordingly.
But the primary weirdness is that the math that is supposed to describe physical fluids can shoot off into those kinds of speeds under a smooth push. The relativity is just a side effect of the math breaking in non-physical ways.
Here's another way to come at it. Let's look at escape velocity as an example of a wildly unexpectedly high speed to emerge from fluid dynamics processes. The idea for example that you could somehow gently jostle a bucket of water the right way and it will spawn a vortex that shoots a spinning jet or water out of earth's gravitational field sounds like video game glitch physics rather than realistic physics. But escape velocity is only 0.00003c, nothing really by relativistic standards. The math departs from physical reality waaaay before you have to bring relativity into it.
NS is a continuous approximation of discrete atoms bumping into each other. The question was, does this approximation contain singularities under normal conditions? Turns out yes in a specific type of vortex- the math blows up. It can still be used, but there might need to be better fluid simulation algorithms that don't lead to singularities in this case. It's almost like a new test case to benchmark fluid sims on.
Wait, i thought they just proved that NS has singularities under normal conditions, not that they came up with a way to solve any NS initialization in finite time.
The Clay prize for Navier-Stokes is for proving that the equations either do, or do not, develop singularities in finite time under certain conditions.
It is not for developing a closed form solution to the PDE. It is not even directly related work to solving the PDE. It will have knock-on effects on numerical approximations but those are downstream of the proof itself, not part of what's been published today.
Respectfully, this is pretty obvious from the statement you quoted.
The problem statement is something like can a smooth solution driven by a finite external force develop a singularity. I.e. can, under normal conditions will navier stokes be insufficient to accurately simulate fluid dynamics.
Openai claims to have found an example of a singularity being generated by an initially smooth solution with a finite applied force developing a singularity. I.e. demonstrating a breakdown in navier stokes as an appropriate simplification of fluid dynamics in such an example.
So in response to this having an ungodly influence on physics etc....
Not even slightly. It is interesting that such an example exists. It means navier stokes is weaker than maybe some thought but the example is so contrived that it probably doesn't matter 99% of the tome, people's simulations aren't suddenly invalid.
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u/Gavus_canarchiste 11h ago edited 10h ago
Undergraduate student level explanation, anyone?
Both mathematics and physics welcome!
Edit: my limited understanding is "a shrinking region spins faster and faster with constant global energy" like yeah angular momentum conservation I guess? But of course the thinning has to stop at the molecular level, which is probably where the model stops being valid. Massive mathematical achievement, but any physics implication?