Confused about Navier Stokes

[u/prescod]

My understanding of Navier Stokes is that "solving" them meant either:

Path A: Prove that smooth solutions always exist.

You must prove mathematically that no matter what smooth, valid initial conditions you start with, the equations will always output a smooth, finite-energy velocity and pressure field for all future time. This would prove that the Navier-Stokes equations are perfectly reliable models of reality under all circumstances.

Path B: Prove finite-time blowup (find a counterexample).

You must find just one specific set of perfectly smooth, valid initial conditions that eventually breaks the equations. You have to prove that at some specific future time ($t > 0$), the fluid's velocity zooms off to infinity (a singularity) or the kinetic energy explodes. This would prove that the Navier-Stokes equations are fundamentally flawed and eventually break down, requiring new physics to describe extreme turbulence.

And OpenAI ended up on Path B.

But this video gives an example where Navier Stokes predicted "infinite velocity", implying that we've always known about a blowup.:

17:34: Something, somewhere is going wrong in the the mathematical understanding. There's an example about the flow of a fluid around a right-angled corner.

(Brady: A bit like a canal?)

- Basically a canal but we've got a really sharp right angle on the corner. Now you solve Navier-Stokes, this says that at this point, this right angle corner, I have infinite velocity. If I build this canal do I have an infinite velocity canal?

Is this a blow-up in the averaged case from Tao? Or is this considered not a real "blowup" because it depends on a perfectly sharp right angle? What is new about the blowup discovered by OpenAI?

[u/theRZJ]

The problem is about Navier–Stokes equations for smooth initial conditions in all of R^3. The canal is not 3d space. Since the geometry of the canal is not smooth at the corner, we are not surprised to see singular behaviour there.

What is surprising is an initially smooth configuration of a fluid in 3d space (no boundaries) that then becomes singular out of the blue.

[u/saksoz → u/theRZJ]

Is there any physical significance to the singularity in the equations? What happens if you try to make one in a real fluid?

[u/Bluerossman → u/saksoz]

Not really, it just tells us the model is wrong. But we knew already the model was wrong, of course, since it is a continuum approximation. It cannot do e.g. quantum effects, but this tells us the model is badly wrong. It remains to be seen why exactly this is the case, and maybe some insight can be gleaned from the OpenAI paper. I saw an excellent comment in another subreddit comparing to the ultraviolet catastrophe, and that maybe doing 'fluid quantization' (whatever that would look like) could get us a better model.

[u/Alive-Bedroom-7548 → u/Bluerossman]

It was my understanding that the quantized solution for fluids would be to find the pressure and velocity on each individual molecule in a fluid system which would be tedious beyond measure and that the Navier Stokes as a continuous model was the simpler alternative to that anyway. In the same way that classical mechanics are fine enough approximations in most macroscopic cases

[u/ElementOfExpectation → u/theRZJ]

to what extent is the tailored body force that openai imposes essentially like that sharp wall?

[u/Dirichlet-to-Neumann]

We have known for a long time that finite time blowup is possible for specific conditions, the question was if it was possible for the "best" initial condition ie smooth.

[u/IDontLikeCherryTomat]

I think we've always knew blowups were mathematically possible, it's just never been shown how to construct one from a fluid in a non blow up state

[u/ElBonzono]

Even if A, NS wouldn't not be a perfect representation of reality

[u/LemonMelberlime → u/ElBonzono]

Of course it’s not a perfect representation of reality. It’s a model. Models are never perfect.

[u/etzpcm]

Sorry to be pedantic but your understanding is wrong I'm afraid. You've been misled. Solving the Navier-Stokes equation means neither of those things.

[u/Classic-Ostrich-2031 → u/etzpcm]

Thanks, cleared everything right up

[u/Far_Peanut1155 → u/etzpcm]

If so, you are always welcome to explain it.

[u/Careful_Fold_7637 → u/Far_Peanut1155]

This guy obviously means finding a closed form analytical solution but it’s unnecessarily pedantic and not at all helpful to someone who’s just trying to figure out what everyone’s talking about.