# The Navier–Stokes result does not solve the million-dollar problem

Markdown copy of https://realalpha.app/s/the-navier-stokes-result-does-not-solve-the-million-dollar/ on Real Alpha, the best minutes from long interviews with the people building AI. Cite and link the page, not this copy.

- Speaker: [Max Tegmark](https://realalpha.app/people/max-tegmark/) · Max Tegmark is an MIT physics professor, cosmologist and author.
- Host: [Brian Keating](https://realalpha.app/people/brian-keating/) · Host, Into the Impossible With Brian Keating
- Show: Into the Impossible With Brian Keating
- Episode: [Max Tegmark: We’re Building A Replacement Species](https://realalpha.app/v/nnYHg0gcWug/), Oct 23, 2026
- Moment: 2:15 to 10:05 of the episode (8 min), playing on https://realalpha.app/s/the-navier-stokes-result-does-not-solve-the-million-dollar/; the episode from there: https://realalpha.app/v/nnYHg0gcWug/?t=135

## Quote

> “They've taken and **solved an easier problem**, which is still super impressive.”
> — [Max Tegmark](https://realalpha.app/people/max-tegmark/) at [2:50](https://realalpha.app/v/nnYHg0gcWug/?t=170) · Into the Impossible With Brian Keating

## Takeaways

1. The proof works only with a continuing external force.
2. The prize problem asks whether blow-up can happen after the fluid is simply set in motion.
3. At atomic scales, the idealized model no longer describes real fluid behavior.

## About this segment

Max Tegmark explains that the new proof allows an external force to keep driving the fluid. The Clay prize still requires blow-up without that force.

From “Max Tegmark: We’re Building A Replacement Species”, Oct 23, 2026. [Watch the full episode](https://realalpha.app/v/nnYHg0gcWug/)

## Transcript

Transcribed by machine (Whisper) from the episode’s audio, so a word may be off here and there.

[2:23](https://realalpha.app/v/nnYHg0gcWug/?t=143) I was actually working a little bit on this problem with a PhD student of mine who's graduated now. So it was extra interesting for me when this story broke. Yeah. How did you react to it? Is it like a Higgs boson, bicep two moment as you wrote, you know, goodbye universe, hello multiverse on March 17, 2014 on your blog? Was that this kind of moment like that? I would say it's a little bit smaller, but it's still huge.

[2:51](https://realalpha.app/v/nnYHg0gcWug/?t=171) First thing first, you know, they have not actually solved the problem that you get a million bucks for from the Clay Foundation. They've taken and solved an easier problem, which is still super impressive. So, you know, the Navier-Stokes equation is the physics equation that can describe the water flowing in your bathtub, making a nice little whirlpool, like my three-year-old son loves to look at, or a tornado, or all sorts of other fluid dynamics.

[3:20](https://realalpha.app/v/nnYHg0gcWug/?t=200) And the question that's being asked is, is it possible to start with a quite normal situation and then start getting the fluid going infinitely fast, which seems completely nuts and would violate the known laws of physics? And they proved that they can do it, but with a kind of cheating where you're allowed to keep pushing on the stuff. so you know when when leo gets this vortex going in the bathtub with the water going out the drain he's not pushing the water it's just flowing there's some gravity trying to pull it down but there's nothing pushing it around to make it spin but they are doing that they're allowed to have a force that acts in some smooth way everywhere in the water so they can actually make it spin faster and faster and faster but not not infinitely fast and then the trick is this you know you know if you look at a figure skater right and they

[4:21](https://realalpha.app/v/nnYHg0gcWug/?t=261) they're spinning and they pull their arms in they will spin faster that's because of angular momentum being conserved of course and that actually implies that if you could take the figure skater and they could like shrink themselves together to infinitesimal size to basically zero size they would go infinitely fast the speed you'd have to go would be infinite But if you look carefully at that vortex in your bathtub, you see there's actually no water in the middle at all, because you have the centrifugal force pushing the water out and instead you get this neat funnel shape.

[4:59](https://realalpha.app/v/nnYHg0gcWug/?t=299) Same with the... Tornado, if you look close, it's actually hollow in the middle, right? That's why it looks like a funnel. And so there's almost no air in the middle, almost a vacuum, which is where you can get sucked up into it. Yes. And so that kind of solution is never going to win you the million bucks from the clay foundation, right? Because it's only when things actually start spinning and get collapsed together to zero radius that it happens.

[5:30](https://realalpha.app/v/nnYHg0gcWug/?t=330) But then you have this ace up your sleeve called viscosity. If you take a real fluid, honey or molasses or something, it's a little bit sticky. And that lets it actually dump some of its angular momentum and make things spin in principle in even smaller circles. This is how black holes eat lunch, for example. You know very well that when black holes start snacking on a cloud of gas the gas will typically pull in and it'll be spinning faster and faster and so it flattens into kind of a pancake shape and if there were no viscosity if there were no stickiness in this gas you would have the same problem as in your bathtub that there's like nothing there in the middle for the black hole to eat but there is some viscosity the gas bumps into gas that's farther in and and so on and then the black hole actually gets to ultimately eat all of that stuff

[6:24](https://realalpha.app/v/nnYHg0gcWug/?t=384) up so so this is the heart of the physics challenge or can you make some sort of spinning vortex eventually spin infinitely fast by making it contract by using them using some of this stickiness but not too much stickiness either because you don't want to like get rid of all the motion then you're never going to go infinitely fast and what they've now proven is that yes you You can actually do that if you're allowed to keep pushing it along with this weird external force.

[6:55](https://realalpha.app/v/nnYHg0gcWug/?t=415) But to get the million bucks, you have to prove that this can happen with no external force. But you can actually just take your fluid and get it started somehow, and then stand by and watch this crazy singularity emerge. Does that make sense? Yeah, and to do so in a finite time. I mean, the thing that's interesting to me is the Navier-Stokes equations don't involve any thermodynamic considerations.

[7:20](https://realalpha.app/v/nnYHg0gcWug/?t=440) Like, we know the laws of physics are, you know, reversible, at least, you know, CPT invariant. But, you know, if you would see a vortex, you should also see an anti-vortex. You know, in other words, all the... fluid kind of, you know, going the opposite direction and then creating some kind of anti-whirlpool, for example. But we never see that, just like we never see, you know, rings of water converging and then making a stone, you know, fly out of it, right? So why doesn't turbulence or, I mean, turbulence appears, but why doesn't thermodynamic dissipation appear in the Navier-Stokes, you know, equations or their solution?

[7:52](https://realalpha.app/v/nnYHg0gcWug/?t=472) So let me say two things to this. First of all, if you started with just water, like, perfectly at rest in Leo's bathtub, but it's not going to be able to start spinning either clockwise or counterclockwise for exactly the reasons you say. There's no angular momentum to start with, so there's never going to be any. But, you know, Leo has always been splashing around a little bit. There's some little random motion.

[8:16](https://realalpha.app/v/nnYHg0gcWug/?t=496) So on average, there's going to be a little bit of spin. It's just like when our solar system was formed, you know, the big gas cloud that contracted and made our solar system wasn't spinning very much, but it was just randomly spinning a little bit on average. And then when it contracted a whole bunch, it became very obvious that the solar system is spinning a lot, and the sun spins around every month, and Earth orbits the sun every year.

[8:38](https://realalpha.app/v/nnYHg0gcWug/?t=518) As long as you don't start at rest, you start in some sort of messy, random situation, you're always going to be able to get a whirlpool going one way or the other. Second part to your question, thermodynamics. so just like all other laws of physics are just approximations we believe of something more profound same thing with the Navier Stokes equation and in that case we know one huge thing we're glossing over which is that all matter is made of atoms so it doesn't make sense for example for in real life for a a gas to suddenly be spinning down and the radius go to zero because then that will be smaller than the separation between the atoms as soon as you're the spacing between the atoms is bigger than what you're modeling the scales you're modeling you know things are broken down so even if if

[9:34](https://realalpha.app/v/nnYHg0gcWug/?t=574) this uh million dollar conjecture is true you would never actually get anything infinite and instead but you would you might be getting things going crazy fast and then those crazy fast moving atoms would bump into other atoms and you would get heat dissipation you know thermodynamics kicks in just like you said the thing that was also very interesting to me and made me want to talk to you uh you know first