The Navier-Stokes regularity problem is difficult because of a 'Maxwell's demon' type possibility — the fluid could conspire to concentrate all its energy into smaller and smaller scales, forming a singularity in finite time.

Tao explains why proving that Navier-Stokes never blows up is so hard: the fluid's energy could, in principle, be transferred into ever smaller scales fast enough to overcome viscosity, like a 'demon' pushing energy into a singularity. ✦ AI generated

Terence Tao · Lex Fridman · 2025-06-15 · original ↗

Short answer is Maxwell's demon. So Maxwell's demon is a concept in thermodynamics. Like if you have a box with two gases, oxygen and nitrogen, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there's no barrier between them, right? Then they will mix. And they should stay mixed. There's no reason why they should unmix. But in principle, because of all the collisions between them, there could be some sort of weird conspiracy like maybe there's a microscopic demon called Maxwell's demon that will, every time an oxygen and nitrogen atom collide, they will bounce off in such a way that the oxygen sort of drifts onto one side and then nitrogen goes to the other. And you could have an extremely improbable configuration emerge, which we never see. And statistically, it's extremely unlikely. But mathematically, it's possible that this can happen, and we can't rule that out. And this is a situation that shows up a lot in mathematics. A basic example is the digits of pi, 3.14159, and so forth. The digits look like they have no pattern, and we believe they have no pattern. On the long term, you should see as many ones and twos and threes as fours and fives and sixes. There should be no preference in the digits of pi to favor, let's say, 7/8. But maybe there's some demon in the digits of pi that like every time you compute more and more digits, it sort of biases 1 digit to another. And this is a conspiracy that should not happen. There's no reason it should happen. But there's no way to prove it with our current technology. Okay, so getting back to Navier-Stokes, a fluid has a certain amount of energy. And because the fluid is in motion, the energy gets transported around. And water is also viscous. So if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy and it will go to zero. And this is what happens when we actually experiment with water. You splash around, there's some turbulence and waves and so forth, but eventually it settles down and the lower the amplitude, the smaller the velocity, the more calm it gets. But potentially there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale. And it will move faster and faster. And at faster speeds, the effect of viscosity is relatively less. And so it could happen that it creates some sort of what's called a self-similar blob scenario where, you know, the energy of the fluid starts off at some large scale and then it all sort of transfers its energy into a smaller region of the fluid, which then at a much faster rate moves into an even smaller region and so forth. And each time it does this, it takes maybe half as long as the previous one. And then you could actually converge to all the energy concentrating in one point in a finite amount of time. And that's an always good finite time blow up.

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