Terence Tao: prediction

Transcript context

…Why is it difficult to prove general things about the set of equations like it not not blowing up? Short answer is Maxwell’s Demon. So, Maxwell’s Demon is a concept in thermodynamics. If you have a box of two gases in 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. Then they will mix and they should stay mixed. There’s no reason why they should un-mix. But in principle, because of all the collisions between them, there could be some sort of weird conspiracy that maybe there’s a microscopic demon called Maxwell’s Demon that will… every time an oxygen and nitrogen atom collide, they’ll 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 which 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 seven over eight. But maybe there’s some demon in the digits of pi that every time you compute more and more digits, it biases one 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. 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 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 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’ll 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 the energy of the fluid starts off at some large scale and then it all sort of transfers energy into a smaller region of the fluid, which then at a much faster rate moves into an even smaller region and so forth. f the fluid starts off at some large scale and then it all sort of transfers 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 scenario is called finite time blowup. So, in practice, this doesn’t happen. So, water is what’s called turbulent. So, it is true that if you have a big eddy of water, it will tend to break up into smaller eddies, but it won’t transfer all energy from one big eddy into one smaller eddy. It will transfer into maybe three or four, and then those ones split up into maybe three or four small eddies of their own. So the energy gets dispersed to the point where the viscosity can then keep everything under control. But if it can somehow concentrate all the energy, keep it all together, and do it fast enough that the viscous effects don’t have enough time to calm everything down, then this blowup can occur. So, there were papers who had claimed that, “Oh, you just need to take into account conservation of energy and just carefully use the viscosity and you can keep everything under control for not just the Navier-Stokes, but for many, many types of equations like this.” And so in the past there have been many attempts to try to obtain what’s called global regularity for Navier-Stokes, which is the opposite of finite time blowup, that velocity stays smooth. And it all failed. There was always some sign error or some subtle mistake and it couldn’t be salvaged. So, what I was interested in doing was trying to explain why we were not able to disprove finite time blowup. I couldn’t do it for the actual equations of fluids, which are too complicated, but if I could average the equations of motion of Navier-Stokes, basically if I could turn off certain types of ways in which water interacts and only keep the ones that I want. So, in particular, if there’s a fluid and it could transfer as energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this one and direct it only into this smaller eddy while still preserving the lower conservation energy.…

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