Transcription
In mathematical physics, we care a lot about whether certain equations, in wave equations, are stable or not, whether they can create, um, these singularities. There's a famous unsolved problem called the Navia-Stokes regularity problem. So the Navia-Stokes equations, equations that govern the fluid flow for incompressible fluids like water. The question asks: if you start with a smooth velocity field of water, can it ever concentrate so much that, like, the velocity becomes infinite at some point? That's called a singularity. We don't see that, um, in real life. You know, if you splash around water in the bathtub, it won't explode on you. Um, or or have have water leaving at the speed of light, I think. But potentially, it is possible. Um, and in fact, in recent years, the the consensus has has drifted towards the uh the belief that uh that in fact, for certain very special initial configurations of of say water, that singularities can form. But people have not yet been able to uh to actually establish this.
The Clay Foundation has these seven Millennium Prize Problems, a million-dollar prize for solving one of these problems; this is one of them. Of these seven, only one of them has been solved: the Poincaré conjecture by Perelman. So the Poincaré conjecture is not directly directly related to the Navier-Stokes problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us understand the Navier-Stokes problem better.
Can you speak to the Navier-Stokes problem? So the existence and smoothness, like you said, Millennium Prize problem, right? You've made a lot of progress on this one. In 2016, you published a paper, "Finite time blow up for an averaged three-dimensional Navier-Stokes equation," right? So we're trying to figure out if this thing usually doesn't blow up, right? But can we say for sure it never blows up? Right.
Yeah. So yeah, that is literally the the million-dollar question. Yeah. So this is what distinguishes mathematicians from pretty much everybody else. Like if if something holds 99.99% of the time, um that's good enough for most, you know, uh for for most things. But mathematicians are one of the few people who really care about whether every, like, 100% really 100% of all um situations are covered by by um yeah, so most fluid, most of the time, um water does not blow up. Could you design a very special initial state that does this? And maybe we should say that this is a this is a set of equations that govern in the field of fluid dynamics. Trying to understand how fluid behaves, and it's actually turns out to be a really comp, you know, fluid is yeah, extremely complicated thing to try to model. Yeah. So it has practical importance.
So this Clay Prize problem concerns what's called the incompressible Navier-Stokes, which governs things like water. There's something called the compressible Navier-Stokes, which governs things like air, and that's particularly important for weather prediction. Weather prediction. It does a lot of computational fluid dynamics. A lot of it is actually just trying to solve the Navier-Stokes equations as best they can. Um, also gathering a lot of data so that they can get they can initialize the equation. There's a lot of moving parts. So it's a very important problem practically.
Why is it difficult to prove general things about this set of equations, like it not not blowing up? Short answer is Maxwell's demon. Um, so Maxwell's demon is a concept in thermodynamics: like if you have a box of two gases, oxygen and nitrogen, uh and maybe you start with all the oxygen on one side and nitrogen the other side, but there's no barrier between them, right, then they will mix, um, and they should stay mixed, right, 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 that that um like maybe there's a microscopic demon called Maxwell's demon that will um 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 the nitrogen goes to the other, and uh you could have an extremely improbable configuration emerge uh which we never see. Um, and and we statistically it's extremely unlikely, but mathematically it's possible that this can happen, and we can't rule it out. Um, and this is a situation that shows up a lot in mathematics.
Um, 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 over 8. Um, but maybe there's some demon in the digits of pi that that like every time you compute more digits, it sort of biases one digit to another. Um, and this is a conspiracy that should not happen. There's no reason it should happen, but um there's there's there's no way to prove it uh with our current technology. Okay.
So getting back to Navier-Stokes, a fluid has a certain amount of energy, and because a 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 it will go to zero. Um, and this is what happens um in um uh when we actually experiment with water, right? You splash around, there's there's some turbulence and waves and so forth. But eventually it it settles down and and and the lower the amplitude, the smaller the velocity, the the more calm it gets. Um, but potentially there is some sort of demon that keeps pushing the uh the energy of the fluid into a smaller and smaller scale, and it will move faster and faster, and at faster speeds the effective viscosity is relatively less. And so it could happen that that it it creates a some sort of um um what's called a self-similar blob scenario where you know um the energy of the fluid starts off at some um large scale and then it all sort of um transfers energy into a smaller um region of of of the fluid, which then at a much faster rate um moves into um an even smaller region and so forth. Um, and and each time it does this it takes maybe half as as long as as the previous one. And then yeah, you could you could actually uh converge to all the energy concentrating in one point in a finite amount of time. Um, and that that's uh that scenario is called finite-time blow up.
Um, so in practice this doesn't happen. Um, so water is what's called turbulent. Um, so it is true that um if you have a big eddy of water it will tend to break up into smaller eddies, but it won't transfer all the energy from one big eddy into one smaller eddy; they will transfer into maybe three or four, and then those ones split up into maybe three or four small eddies of their own, and so the energy gets dispersed till the point where the viscosity can can then keep the thing under control. Um, but if it can somehow um concentrate um all the energy, keep it all together um and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blob 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 blow up, that velocity stay smooth, and it all failed; there was always some sign error or some subtle mistake, and and it couldn't be salvaged. Um, so what I was interested in doing was trying to explain why we were not able to disprove um finite-time blow up. I couldn't do it for the actual equations of fluids, which were too complicated. But if I could average the equations of motion of Navier-Stokes, basically if if um if I could turn off certain types of of ways in which water interacts and only keep the ones that I want. Um, so in particular, um, if, um, if there's a fluid and it could transfer 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 this one and and direct it only into um this smaller eddy while still preserving the law of conservation of energy.
So you're trying to make it blow up. Yeah. Yeah. So I I I basically engineer um a blow up by changing the laws of physics, which is one thing that mathematicians are allowed to do. We can change the equation. How does that help you get closer to the proof of something? Right? So, it provides what's called an obstruction in mathematics. Um, so so what I did was that uh basically if I turned off the um certain parts of the equation, so which usually when you turn off certain interactions, make it less nonlinear, it makes it more regular and less likely to blow up. But I found that by turning off a very well-designed set of of of of of interactions, I could force all the energy to blow up in finite time. So what that means is that if you wanted to prove um global regularity for Navier-Stokes um for the actual equation you had, you must use some feature of the true equation which which my artificial equation um does not satisfy. So it it rules out certain um certain approaches. So um the thing about math is is it's not just about finding, you know, taking a technique that is going to work and applying it, but you you need to not take the techniques that don't work. Um, and for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem. But uh it's only after a lot of experience that you realize there's no way that these methods are going to work. So having these counter-examples for nearby problems um kind of rules out um uh it saves you a lot of time because you you're not wasting um energy on on things that you now know cannot possibly ever work.
How deeply connected is it to that specific problem of fluid dynamics or just some more general intuition you build up about mathematics? Right. Yeah. So the key phenomenon that uh was my my technique exploits is what's called supercriticality. So in partial differential equations, often these equations are like a tug-of-war between different forces. So in Navier-Stokes there's the dissipation um force coming from viscosity, and it's very well understood. It's linear. It calms things down. If if viscosity was all there was, then then nothing bad would ever happen. Um, but there's also transport um that that energy from in one location of space can get transported because the fluid is in motion to to other locations. Um, and that's a nonlinear effect, and that causes all the all the problems. Um, so there are these two competing terms in the Navier-Stokes equation: the dissipation term and the transport term. If the dissipation term dominates, if it's if it's large, then basically you get regularity. And if um if the transport term dominates, then uh then we don't know what's going on. It's a very nonlinear situation. It's unpredictable. It's turbulent. So sometimes these forces are in balance at small scales, but not in balance at large scales or or vice versa. Um, so Navier-Stokes is what's called supercritical. So at at smaller and smaller scales, the transport terms are much stronger than the viscosity terms. So the viscosity are things that calm things down. Um, and so this is um um this is why the problem is hard. In two dimensions, so the Soviet mathematician Ladyzhenskaya, she in the 60s shows in two dimensions there is no blow up, and in two dimensions the Navier-Stokes equations is what's called critical: the effect of transport and the effect of viscosity are about the same strength even at very very small scales, and we have a lot of technology to handle critical and also subcritical equations and proof um regularity, but for supercritical equations it was not clear what was going on, and I did a lot of work, and then there's been a lot of follow-up showing that for many other types of supercritical equations you create all kinds of blow up examples. Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen. So this is sort of one of the main insights of this this line of work is that supercriticality versus criticality and subcriticality. This this makes a big difference. I mean that's a key qualitative feature that distinguishes some equations for being sort of nice and predictable and you know like like planetary motion, and I mean there are certain equations that that you can predict for millions of years or thousands at least. Again, it's not really a problem, but but there's a reason why we can't predict the weather past 2 weeks into the future because it's a supercritical equation. Lots of really strange things are going on at very fine scales. So, whenever there is some huge source of nonlinearity, yeah, that can create a huge problem for predicting what's going to happen. Yeah. And if the nonlinearity is somehow more and more featured and interesting at at small scales. Um I mean there there's many equations that are nonlinear, but um in in many equations you can approximate things by the bulk. Um, so for example, planetary motion, you know, if you want to understand the orbit of the moon or Mars or something, you don't really need the microstructure of like the seismology of the moon or or like exactly how the mass is distributed. um you just basically you can almost approximate these planets by point masses, and just the aggregate behavior is important, um but if you want to model a fluid um like like the weather, you can't just say in Los Angeles the temperature is this, the wind speed is this. For supercritical equations, the fine-grained information is is really important.
If we can just linger on the Navier-Stokes uh equations a little bit, so you've suggested maybe you can describe it that one of the ways to uh solve it or to negatively resolve it would be to sort of to construct a liquid, a kind of liquid computer, right? And then show that the halting problem from computation theory has consequences for fluid dynamics. So uh show it in that way. Can you describe this this? Yeah, so this came out of of this work of constructing this this this average equation that that blew up. Um, so one um as as part of how I had to do this. So there's sort of this naive way to do it. You you just keep pushing um um every time you you get energy at one scale you you push it immediately to the next scale as as fast as possible. This is sort of the naive way to to to to force blow up. Um, it turns out in five and high dimensions this works. Um, but in three dimensions there was this funny phenomenon that I discovered that if you if you keep if if you change laws of physics, you just always keep trying to push um the energy into smaller smaller scales. Um, what happens is that the energy starts getting spread out into many scales at once. Um, so you you have energy at one scale, you're pushing it into the next scale, and then um as soon as it enters that scale you also push to the next scale, but there's still some energy left over from the previous scale. um you're trying to do everything at once. Um, and this spreads out the energy too much. Um, and then it turns out that that um it makes it vulnerable for viscosity to come in and actually just damp out everything. So um so it turns out this this direct push doesn't doesn't actually work. There was a separate paper by some other authors that actually showed this um in three dimensions.
So what I needed was to program a delay. Um, so kind of like airlocks. Uh, so um I needed an equation which would start with a fluid doing something at one scale. It would push this energy into the next scale, but it would stay there until all the energy from the from the larger scale got transferred, and only after you pushed all the energy in then you sort of open the next gate and and then you you push that in as well. So um by doing that it kind of the energy inches forward scale by scale in such a way that it's always um localized at one scale at a time. Um, and then it can resist the effects of viscosity because it's not dispersed. Um, so in order to make that happen um yeah, I had to construct a rather complicated nonlinearity. Um, and it was basically like um you know like was constructed like an electronic circuit. So I actually thank my wife for this because she was trained as an electrical engineer. Um, and um you know she talked about um uh you know she had to design circuits and so forth. And you know if if you want a circuit that does a certain thing, like maybe have a light that that flashes on and then turns off and then on and then off, you can build it from from more primitive components, you know, capacitors and resistors and so forth, and you have to build a diagram, and you um and these diagrams you can you can sort of follow up with your eyeballs and say oh yeah the the current will build up here and then it will stop and then it will do that. So I knew how to build the analog of basic electronic components, you like resistors and capacitors and so forth, and and I would I would stack them together um in in such a way that that I would create something that would open one gate and then there'll be a clock that once the clock hits a certain threshold it would close it, kind of a Rube Goldberg type machine but described mathematically, and this ended up working.
So what I realized is that if you could pull the same thing off for the actual equations. So if the equations of water support a computation, so um like if you can imagine kind of a steampunk but really waterpunk uh type of thing where um you know so modern computers are electronic, you know, they they they're powered by by electrons passing through very tiny wires and interacting with other electrons and so forth. But instead of electrons, you can imagine these pulses of of water moving at a certain velocity. And maybe it's there two different configurations corresponding to a bit being up or down. Probably if you had two of these moving bodies of water collide, they would come out with some new configuration which is which would be something like an AND gate or OR gate. You know that it the the output would depend in a very predictable way on on the inputs. And like you could chain these together and maybe create a Turing machine, and and then you could you have computers which are made completely out of water. Um, and if you have computers then maybe you can do robotics. So you know hydraulics and so forth. Um, and so you could create some machine which is basically a fluid analog of what's called a von Neumann machine. Uh, so von Neumann proposed: if you want to colonize Mars, the sheer cost of transporting people and machines to Mars is just ridiculous. But if you could transport one machine to Mars and this machine had the ability to mine the planet, create some more materials, smelt them and build more copies of the same machine. Um, then you could colonize a whole planet um over time. Um, so uh if you could build a fluid machine, which uh yeah, so it's it's it's a it's a rob it's a fluid robot. Okay. And what it would do, its purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state. It wouldn't start just yet. Once it's ready, the big robot configuration of water would transfer all its energy into the smaller configuration and then power down. Okay? And then like clean itself up, and then what's left is this newest state which would then turn on and do the same thing but smaller and faster. And then the equation has a certain scaling symmetry. Once you do that, it can just keep iterating. So this in principle would create a blow up for the actual Navier-Stokes, and this is what I managed to accomplish for this average Navier-Stokes. So it provided the sort of road map to solve the problem. Now this is uh a pipe dream because uh there are so many things that are missing for this to actually be a reality. Um, so um I I I can't create these basic logic gates. Um, I I don't I don't have these in these special configurations of water. Um, I mean there's candidates: there are things called vortex rings that might possibly...
Work, but um, um, but also, you know, analog computing is really nasty, um, compared to digital computing. I mean, because there's always errors, um, you you have to you have to do a lot of error correction along the way. I don't know how to completely power down the big machine so that it doesn't interfere with the the the running of the smaller machine, but everything in principle can happen, like it doesn't contradict any of the laws of physics. Um, so it's sort of evidence that this thing is possible.
Um, there are other groups who are now pursuing ways to make nodes blow up, which are nowhere near as ridiculously complicated as this. Um, um, they they actually are pursuing much closer to the the direct self-similar model, which can it it doesn't quite work as is, but there could be some simpler scheme than what I just described to make this work. There is a real leap of genius here to go from Navia Stokes to this Turing machine. So it goes from what the self-similar blob scenario that you're trying to get the smaller and smaller blob to now having a liquid Turing machine gets smaller and smaller and smaller and somehow seeing how that could be used to say something about a blowup. I mean, that's a big leap. So there's precedent. I mean, um, so the the thing about mathematics is that it's really good at um, spotting connections between what you think of what you might think of as completely different um, problems. Um, but if if the mathematical form is the same, you you can you can you can draw a connection.
Um, so um, there's a lot of work previously on what's called cellular automata. Um, the most famous of which is Conway's game of life. There's this infinite discrete grid, and any given time the grid is either occupied by a cell or it's empty. And there's a very simple rule that uh tells you how these cells evolve. So sometimes cells live and sometimes they die. Um, and this um, you know, um, when I was a a student it was a very popular screen saver to actually just have these these animations going and and they look very chaotic. In fact, they look a little bit like turbulent flow sometimes. But at some point people discovered more and more interesting structures within this game of life.
Um, so for example, they discovered this thing called a glider. So a glider is a very tiny configuration of like four or five cells which evolves and it just moves at a certain direction and that's like this this vortex rings this. Um, yeah. So this is an analogy. The game of life is kind of like a discrete equation and and um the fluid Navier-Stokes is a continuous equation, but mathematically they have some similar features. Um, and um, so over time people discovered more and more interesting things you could build within the game of life. The game of life is a very simple system. It only has like three or four rules um to to do it, but but you can design all kinds of interesting configurations inside it.
Um, there's something called a glider gun that does nothing but spit out gliders one at a one one at a time. Um, and then after a lot of effort people managed to to create um and gates and or gates for gliders, like there's this massive ridiculous structure which if you if a if you have a stream of gliders um coming in here and a stream of gliders coming in here then you may produce a stream gliders coming out. If so maybe if both of of the um streams um have gliders then there'll be an output stream, but if only one of them does then nothing comes out. So they could build something like that, and once you could build an um these basic gates then just from software engineering you can build almost anything um you can build a Turing machine. I mean, it's again enormous steampunk type things. They look ridiculous. But then people also generated self-replicating objects in the game of life. A massive machine, a bon machine which over a huge period of time and it always look like glider guns inside doing these very steampunk calculations. It would create another version of itself which could replicate. It's so incredible. A lot of this was like community crowdsourced by like amateur mathematicians actually.
Um, so I knew about that that that work and so that is part of what inspired me to propose the same thing with Navier-Stokes. um, which is a much as I said analog is much worse than digital, like it's going to be um you can't just directly take the constructions in the game of life and plunk them in, but again it just it shows it's possible.