The Navier-Stokes existence and smoothness problem asks whether a smooth three-dimensional incompressible fluid flow can develop a singularity, a point of infinite velocity, in a finite amount of time. The explanation centers on two competing terms in the equation: the convective (nonlinear) term, which pushes energy into smaller and smaller structures, and the viscosity term, which smooths those structures out. Using the Taylor-Green vortex as a worked example, it shows how the convective term creates finer motion by leaking a flat, circulating flow into a third dimension, and why in every simulation tried so far viscosity eventually wins and dissipates that energy before anything blows up. The problem remains open only in 3D, where a mechanism called vortex stretching can concentrate vorticity, and the singularity being hunted for is described as a purely mathematical feature of a vector field rather than a physical prediction about real fluids.
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Soviet mathematician Olga Ladyzhenskaya proved in a 1969 book that in two dimensions every smooth initial condition stays smooth forever, so the open problem is specifically about three dimensions.
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Vortex stretching is explained through an hurricane-to-tornado analogy: squeezing a vortex's radius down forces it to spin faster by conservation of angular momentum, the same principle behind a figure skater pulling in their arms.
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A simple nonlinear equation, dy/dt = y-squared starting at y=1, is used to show how a finite-time blowup (reaching infinity at t=1) differs from the infinite-time growth of compound interest.
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Terrence Tao is cited as describing the Navier-Stokes equations as one of the simplest 'supercritical' differential equations because of this competition between nonlinear convection and diffusion.
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The equations under study don't even apply to real-world fluids in many regimes, such as hypersonic flight, molecular-scale flow through proteins, or compressible weather modeling, reinforcing that the blow-up question is a mathematical rather than physical one.
5,408 words · auto-generated from the episode video
1:44:59solutions always well behaved? >> Okay, that's the question. Here's what I mean by well- behaved. Let me give you an example of why they shouldn't be well behaved. It really has to do with these two terms that are in complete competition to one another. >> And let me see if I can get this right. We're talking about the convective term on the left >> which accounts for our acceleration and we're talking about the viscosity term on the right which is a part of our force. >> Yes. And these two things do very different qualitative transformations to our velocity field. The viscosity term as I told you smooths things out. Right? If
1:45:40there's a tiny bit where there's a lot of velocity and it's surrounded by not that much velocity then it's going to smooth out. the velocity is going to leak through and you're going to smooth out the velocity function. The convective term on the other hand, what it does is transfer energy from large systems to smaller and smaller systems. So the convective term is making smaller and smaller structures while the viscosity term is making those smaller structures smooth out. M there's a constant competition between these two terms in the equation >> because depending on the size of the
1:46:20structure you're referring to the viscosity will be different and the convective term will be different and it's different across every scale >> yes and the question is does one win out over the other specifically does the convective term ever win out over the viscosity term can I get smaller and smaller structure to such an extent where I can have theoretically infinite velocity somewhere because the convective term is just creating smaller and smaller structure. Now, I want to be a bit more concrete here when I say the convective term is creating smaller and smaller structure. Okay, so we're going to talk about um this competition in a very >> real sense with something called a tailor green vortex. This is a standard test case that is used in computational
1:47:01fluid dynamics because what you can do is actually like solve this thing analytically and then you can solve it in a computer and compare the two. Okay. >> And then that'll tell you how good your computer simulation is because you have a ground truth that's like a functional mathematical form. Here's what the tailor green vortex looks like. This is the initial condition condition. On the right hand side, you're seeing the velocity field at time t equals z. So this is what you're starting out with. It's a bunch of ss and cosiness for the x velocity. It's a bunch of ss and cosiness for the y velocity. And it's zero on the z velocity. Meaning all of the particles are are only moving in the xy plane. No one's going up and down.
1:47:42Everything is moving side to side. >> Flat land. >> Flat land. In flat land, right? But there's bunches of flat land. Like it can still be a 3D box, but like the the stuff at the top of the box is still going side to side. And the the stuff at the bottom of the box is still going side to side. That's how we start out with. And crucially, all of these terms have signs and cosiness. So what that what that amounts to is you have vortices. >> Okay? you have a the signs and cosiness always means circles. Okay, if you if you if you come out with one thing from this podcast, signs and cosiness mean circles. But in this case, what's happening is you've got these vortices that are circular. Okay? And so all of the particles are moving around in circles like this. Like the ones up top
1:48:22are moving like this. Down here, they're moving like this. Down here, they're moving like this. Imagine a 3D box. And now I press play. >> Mhm. >> On this thing, >> what is going to happen? Well, let's see what happens. Um, we've got a bunch of math here that I'm going to very quickly go through. Okay, for those who are not watching, my my eyes just expanded to a very large size. >> Yeah, but it's it's not that bad. And and I want I I'll tell you what to focus on. >> Okay. >> On the left hand side is our initial conditions. These are our ss and cosiness. It's just sin x and cosine cosine y sin cosine z. So this tells you
1:49:03that like the the size of my vortex is 2 pi in size. Okay. Okay. >> Now I apply the convective term to these equations. Right. These equations tell me what the velocity field is. I can find the gradient of the velocity field and I can multiply it by itself. And then I'll leave this as an exercise to the reader or in this case the podcast listener as all of the famous science and math textbooks do. Um, but you can go through and do the derivatives and then you'll have to remember your double angle identities and trig. But what ends up happening is >> those derivatives are going to create terms with a sin 2x and a cosine 2x.
1:49:44That's going to percolate into the into the pressure terms. Mhm. >> You're going to get sin 2x and cosine 2x. And one thing that you should remember is sinx is a wave. >> Mhm. >> Sin 2x is that same wave squished. >> Yeah. >> You're creating smaller and smaller waves. The wavelength is getting smaller. >> The oscillation >> is getting squished. Okay. Now, what does that do to our structure of the vortex? What happens is remember before I told you that yeah that's that's the so remember before I told you that the Z component of my velocities was zero everything was
1:50:25happening yep in side to side right but now because of that convective term and because of those higher wavelengths sorry I should say smaller wavelengths there's going to be picked up Z velocities there's going to be interaction that are happening in the Z direction. You're going to push up stuff and you're going to push down stuff. And the way you pushing up push up and down, those structures are going to be half the size of your original vortex >> of and this is direct coming from the squishing. >> Mhm. It's coming from the squishing and the fact that the squishing is forcing stuff into the third dimension. >> So on the left hand side you see these vortices. Those blobs sort of represent
1:51:07where the vortices are. You can imagine the orange blob is a vortex going in one direction. blob is a vortex going in the other direction, but all of those vortices are X and Y. >> Okay, they're they're revolving in this way. >> And they sort of have sort of, for lack of a better term, a consistent nondisturbed structure for time t equals z. Like that's what we started with. And when we press play, all of a sudden, you start getting these vertices. You're getting these interactions between the vertices up and down. >> Mhm. And the interactions those structures that are created in that interaction have a smaller and smaller size >> because of that convective term.
1:51:47>> Okay. >> Okay. >> I see. I see. >> And if we continue this at infinitum, it's going to get even smaller >> and even smaller and even smaller to at some point you would think you would think, right, that like it's just going to get smaller and smaller to infinity. You're going to get smaller and infinite decimally smaller structures. That's not the case. Viscosity at some point wins out in this scenario. >> So coming back to how we started this, >> we're talking about the viscosity term on the right side of our Navier Stokes equation. >> And then we're talking about the convective term on the left side of our Navier Stokes equation. >> And these two things are in competition. >> Yes. >> As we looked at in this example.
1:52:28>> Yeah. >> And at at time equals zero >> when we have those blobs, >> we have the blobs in the top left. And as time goes by, the the left side of the equation is convective is winning >> is winning. Yeah. Effectively >> effectively because it's creating smaller and smaller structure that's leaking into the Z direction. >> And so our our sort of nice discrete blob structures at the top become a more chaotic system >> as we get these smaller and smaller structures. >> But at some point and the idea the qu the fundamental question is does this go all the way to infinity in smallness? And do I get smaller and smaller smaller and smaller and smaller structures? >> Does the is the convective term the
1:53:08goat? >> And does it always win? >> Yeah. >> And what we're saying is no. >> In this case, no. >> In this case, at some point, the viscosity starts to matter once you get to a sufficiently small scale. >> Yes. Because and for technical audiences, the the convective term goes up kind of linearly, but it like grows very fast in the beginning. The viscosity term grows like the square. >> So at some point it's going to catch up and it's going to diffuse everything and then everything will just sort of like be chill. >> Yeah, that makes sense. >> Okay. Um, so the Taylor Green vortex, I think it puts this conflict right in front of us. >> You've got the nonlinear term, that convective term that takes an organized flow
1:53:49>> and it creates smaller and smaller spatial structure. And finally, we've got this viscosity that becomes increasingly more powerful at those smaller and smaller spatial structures is the convective term going like this and the viscosity is going like this. >> It's like it's like this versus this. Okay, this so so eventually at like at some this meeting point of of structure size is when >> is when the viscosity is like okay that's enough >> that's enough let's let's calm down and so the viscosity waits for those smaller scales to convert all of that energy into just like dissipation and heat right and the where where it happens depends on the viscosity of the fluid okay if the fluid is very very viscous
1:54:31it's going to cut it immediately >> but if it's like not very viscous >> it'll take time Um, it'll take some time, but it's going to get there with for the Taylor Green vortex. It is going to get there. >> That's fascinating. >> Right. And now in every single simulation that we've done, viscosity always wins out. >> Okay. >> Okay. At some point, viscosity is always going to win out. >> V for vendetta, one would say. >> That's right. And the millennium problem is asking, is that actually true? For every single scenario that I can come up with, I just showed you two, right? We showed the coete flow and then we showed this Taylor Green vortex. In both of those cases, pretty fine. Everything's smooth. Everything's normal. >> Things aren't going crazy.
1:55:12>> So, is this the idea that smoothness always wins? >> Yes. >> Like to describe it. >> Is the the solution is always going to be smooth because of the viscosity. >> Because of the viscosity winning in this race. >> Yeah. >> Okay. >> Is it always going to win? >> Is it always going to win? Right. We've shown that it can win sometimes. >> Yes. And for those two times, it's going to win. And actually for every single simulation that we've done, we've shown that it's going to win. >> Every single simulation in the history of aerodynamics, we've shown that it's going to win. Now, that could be an artifact of the fact that it's a simulation, right? And there's like discretized grids and so we can't go infinite decimally small at some point. Maybe it's an artifact of the fact that like you you literally can't physically
1:55:53like there's not enough compute to go down to what the plank length or something like that, right? >> Where it could start falling apart. like like so so perhaps that's that and that's what the millennium problem is asking. It's asking something stronger. It's saying for >> every single initial condition that I could come up with, is viscosity always going to win? >> This is the is smoothness global? >> Yes. >> Idea. >> Mhm. And and that that actually it's so funny because now it's it kind of reminds me a little bit of was it the uh was it the Remon hypothesis episode where we talked about uh the non
1:56:34>> trivial zero >> the non-trivial zeros and it's similarly like >> you know we want to take the question to its ultimate >> yeah it's like is every single zero on the critical line >> on the critical line right every single one >> no every single one Because you can still have 99.9 you could even have 100 as we were saying as we were saying and it does that doesn't mean that >> you can have 100% and still have one off >> and anywhere in all possibility that one is off it means just concept >> that's what mathematicians are worried about right it's like no no no no in every universe right that Doctor Strange has ever visited >> is this true >> it feels similarly in that direction >> yeah it's it's a very mathematical thing
1:57:15to to worry about. >> Now I understand the mindset. >> Yeah. Now um crucially this is now an open problem only in 3D. >> Okay. >> In 2D there was a very famous mathematician. She was a Soviet mathematician um Olga Ladi Jenkaya. Olga Ladenskaya. She wrote this 1969 book, The Mathematical Theory of Viscous Incompressible Flow, where she showed that in two dimensions, yes. Every single initial condition, guaranteed viscosity is going to win. >> If we're in flat land, >> Mhm. Yes. If we're in flat land and we've got like fluids flowing in flat land >> and there's no a Z upz down like we just
1:57:56described >> as as we saw actually like even in the Taylor Green vortex, right? We saw that like the Z dimension was the one that was causing that >> weirdness because because the flow was escaping into the Z dimension >> and then creating those smaller and smaller structures. So Olga showed in 1969 in her in her very famous book that indeed in two dimensions it's totally fine. Every single starting point that you could ever think of you're going to you're going to come up with a smooth solution. Okay. So for for 2D the Navia Stokes global smoothness is known. >> It's fine. That's not the question we're asking. >> Yeah. For 3D >> for 3D is is
1:58:36>> it is not known. >> And is this and I just have a quick question. So when >> when I see the term blow up >> as it relates to how people describe this, this is what we're is this kind of what we're talking about is like does the smoothness does um >> does viscosity lose? >> Yes. which is effectively blow up. Yeah. Like for lack of >> does viscosity lose? Because if viscosity loses then that can effective term is going to keep going on this runaway reaction to create smaller and smaller structures of higher and higher velocities and at some point you're going to get like stuff that's moving at infinite speed, >> right? >> Mhm. It's it it's kind of like is is
1:59:17there a singularity in fluids? Not really, but like >> Well, no, but that No, no, no. That's literally what they're asking. >> Okay. >> Is there a singularity? like can I create a singularity? That's what that's that's what they're asking. >> Okay, fair enough. Fair enough. >> That's exactly what they're asking. Now, why is it so hard in three dimensions? Well, we all already saw like kind of oluded that with the Taylor Green vortex, right? It's the fact that like stuff can move up and down. And crucially, that had to do with a vortex that I had started, right? The initial condition was a vortex. The signs and cosiness described stuff going around in a circle. And we had these mini tornadoes inside of our thing that were interacting with one another and leaking into one another. Mhm. Mhm. >> Vorticity is the problem in 3D. Okay. Vorticity is the curl. Um for those for
1:59:59those who have taken calculus, the curl of the vector field that dell operator is back again, but this time we're taking a crossroduct of the dell operator with the velocity field. It's just a way of saying is the velocity field curling? Like if I put a pin wheel in the velocity field, is it going to turn? >> If it's going to turn, then that means that there's a curl that's happening over there. Okay. Now, in two dimensions vorticity has nowhere to go, >> right? >> Right. It's just spinning and then it's got to like dissipate in the flat land. >> Yeah. >> In three dimensions, you can have something called vortex stretching. Okay. No, this is in two dimensions. You've got vorticity that's like kind of just spreading out and like going out
2:00:42about like um >> you know, it's a small vortex and it becomes a bigger and bigger vortex, right? And that's all it can do in two dimensions >> because there's nowhere to go. >> It's got nowhere to go. In three dimensions, you get something called vortex stretching. >> Okay? And this could be a problem. >> Imagine a vortex that has a certain size. It's got a certain radius like a hurricane that's or yeah, let's say a hurricane. It's quite big in space and it's moving but the the the velocity is not that high. Okay. And now that hurricane, all of that energy gets squished into a tornado. M >> what's going to happen? Well, the tornado is going to get very very tall
2:01:22because all of that energy from the hurricane has to >> divergence has to equal zero. >> Yeah, very good. This is exactly it. >> It has to equal zero. >> It has to so it's got to go somewhere. >> It's got to go somewhere. >> So, it's going to go up and but because it's getting tall again, the radius is going to decrease. But if the radius decreases, >> now we have another principle of physics which is the conservation of angular momentum. Mhm. >> And so just like um you know um in the the figure skaters when they're twirling around they pull their arms in they spin faster. Same thing's going to happen here. The thing is going to spin faster. And this this kind of reminds me in a different context of our sideways wine
2:02:03bottle and then it getting narrower and so it increased the acceleration. Just conceptually there there's a sort of similar. >> Yeah. packing stuff in here, but here it's the vorticity that's increasing because of the conservation of angular momentum. Again, it's like it's kind of similar because there's something being conserved. In that case, it was the amount of fluid. In this case, it's the amount of angular momentum. And >> this is called vortex stretching because you're taking a vortex, you're stretching it up, but as you stretch it up, you thin it out. And if you thin it out, it's like the figure skater bringing in her arms. The thing is going to move faster and faster. And this kind of relates to what we just talked about about in the Taylor Green vortex why we're now leaking into the >> into those higher dimensions. Right. And the question is maybe that with this
2:02:44vortex stretching >> you can actually um create a kind of singularity. >> Mhm. >> Okay. Now let's get into what we mean by singularity in the first place. We we had already done our first toy differential equation, right? Yes. Which is the compound interest differential equation. Now this thing does go to infinity, right? I mean, it's e to the x or e to the t for infinite time, I'll get an infinite amount of money. Infinite money. If I wait long enough, even if I have a dollar in my bank account, that dollar is going to become infinity if I wait long enough, right? Um, it just turns out maybe AI is going to end the world. So, I The point is that's not a real singularity. A real
2:03:25singularity is an infinite infinite time. I shouldn't say infinite because that sounds like infinite. So I should say um before no in pause >> pause pause >> finite time >> can I get to infinity >> in some constrained amount of time >> there we go yes for our listeners out there that that'll be good in in a constrained amount of time can I get to infinity here's another differential equation that does that here the derivative my derivative with respect to time is equal to y^2 so dydt is equal to y^2. Um, initial condition, you start at 1 when time is zero. M
2:04:07>> and the solution to that is 1 / 1 - t. If I plug in t equals 1, I get infinity. >> Mhm. >> Right. So, this is a differential equation that creates a singularity in >> but the time is not infinity. The time is just one. At 1 second, I have a singularity. >> You reach you reach the singularity. Another thing I want to point out here, what's special about this differential equation? It is nonlinear. >> You see, it's y^2. >> It's the simplest nonlinear differential equation. And already I am getting a singularity, right? The other one was just dydt= y. That's a linear equation. I can just add up the terms,
2:04:47>> right? But here the nonlinearity is creating that cascade that is creating a singularity before time goes to infinity at a at a finite time. So it's conceivable to extrapolate from this. >> Yeah. >> It would be conceivable based on how we've constructed the nonlinearity of trying to look at this incompressible fluid >> and look at a vector field. >> Yeah. It's like there's a chance >> there. There's a chance because we got a lot of nonlinearity going on. >> Yeah. Yeah. It's just that there's the viscosity term, right? In in this one that I showed you, there's nothing. There's no there's no smoothing. There's no counting. It's just going it's just going, >> right? But but in the Navier Stokes and that's why it's so interesting >> because the Navier Stokes has this
2:05:28nonlinearity, but it also has this smoothing term. >> It has this counter force which is viscosity. >> And so you're you're asking who's going to win and are they always going to win? So far it seems viscosity is always wrong. >> That's actually that makes so much sense now. It it's it's ah that's so good right >> that's so good >> it's quite nice >> yeah it's quite nice so it is a very interesting problem >> yeah no it it because because you can understand you could >> there's good arguments for both sides >> one in which viscosity always wins feels true seems legit >> because we've been able to show that >> overations in a variety of context as
2:06:09being true >> uh but being fundamentally true >> is another thing >> is another thing. And because of again the nonlinear nature of the convective term in this case >> that is challenging the viscosity as the counter force. >> Mhm. >> There's an argument that both of them have a reason that they could win. >> Exactly. And in finite time. >> Infinite time. Exactly. And so >> it's it's it's an interesting problem. I have to give it to him. Right. like it's it's it's something that definitely um makes you think and there are other things in physics that have singularities in finite time. I mean the the most famous one would probably be a black hole as a result of the Einstein field equations. The Einstein field
2:06:51equations are also very famously a nonlinear differential equation because you've got this um you've got these tensors and these tensors are multiplied to themselves and also multiplied to the reciprocal of themselves twice over and so you get these squared terms um you know for example I mean very in a in a very simple case like the curvature of space and time causes gravity right but gravity itself has an energy that causes the curvature of space and time, right? So, so what and so, so now you get these nonlinear actions on space and time that create the richness of the phenomenon that we see. And on the right
2:07:32hand side, um I mentioned Lake Geneva that I was on like I was actually at the Observatory on Lake Geneva which is known as the birthplace of modern astrophysics. We'll do an episode on them. They had a nice sculpture of um Einstein's general relativity because Einstein had visited the Yorks observatory many times and there on the on the slab of marble was written out Einstein's field equation. So I just wanted to show that um when I when I received this news I was looking at a nonlinearity >> live and I was like damn >> right >> here we go again. >> Here we go again. >> You know >> like the meme we started the episode with. >> Exactly. So singularities can exist in
2:08:12physics. Right. The center of a black hole is a singularity. But I do have to say that in this case, in the Navier Stokes case, this singularity is not a physical singularity. And physicists do not care about this singularity. >> It it's purely for the math. >> It's purely a mathematical question. Okay? Physicists do not care about the millennium problem in Navier Stokes because fluids are never going to achieve an actual physical singularity. There are plenty of cases where the Navier Stokes equations don't even apply. And I've got I've got some examples over here. I mean, for example, um hypersonics and supersonics even Navier Stokes equations don't apply.
2:08:54Navier Stokes equations are both of those equations. One of the key ones is that second one, the incompressibility, the fact that the density is always >> constant. Well, when you're approaching the sound barrier, all of a sudden air becomes compressible. Right. Right. the the sound wave is catching you are catching up to the sound wave in front of you and that is going to compress the air when it comes. So that's the bottom there where you get a sonic boom and you get like a cloud because like literal condensation forms around your aircraft. >> Hypersonics is when you're going so fast that the chemical nature of the air starts mattering and you get thermal effects. That's not in the Navier Stokes
2:09:34equations. um in the middle that's an aquaporin protein. Those things are so small that they let in only water molecules. At a molecular scale it doesn't velocity fields don't matter at a molecular scale. Like what are we talking about? Right? So again at the molecular scale Navier Stokes equations don't apply and for a lot of weather patterns even Navier Stokes equations at least the ones that we have up top here are not the ones that are used. We use a compressible version of Navier Stokes equations where the air can expand and contract and change densities. That's how we predict weather. The Millennium problem which is the mathematical
2:10:16problem has to do with these two specific equations >> which are specifically for incompressible fluids. >> Yes. And so I would even say that the Millennium problem has nothing to do with fluids. M >> it has to do with a vector field that is governed by this equation. >> It has to do with an object, a mathematical object that is a vector field that in normal circumstances would describe normal fluids. But if you're trying to hunt for singularities, you have left the real world and you have entered the world of mathematics. And it's totally legitimate >> 100%. >> Right? And but but it's something that is outside the realm of physics. And I want to make that very clear because
2:10:57there's a lot of halaloo and I think there's a lot of like stories out there being like oh this is like fluids right can a fluid achieve singularity no actually the answer is no right infinite speed again Einstein would be rolling over his grave you can't have infinite speed because as like if you have a fluid right that is like going faster and faster and faster then all of a sudden Navier Stokes better account for relativity and the fact that the mass of the fluid >> like that density term is going to have a lorren factor in there or something because the the mass of the fluid is going to infinity the closer I approach the speed of light, right? I can't go to infinity. So, all I'm saying is this is a mathematics question, completely
2:11:38legitimate, but let's not confuse this for something that is physically relevant. It is not. And I I think we've now built up a full understanding to be able to specify what exactly the Millennium Prize problem as it relates to Navier Stokes >> is specifically asking >> about incompressible fluids >> and sort of do you get blow up in finite time of an incompressible fluid and in this context what we're saying is we're hunting for singularity. in these mathematical constructs that don't apply to the real world but still have a
2:12:20fundamental value to our mathematical understanding of the world of just ma maths >> which again then can bleed into like I was mentioning earlier to ways to think about other problems because it's it's this human understanding that creates look at how we went from Newton's >> second law of motion >> to the Navier Stokes equations because it created a framework and we said, "Can we apply apply this framework to fluids?" Yes. And then it created this whole centuries long journey. >> Yeah. I mean, all a lot of the stuff that led up to the Navier Stokes stuff had to do with just analyzing what vector fields are like and what it means to do calculus on vector fields. Like
2:13:00things like Stokes theorem. That's from the Navier Stokes guy, right? That's a that's a mathematics theorem, Stokes theorem that like then oh, it like works here, right? Um, so definitely something that is super interesting. And Terry Tao actually said that like the never So, >> oh, we're calling him Terry now. Look at this guy. Oh, my buddy Terry. >> We need to edit that out. There you go. If he if somebody All right. Terrence Tao, what? No, no, no. Keep it in. But like Terrence Tao, he said, um, the Navy Stokes equations are the are one of the simplest supercritical differential equations. So in the theory of differential equations, it's very interesting to think about because you've got this nonlinear term and
2:13:41you've got this competition with the diffusion term, right? You've got a lelassian and you've got this like convective thing and they're competing and mathematically it's a very interesting question to answer. >> Mhm.