What OpenAI Actually Did to Navier-Stokes (EP 58)

0:00 But in the Navier Stokes, and that's why it's so interesting. Because the Navier Stokes has this non-linearity, but it also has this smoothing term. It has this counterforce, which is the viscosity. And so you're asking, who's going to win? And are they always going to win? There are questions about credit, commercial incentives, accountability.

0:18 Yes. But I do believe that the risks deserve real attention without being mired in those other things. Yeah. Those other things can be true and the risk can still be real. Okay, so just turn it off. Yeah. What does just turn it off actually mean? Yeah. Hello, Internet. This is your captain speaking, Lester Nare, joined as always by my co-host and our resident PhD, Krishna Chowdhury.

0:43 We are about to get into what will be our most requested episode in the history of From First Principles. OpenAI announced a solution to one of the Millennium Problems. Many of you have either seen that announcement or all of the fallout and drama problems. as it relates to AI that has arisen since that announcement.

1:05 And so a quick overview of how we're going to structure this deep dive episode. We'll be lucky to get it under three hours. So we're going to do something a little bit different than some of the mainstream coverage that starts with the drama. First, we're going to ground ourselves in the fundamental understanding of the mathematics as it relates to the OpenAI solution.

1:26 What are the Navier-Stokes equations and what is the existence and smoothness problem? And how exactly did they solve it? And then we will obviously move into some of the drama and tragedy that has arisen as a result of this and other progression in AI over the summer into the early fall. This is going to be a nuanced episode.

1:51 We're going to talk about the science and some of the larger complexities. If you want to understand science headlines, be sure to subscribe. As always, we are going to talk about the science from the ground up today because this is From First Principles. It's not a question of if, but when. All of the people that have been paying attention to what AI has been doing in mathematics, that is what they are asking.

2:29 When is AI going to solve one of the six open millennium problems? There's six of them that are left. They're kind of like the infinity stones. And I'd say AI is kind of like Thanos, you know? AI is coming. And the question was always, when is he going to get the first stone? Well, it seems he might have got it.

2:48 Okay. The Navier Stokes existence and smoothness problem might have been solved. And naturally we were on vacation. I was in beautiful Lake Geneva in Wisconsin. Yeah. And somebody forwarded me a tweet and this is what I was feeling. I made a meme about it. You know, we're, we're also anticipating GTA six.

3:06 Yes. And this is the, ah, here we go again, because honestly, I didn't, I didn't want to keep talking about AI and math. We'd already done the Jacobian conjecture. We'd already done the Riemann hypothesis or a problem related to that, both by anthropic, but this, this, this is history. If it's true, you know, This is a big deal.

3:28 And it's not anthropic this time. It's open AI. And the announcement was met with a different kind of reaction than the other two. In the other two, there was a reaction of, oh my gosh, AI is so good at mathematics. This is incredible. There's a bit of an existential crisis in mathematics and so on and so forth.

3:48 But here, there was also something else. There was a fight. There were allegations of academic misconduct. There were questions about who deserved credit. There's questions about what OpenAI proved themselves and what they maybe stole from other people's work. That's someone else being the NYU mathematician Tristan Buckmaster and anthropic researcher Levant Alpoghe, who was the hero behind the Jacobian conjecture a few episodes back.

4:18 So there's a lot to unpack. But before we get into any of that, as you said, we're going to do something very deliberately. We're going to talk about the mathematics. Because at the end of this two-page document that Buckmaster published, which really kicked off the controversy, this document where he talked about all the shenanigans that had happened behind the scenes, he said, He makes the argument that if OpenAI has really solved Navier-Stokes, then they should just say so.

4:43 And he'd rather be discussing the mathematics than anything else. And honestly, same. For me, same. So that's where we're going to begin. We're going to do the math first. We're going to do the drama second. So here's the roadmap for the episode. We're going to start from first principles and try to build up the Navier-Stokes equations.

5:01 They're a set of two equations. And at the end of that segment, I want you to have an intuitive understanding of every single term. in that equation. So it's not just a bunch of words and fancy symbols. At the end of that, we'll confront the actual millennium problem, this question of existence and smoothness.

5:18 There is going to be some calculus involved. There's going to be some equations, but I have a lot of visuals. So hopefully it's not going to be that bad. And we'll describe it for our audio listeners. But I do suggest this is a very visual heavy episode, as most of our episodes are. Then we'll take a tour through the mathematics over the last several decades and get to the main event, which is OpenAI's 165-page proof.

5:42 The claim is that they have solved it. Now, I am not a fluid dynamicist. And I'm certainly not an expert in the kinds of linear partial differential equations that the Navier-Stokes problem is. This is a problem that has attracted some of the greatest mathematicians of our time, including Terence Tao, among others.

6:02 So I'm approaching this from a perspective of a physicist who's trying to understand fluid mechanics. And so I went back to our trusty old Landau-Lefschitz course in theoretical physics, volume six, Fluid Mechanics. Landau-Lifschitz goes through some amazing, amazing mathematics here. And Navier-Stokes is in the early chapters because there's so much more to fluids than just Navier-Stokes.

6:26 Shout out to my professor, Robin Bruinsma, who taught me this course. It was an amazing course where we got to apply some of my favorite mathematics, i.e. complex analysis, to calculate the lift on a wing of an airplane. using a complex integral, which was one of the most mind-blowing things that you could use complex mechanics to talk about real things, not just prime numbers, as we've been known.

6:53 So at the end of all that, we're going to be understanding the solution, and then we'll talk about the drama, and then all of the existential stuff about what all of this means. Because we're going to have to zoom out. This is not happening in a vacuum. And there is a lot of crazy developments that are honestly happening daily.

7:12 It is really happening daily. And once Krishna has kind of walked us through the mathematics, I kind of want to look at the larger world that this announcement is, happened in. There is legitimate disputes over the credit and how we think about acknowledgement in scientific discovery and understanding.

7:31 And this is also happening in the context of an already existing and accelerating battle between multiple stakeholders as it relates to this issue. You have the frontier labs, you have the investor and venture capital community You have politicians beginning to finally weigh in on this. Obviously, you have the general public who, for many people, don't really see real-world impact on this in their everyday life.

7:58 And for context around some of these AI tools, I personally have been a user of them lately. just as a technologist and someone who's interested in these subjects in the way that we cover on the show, since GPT-2. And I've seen what the progress looks like from a very direct personal context. And I do think that there is a challenge in discussing this issue right now because you have to fit into one of two boxes, which is this is all hype and this is going to cause human extinction.

8:33 Yeah. Yeah. And in my view, there's a lot of space in between this is all hype and this is going to cause human extinction. And I would love for us to be better able to have conversations within that gap. And so we're going to talk through some of the larger context. to help frame this because this will be a technology, a transformation that will be with us for the foreseeable future.

8:58 And I think it's going to be important to look at these two things. And we'll look at some concrete examples of other things that have happened, like the hugging face incident, where there were some issues with open AI's models as it relates to the company hugging face, because I think that'll help, Really ground us around this shift from this is fancy autocomplete or this is just a better way to do search.

9:22 And this agentic world that has been actually a driver behind how OpenAI works. got to this solution. So we're going to get into all of that. A real brief moment of housekeeping. For those of you who may be joining us for the first time, welcome to the best science show on the planet. You can find us on all platforms where you listen to your favorite podcasts, as well as social for clips.

9:47 Our website has all of the research papers, background material, and other ways to look into what's happening in Frontier Science Research at ffppod.com. Any engagement that you do on our videos on all platforms help us reach more people with what we think are some of the most fun and fun things. and informative conversations about science.

10:08 And we have some great merch. So with that, we're going to jump right in to what is the meat and bones of what's going to be, I think, a very fantastic episode. Yes, and we will start with the mathematics. So these are the Navier-Stokes equations. There's two of them. And our goal is to understand these equations.

10:26 These equations are an equation of motion that describes fluids. How fluids behave, what are the forces that they exert. If I know what the fluid is doing right now, can I predict what the fluid will be doing at some later time? These are the equations that allow that to happen. And when I say they're an equation of motion, that's what I mean.

10:48 They're like, they're telling you how the equation, how the fluid is going to move around in some space. Now, in classical systems, the starting point for all equations of motion is always Newton's laws, okay? Newton's first law is an object at rest stays at rest, and an object in motion will keep going unless it's acted on by another force.

11:10 So, you know, if you're in empty space and you just have a satellite that's moving in one direction, it's going to keep moving. What happens when there's a force? Well, when there's a force, there's an acceleration. It's not a velocity. It's not like you need a force to keep it going. If it's going, it'll just keep going.

11:27 Okay. If there's a force, it'll change how it's moving. So that's called an acceleration. It's a change in velocity. And the relationship between the amount of force that I put on an object and how fast it's going to accelerate is dependent on the mass. Mass is a kind of inertia. It's like, how much resistance to a force do I have, is literally what mass is.

11:52 And there's a pretty common demonstration that you see in physics classrooms in high school, where you've got two rubber bands, they're the same rubber band, but one of them is attached to a block of wood, and the other one is attached to two of the same blocks of wood. And you can see that the single block of wood accelerates a lot faster than...

12:10 than the one with two blocks of wood. That's because the two blocks of wood has twice the mass, so it's going to have half the acceleration. There's Newton's second law. And Newton's third law is something about conservation of momentum, which is like, you know, if I'm pushing on an object, then that object pushes back on me.

12:26 So those are the three laws. And if we want to understand how something moves around, all we have to do is figure out what is the nature of the forces on this object and, And if we're good enough at the math, we can figure out how this object is going to move forever. This is partly what enables us, for example, to land on the moon.

12:45 Everything. Or Mars, all of these kinds of, obviously, much more surface area than that. But this is why we can make predictions over such long periods of time. Yeah, really, in order to get to the moon, you just need Newton's laws. I guess you need electricity to work the capsule or whatever. But if there was some analog version where you had fluids that did computation or something, you could land on the moon with just Newton's laws.

13:09 So let's take, for example, the mass on a spring. I want to go through a singular example of a very simple problem where I can have an equation of motion derived from Newton's laws, and I can then figure out what that thing is going to do forever. A common example is a mass on a spring. A spring is a substance that obeys Hooke's law, meaning if I pull on the thing, it's going to resist and it's going to pull back.

13:34 And how much it's going to pull back is dependent on how much I'm stretching it. If I stretch it twice as far, the pullback, the force backwards, is twice as much. That's called Hooke's Law. And you usually see it as the force is equal to negative kx. k is how stiff the spring is. x is how far I've pulled.

13:54 And so, you know, the larger I pull, the more it pulls back. And the negative is because it's pulling in the opposite direction of how I'm distorting the spring. Okay? Mm-hmm. So that's Hooke's Law, f equals negative kx. Now, force, according to Newton's second law, is mass times acceleration, right?

14:10 And the acceleration is really a second derivative of the position, of the x, right? The acceleration is a change in velocity, and the velocity is a change in position with respect to time. So it's really a double change, which is why I have mass times a second derivative in position is equal to negative kx.

14:30 Notice, this is an equation of motion. Mm-hmm. I've got a single variable that I need to solve for, which is x, the position. And the key thing is, the force is dependent on the position. But the force is related to how fast the position accelerates. So now I've got an equation that is in a single variable, x, and it's a differential equation with respect to time.

14:54 This is something I can solve. And that's what most people do. You can solve this thing. It's a differential equation. And if you solve it, you get a very nice solution. Right? you get a very nice solution that is an oscillation. And I think we've got a little video of that. This is what happens. You take your equation of motion and you say, what is something where if I take a second derivative, I take a derivative once and I take the derivative again, I get the negative of the thing back.

15:22 And that's cosine and sine. The derivative of cosine is negative sine. The derivative of sine is cosine. So I'd go from cosine to negative sine to negative cosine. Well, that's just the negative of what I started with. And there, I have an oscillatory motion. So this is the paradigm of how we do physics in general.

15:40 We have Newton's laws. We analyze our system. And we figure out what is the differential equation that the system obeys. And then we solve this. And if we're lucky enough, the differential equation is simple enough that the mathematicians have figured out how to solve this differential equation. And the physicists can then borrow that intuition.

15:59 And boom, you've got a solution that tells you, for any time, I can plug in the time and I can tell you where the mass is going to be. And it's a way in which we can now simulate and create replications such that we can predict how a system will act under certain circumstances. Yes, exactly. And you can make things even more complicated.

16:21 For example, you can take the same spring and you can add friction. In this case, there's like some kind of damping force. And what that means is that in your differential equation... you've got a term that's related to velocity. Because the faster I move through some... Let's say I dunk this thing in honey.

16:39 The faster I move through this thing, the more the honey is going to resist my motion. So that is dependent on the first derivative. of position, just the velocity. And if I add that term, then I get a new solution, which is the damped harmonic oscillator. And that's the red that you see over there. So in the undamped case, you've got an oscillation that just bounces over and over and over again.

17:01 And then in the damped case, you've got an oscillation that dies out very quickly. For those of you who are listening to us in the car, you can thank damped harmonic motion for your suspension because there's a particular value of the damping that makes all these oscillations go to zero very fast. It's called critical damping.

17:19 And a lot of times car manufacturers want their springs, the suspension in their springs, to have critically damped so that you get bounced on a... What are those called? The holes? The potholes? You bounce on a pothole, but you don't bounce up and down forever. You just immediately bounce and then you go right back.

17:35 And there's no bouncing whatsoever. And that's because the spring is critically damped. That has to do with how much damping is happening. My point is you can make the... equations of motion a bit more complicated, and still be able to solve stuff. This is the visual I think of is seeing those meme videos of those giant trucks with giant suspension.

17:55 That's visible going over, you know, this hump dirt, you know, dirt track. And they're chilling. Yeah. Yeah. Yeah. There's all these like Chinese propaganda videos too, where they show like the luxury cars of China versus the luxury cars of Germany. And it's like the luxury cars of China are just like, it's, they're like on a cloud.

18:13 And then the ones like the BMW is like, it's like, okay, well, is that a new BMW or like, You never know with those propaganda videos. Or is it just straight up AI? Anyways, for every case, the logic has been pretty simple. The point is we start from Newton's laws. We understand what the forces are. And then we try to set up the equations of motion.

18:31 This works for springs. This works for planets. For a planet, the force is proportional to 1 over r squared, how far away my thing is. and it's pointing directly at the central body. And again, that's a force that is a function of position. And now I can solve for what the position of the planet is because I can set up a differential equation that does so.

18:53 Okay, I can set up a differential equation for a three body problem, but crucially I cannot solve that. So I can set up the equation of motion, but I don't have a solution for it. So there's caveats, right? Okay, so now let's try to understand fluids. I see the setup we're going with here because we're saying we can understand the world around us and make predictions using math.

19:15 Yeah, and Newton's laws. And Newton's laws for certain physical systems. Yes. And we just defined examples of those physical systems in which we can do so. Yes. Now, I think where we're going is fluids maybe have some unique aspects that make that different. Yes, it makes it different and difficult. Okay.

19:38 Okay? It's different in the... So, first of all, it's same, same a little bit because each part of the fluid still obeys Newton's laws. Yes. If I were to look at a single... particle of fluid, let's say I drop a dust grain into a moving river, right? That dust grain is going to experience forces because water molecules are pushing on the thing.

19:58 And those forces are going to make it accelerate. Okay, so Newton's law is still a work. 100%. Right? The problem, though, is that Newton's laws are tenable and they're, like, tractable if it's a single thing that's, like, moving around. In a fluid, for example, even in this cup, right, of water, I have 10 to the 3 molecules.

20:21 No, no, no, it's not 10 to the 3. That's only 1,000. 10 to the 23 molecules of water moving around. Okay? So what? Do you expect me to apply Newton's laws to every single molecule that's in this water? It doesn't make any sense. Sure, that's one way of doing it. If I had infinite compute, which I don't.

20:40 I'd like to understand this thing at a level of granularity where I can make predictions, but maybe I don't have to worry about every single molecule and what it is doing. We don't want to have to simulate everything because the system is so large as compared to what we were doing before. Yeah, the system is large and it's self-interacting is the other point, right?

20:58 The water molecules are interacting with other water molecules. And like, it's just a whole mess, right? So it's not a new kind of physics. It's the same physics, but it's... being applied everywhere in the system. Everywhere all at once. And all at once. Exactly. And so we need a new language. Okay.

21:17 And perhaps here's the point. What do we want to actually describe about this fluid? We don't want to describe the individual molecules, as I just said. It would be crazy to try and describe the position of every single molecule as a function of time, like we could with that mass on a spring, right? That doesn't really make sense here.

21:36 So instead of tracking individual molecules, we are going to track the velocity of the fluid itself, okay, at different spots. Okay. And it turns out this has all of the necessary information that we need. Interesting. Okay? This is what we're doing. On the left-hand side, you see a video of sort of dust particles, let's say, that are in a chunk of a river.

21:58 Okay? So we can see the dust particles that are moving from left to right. Mm-hmm. Now, I don't want to worry about all of the dust particles. So instead, what I worry about is what is the velocity of the particles at that point. And those can be represented by little arrows that just represent where the dust particles are going at that point.

22:18 Now, crucially, the dust particles are moving. But this velocity field that I've made of a bunch of arrows that are representing particles, all of the velocities, that thing is stationary. You see? Because it just represents, this is the direction in which I'm moving. Okay? This is the central element that we are worried about.

22:38 Okay. Okay? This is a velocity field. It's a field because it has a value at every point in space and time. and it's a velocity, so it's got a vector. It's a vector field, okay? And usually this thing is represented as a U. We don't want to use V for some reason. I think it's a historical reason. So in every case, the U that we're going to see later on in this episode, that is a velocity field.

23:03 It's a vector field everywhere in space, and it represents what the velocity is at that particular moment. So if I have a velocity vector that's pointed in this direction, that means water is moving past in this direction. If, on the other hand, it's over here, towards that arrow, towards where the arrow is pointing.

23:20 And the component parts that make this velocity field are... Some position in 3D space, which is our X, Y, and Z. Yeah. So we know like where in this cube, the velocity field that we're specifically referring to exists. And then time allows us to create the arrow of time. Yeah. Of the direct, like at that point.

23:39 At that point, it's moving in this direction. Yeah. Yeah. Right. And it could change, right? Maybe I introduce the flow and then it starts moving in some other direction. Right. And so the velocity field can change. This is a very simple case, right, of just like particles, water moving from left to right.

23:53 There can be other more complicated cases. For example, this is the velocity field of a tornado. If I were to look like up the axis of a tornado, it's rotating around. And so imagine if you're looking like up through the eye of the tornado, you would see particles and clouds and everything moving around.

24:13 around you, right? Like in a circle above you. And the velocity field for that looks like a bunch of arrows that are sort of circulating around. Okay? And the color of these arrows represents how strong the velocity is. Okay? So the vector is big near the eye, and as you move farther and farther away from the axis of the tornado, the vector field diminishes in magnitude because things are moving slower.

24:39 This makes sense. This is why when you have a tornado or a hurricane approaching you, if you live on the coast, the outer parts, you can feel it coming. Yeah. Yeah. Because the outer parts are a little bit slower than the inner parts. Yeah, when you get to the inner parts, then it's like, okay, then the palm tree is like... On the side.

24:53 On the side, right? And so this is the basic object in modern fluid mechanics. It's not the velocity of water, as though the entire ocean is a single baseball or entity that's just moving as one, right? No, it's different parts of the fluid are moving in different ways, and we can represent that using these little arrows that are in every single part of the fluid.

25:14 that's actually a really important mental picture for me to kind of reframe how to think about it because my initial, um, mental model is this is one discrete thing and we're trying to look at it as one discrete thing yeah and what we're sort of saying is in order to get back to this idea of we want to be able to make predictions about a system uh we need to look at it from this different vantage point which is uh velocity fields which help us look at a points in space and the velocity at those points in space which then are also going to interact with each other over time exactly and Yeah, and so if we wanted to, for example, calculate what a dust particle is doing in that velocity field, all we have to do is follow the arrows.

25:58 We can still recreate... What we want, which is like if I were to drop something, what is the position of that thing, right? All we have to do is follow the arrows. But the velocity field is something that holds all of the relevant information. To be able to do that. That's the point, yeah. And it becomes something that is a tenable object.

26:14 It's something that we can understand and we don't have to worry about the 10 to the however many stuff that the fluid is made out of. It's less compute constrained for lack of a better way to put it. Yeah, yeah, yeah. It's one way to put it. Yeah, that's certainly one way to put it, yeah. Let's try to understand this velocity field a little bit further.

26:34 I just showed you two examples. The first one was water is moving from left to right, so all of the arrows are pointing from left to right. Or the tornado, where the clouds are moving in a circle above you, and so all of the arrows are pointing in kind of a circle, right? They're like circulating. In both of those, we saw the particle picture, and we were like, okay, that seems reasonable.

26:55 Now, I can imagine a fluid doing that, right? I can imagine the tornado, and I can imagine stuff like this. Now, are there velocity fields? Are there setups of arrows that we can make up that don't make any sense in the real world? Okay? The answer is yes. Let's look at these cases. Here are two cases.

27:14 The first one on the left... is a bunch of particles that are just like emanating from the center. The expansion of the universe. Yeah, it kind of looks like the expansion of the universe. Exactly, right? It's like all of the fluid is coming from somewhere in the center, and then it's just like expanding out.

27:31 Okay? How is that possible? Where is the fluid coming from? Yeah. Right? That's not possible. Just like in empty space, it's just like there's like just stuff. Right. Like what is the source? Right. Like if there was a hose, if I saw a bunch of particles come from the bottom and then there was like a sprinkler head or something and then it moved.

27:50 Then I'd be like, okay. But here I don't see that. It's just the things are just coming out. I understood. Right. Doesn't make sense. Okay. Okay. First thing doesn't make sense because somehow the fluid is being created in the center and it's just moving out. Right. Okay. Second one on the right. All of the fluid is moving into a single sheet, that yellow sheet.

28:10 It's like a sheet of paper. And the fluid is moving from the top into the sheet of paper. And the fluid is moving from the bottom into the sheet of paper. How is that possible? Where is the fluid going? Yeah, right. Right? Right. Because you would imagine it should continue. I mean, if I saw it shoot out to the side, then I'd be like, okay, that's possible.

28:29 But I'm not seeing that. It's hitting the surface of a wall or something. Yeah. In that vector field, all of the arrows are pointing down, and there's no arrows pointing out. In the vector field on the left, all of the arrows are pointing out, and there's nothing that's supplying the fluid to have those arrows point out.

28:44 And so just to be clear, we're sort of using these as sort of like an abstract theoretical or mental model to test the... Four corners of the sandbox of how to do this. Yeah. We're trying to understand, like, we've created this concept of a velocity field. Yes. Now, is any arbitrary vector field okay?

29:05 Right. And the answer is no. Because on the left-hand side, that's an arbitrary vector field. Like, if I were to create those arrows that are just emanating from the center, if that was an electric field around a positive charge, totally fine. Mm-hmm. But for fluids, that's not fine. It doesn't work for fluids.

29:21 And so we can formalize this with mathematics. The property that encodes this is something called the divergence of an electric field. Okay. On the left-hand side, we've got something that has a divergence of negative, negative divergence. That means stuff is coming in into a singular point. Mm-hmm. On the middle, we've got a positive divergence.

29:41 I think like the collapsing of a black hole. Exactly. Just for my own visual. Yes, 100%. On the middle, that's a supernova type. Like the positive divergence stuff is moving out. And on the right, we've got a zero divergence. Okay. Okay? The only types of vector fields that are allowed for fluids are the ones with zero divergence.

30:01 Okay? Because that means there is neither a sink nor a source. You can't create fluids out of nowhere. So... Whatever fluid you have, it's got to come from somewhere and it's got to be going somewhere. Okay? So this is a key constraint. For our vector field. For fluids specifically. For fluids specifically.

30:21 For other things, it's totally fine. Like, for example, if we were talking about the electric field, on the left would be a negative charge. In the middle would be a positive charge. Fine. Fine. Totally fine. But for fluids, that can't be possible. So I think this is an important point here. The idea is this concept of divergence applies differently based on the underlying subject matter that you're speaking to.

30:43 And so as we've gone through this progression over... of testing vector fields, it only can work when the divergence is zero. Is zero, yeah. Any non-zero divergence and it doesn't make sense. Another way of putting this is something called incompressibility. You cannot compress the fluid. Okay. Okay.

31:01 Those are the fluids that we're talking about. And if we go to the next... Slide. Those are our two Navier-Stokes equations. You can already see that we've already got one of them. Okay. Okay. The second one is that the divergence of this fluid velocity field is zero. Yeah. That's already the second Navier-Stokes equation.

31:17 Okay. So we're already, you would say, halfway through. Okay. And can you remind me again, we have our, I guess, delta symbol and our U symbol equals zero. That's what you're referring to. Yeah, that's the second one. The second one there. You would say the divergence of U is zero. Right, which was that diagram where we saw the center point, everything moving in one direction.

31:37 Everything moving in one direction. Stuff came in, stuff is going out, and everything is fine. You're not creating or destroying fluid out of... I don't know where. Yeah. And for engineers, you might you might understand this with the continuity equation. Like, for example, this is applied very directly if you've got a pipe that has a large cross section and you've got water coming in and then the pipe like narrows into a small cross section.

32:05 For our audio listeners, imagine a wine bottle on its side. Yes, exactly. Wine bottle on the side, but its base has been cut off. Yes. And then now you've got water or wine coming in from the base, right? As it exits the nozzle or as it exits the tip, it's going to be moving faster because all of that fluid has been jammed in.

32:23 You cannot create or destroy fluid. So the amount of fluid that's coming in has to equal the amount of fluid going out. But that means that if I've got a larger cross-section, then in a single amount of time, I've got some amount of volume coming in. And with a smaller cross-section, that same amount of volume has to go out, which means the velocity has to increase.

32:40 This is the continuity equation in action. And a mathematically succinct way of putting it is that del.u equals zero. The divergence of the vector field is always zero. And it kind of makes intuitive sense for someone like me, which is when I look at the sort of the area of the cross section in the wider part, there's just more space for stuff to move.

33:02 And we have to move the same amount of volume through a smaller space. Yeah. So it necessarily needs to accelerate right at that point in order to for the divergence of you to equal zero. Yeah. For that fact. To be true, you have to account for this difference in velocity. Right. Now, crucially, this is an independent thing from that first Navier-Stokes equation, which is going to be the topic for the next segment.

33:27 Okay. This is an independent criteria. Okay. If we didn't have this, you could have different types of fluids. You could have, for example, compressible fluids. Like air, for example, is kind of compressible. In the sense that if you heat it up, it's going to expand, and if you cool it down, it's going to contract.

33:46 Now, that's not a divergence-free fluid. The Navier-Stokes equations, these two, have to do with fluids in the sense of water that is really not compressible. The density is the same everywhere. You can't adjust the density. So a key takeaway for – one of the key takeaways for this is the concept of this idea of incompressible.

34:05 Yes. Which we've just defined from mathematical first principles. Yeah. But it really just means you can't – you know, you can't pack stuff in. Right. The density is always constant. Right. Right? It's like one of those – have you seen those like stress thingies? Yeah, the little stress ball. Yeah, where you like squeeze it and then the thing is going to like shoot out somewhere else.

34:24 That's an incompressible fluid. Right? Because if I squish it, it's got to go somewhere else. Mm-hmm. Right. Okay. It's not like one of those foam things where you can just compress it. Right, which is actually an important distinction. Okay, so this is helpful because that's one half of what people say when they say the Navier-Stokes equations.

34:43 And it's going to be the basis by which I'm guessing we're now going to build an understanding of the second equation. Of the second one, right? We've sort of gotten an intuitive understanding of what types of velocity fields are allowed, first of all. Right. And kind of what the velocity field represents, which is a bunch of these arrows that tells you how the velocity…

35:05 is at that moment in space and time. In a fluid. In a fluid, right? In an incompressible fluid. In an incompressible fluid. Yeah. Okay, so now let's try to apply this velocity field to Newton's laws. Okay. Or the other way, actually. We want to apply Newton's laws to this velocity field. Now, that's easier said than done.

35:23 The challenge now is how do we talk about Newton's laws, which has to do with individual particles in the fluid, right? It's like this water molecule is getting forces from other water molecules, so it's moving around. Newton's laws has to do with the stuff. It doesn't have to do with the velocity field.

35:39 So we have to now put Newton's laws in the language that we've just invented. And just to take a quick step back, just from my brain, part of, like, it's like, you know, if you have a bathtub with a little boat, toy boat in it for a kid, right? We talked about Newton's laws in the context of, like, the boat in the water.

36:01 Yeah. Yeah. And then when we talked about velocity fields, it's like the medium of the water. Yeah, it's the water itself. It's the water itself. And these are different. So it's like the discrete object versus the medium in which an object or many objects may exist in. Yeah. And the lens in which we look at them is from different doorways.

36:18 Yes. In the first one, I could easily just be like, what is the position of the boat? Right. Oh, it's like right there. Right. Right. What is the position of the water? What do you mean? Which water? Yes. There's an Avogadro's number of water molecules that we're talking about. Right? So instead we have to reinvent this new language of the velocity field.

36:36 And now what we're going to do is build this step by step. We're going to apply Newton's second law, which is force equals mass times acceleration. We're going to apply that to our velocity field. The first thing we're going to do is try and describe what is the acceleration of a fluid particle. Okay.

36:53 Like in this velocity field. Okay. Okay. Okay. And we're going to use traffic as an analogy. If you live in LA, you very much are intuitively familiar. Yes. And you're probably listening to this while you're in traffic. Okay. So traffic is kind of a nice analogy for fluids because you've got these working agents and in traffic, i.e.

37:21 the cars, right? The cars can be the individual fluid particles, and the highway is like the receptacle, and the entire flow of traffic is the fluid, okay? The challenge now is that we want to have a description for how the individual cars accelerate, or de-accelerate if they brake or if they press on the pedal, based on the stuff that we can see from a news helicopter.

37:45 Because the news helicopter is up there, and it's seeing the velocity field of these cars, right? It's seeing, like, oh, over there, cars are moving fast. Over here, the cars are moving slower. And, like, that's how they, like, if you're on the radio, they tell you, oh, the 101 is backed up from Sepulveda, blah, blah, blah, blah, blah, right?

38:04 Um... So they're actually watching the cars, but they're watching an aggregate mass of cars. It's not like they're following a single car and seeing how stuff is happening. So what we want to do is describe the acceleration of a single car based on what we see. from a chopper, right? Now, first, let's think about how cars can accelerate.

38:25 Cars can accelerate and deaccelerate as a response to the traffic itself changing at a specific spot. For example, suppose you're in the chopper and you notice that everyone is moving at like 80 miles an hour, right? It's a nice day and everyone's moving at 80 miles an hour and all of a sudden it starts raining, right?

38:44 I was going to say 80 miles an hour in LA never happens. Yeah, yeah, never happens. This is an unrealistic example. Yeah, yeah, yeah. That's totally fair. Anyways, that's what I'm stuck with in the visuals. So we got 80 miles an hour, and then all of a sudden, it starts raining. Okay? And within 10 seconds, everyone slows down to 60 miles an hour.

39:03 Obviously, the cars have deaccelerated, right? They've used their brakes to slow down. This 20 mile per hour reduction happened in 10 seconds, let's say. And so the cars accelerated, de-accelerated at negative two miles per hour per second. They're slowing as a response to what the traffic itself is doing.

39:24 The entire traffic slowed down. And so we slowed down. It's kind of like seeing, like, you know, in the traffic map of LA, everything was green, and then rain started happening, everything turned yellow. Well, that means all of the individual cars also deaccelerated. So that's one way that you can deaccelerate.

39:42 And this is called the Eulerian perspective, because it's asking how the field itself is changing as a function of time at a given spot. And this is represented by the derivative of the velocity field with respect to time. This is... It makes sense, right? It's a velocity field. If I take the derivative of velocity, I get acceleration.

40:01 And there you go. This is an acceleration of my particles, right? In this case, the thing is slowing down, but you can easily imagine after the rain ends, the things go back to 80 miles per hour, and it's going to speed up again. This is represented by a derivative of my velocity field. And so the idea is, here's a way to describe the velocity field as...

40:23 as this system unique and different from the discrete individual car in traffic. Yeah. Yeah. And we can use the description of the velocity field and how the entire field is changing in time to describe how an individual car is experiencing acceleration. That's the key, right? So good. We've got at least some of acceleration down, but that can't be the whole story.

40:44 Okay. That cannot be the whole story. For example, let's take a look at this airfoil. This is a airplane wing. where the air is coming in from the left and it's going over the wing and under the wing and then to the right. Okay? Now, crucially, on the lower right, you're seeing the velocity field representation of this airplane wing.

41:05 Mm-hmm. Okay. So the arrows above the wing, the velocity is very high because the air is like getting pushed up and then there's high pressure. So it's like, these are the red arrows. Yeah. Those are the red arrows above the wing. So there's higher pressure above the wing, lower pressure below the wing.

41:20 And so there's slower fluid below the wing. And just the length of these arrows is meaningful here in that there's more. Yeah. It's faster. Faster. Yeah. And okay. I'm already seeing where this is going. Okay. Okay. So this is the, on the left is your particles that are moving. Right. Like our flow of traffic.

41:39 Yeah, our flow of traffic. And on the right-hand side is the velocity field. But if I were to take a derivative of that velocity field, if I were to ask, how is this velocity field changing with respect to time, the answer would be zero. Hmm. Because this is a static velocity field. If you go back to that picture that I had with the tornado, even though the particles were circling around, the velocity field was static because in this point, the velocity is this way, and then as it goes around, it's like coming around, right?

42:12 And so what? Does that mean that the particles are not moving around? Another way of putting it with the chopper idea is like, you know, you have, let's say, an accident over here. Okay? On the 101. Way behind, way ahead of the accident... the speed is very high. But as you approach the accident, it's very low, right?

42:38 But the accident isn't going anywhere because the police are very far away. It just happened. And so the police haven't gotten there. So the accident is not going anywhere, which means that ahead of the accident, the speed is very high. Before the accident, the speed is very low. And that's not changing.

42:54 The traffic pattern is not changing. It'll still be red here, green here. If I come back five minutes later, because of how slow the response is, it'll still be red here and green here. So the traffic pattern has not changed in time. And yet... I think you would be remiss to say, yeah, the cars are accelerating.

43:16 Once you get past the accident and you're done rubbernecking, yeah, you're like, okay, I got to get on with my life. Like, let's go, right? Okay. And so I just want to make sure that I'm following this here because part of what we're trying to say is – We're trying to go from the velocity field and then say, can we describe an individual object within that velocity field?

43:37 And how it accelerates. And how it accelerates based on the mapping we see in the velocity field. Yes, exactly. And part of what you're pointing out here is there are contexts like the airfoil or the traffic accident example you just described where the velocity field… equals zero. And so that means you can't...

43:55 The derivative of... The velocity field is not changing. Sorry, right. So the velocity field, it's static. Yeah. It's not changing, which means you can... In some cases, you may not be able to accurately describe certain particles within that context. Yeah, using just the time derivative of the velocity field.

44:14 You have to figure out something else about the velocity field that's doing this. And let's just think about this. I think we've got in the next one, you know, the highway slowdown. Let's say there's a – let's say there's – Next one here. And the next one, exactly. So let's say there's road work ahead, right?

44:29 Okay. A car is going to slow down. Now, all of the cars near the roadwork are going to be going slow, and the cars before the roadwork are going to be going fast. The roadwork doesn't change where it is. But what is making the car slow down? It's the fact that it is approaching the roadwork. There is a spatial difference between...

44:49 In the velocity field. You see? There's the time difference that we talked about earlier, where the rain came in, so everyone slowed down. People slowed down here if there's only a patch of rain here, and everyone else is fine, but this patch of rain, people slowed down. So that's a single spot. Stuff is changing.

45:06 But here, I am accelerating or deaccelerating because I'm moving through the traffic pattern. If I go from a red to a green, I'm going to accelerate in my map. If I go from a green to a red, I'm going to de-accelerate, even though where the green and red are, are staying exactly the same. The fact that I'm moving through it is making me accelerate or de-accelerate.

45:31 And so I think part of what you're trying to get at here is that the time derivative of the velocity field is not sufficient to describe the acceleration of an individual object. Yeah, it's part of the story. But it seems like the distance is potentially the other ingredient in the pie, at least based on the example that you just brought up.

45:50 Exactly, yeah. And that's a big part of it. It's the spatial difference. It's how the velocity field changes as I move from one space to another. You might think, well, what if I took a time derivative before? What if I take a spatial derivative of the velocity field? So that tells me how much the velocity is changing from one spot to another, right?

46:12 If I move here and then I move like 100 feet over there, if I've slowed down, that tells me how much I should slow down. That's not everything. Okay. Okay? There is a secondary effect. Okay. And to understand that, let's look at that scenario again where you've got road work ahead. Okay. Okay? Okay. But now I've got two different cars that are approaching.

46:32 The first car approaches that road work at 100 meters per second. Okay. Okay? And the second car approaches that road work at exactly half the speed, 50 meters per second. Hmm. I think you'll agree that the guy that is moving faster is going to have to break harder. Yeah. Yeah. Because he's going to come upon that road work way faster than the guy who's slowing.

46:57 He's going to have to be like, oh, there's road work. Fines are doubled. I better slow down a lot. So there's a secondary effect. It has to do not just with how fast the traffic pattern is changing. but also how fast the individual car is moving. Yes? Yes. Because, so, and again, trying to use the terminology again, we talked about the time derivative matters.

47:23 We talked about the spatial matters. And also the individual object's velocity. Velocity. The speed. The speed itself. is the third item in our cookie recipe here. Like all of these three things are relevant in order to be able to go from looking at a velocity field and describing an individual object within that velocity field.

47:50 Acceleration. Acceleration, exactly. And so for the second scenario, we have to take into account both the velocity and we have to take into account the change in velocity. Right. Okay? Right. The change in velocity as a function of distance is like how far away is the road work? Right. And my own velocity as I approach that also has something to do with how fast I need to break.

48:16 Yes. So it's going to be a multiplication of the velocity field times the spatial derivative. Yeah. That's the upside-down triangle. It's called Nambla. I call it del. But it's u dot del times u. Okay? This is called the convective term. And what this convective term is doing now, it's giving us in the example we just talked about here, it's giving us the 100 meters per second, 80 meters per second.

48:43 Right. Okay. Multiplied by the del u, which is the spatial delu, How much time do I have before the road work? I mean, how much distance do I have before the road work gets there, right? How much time do I have is dependent on how fast I was getting there. Yeah, yeah, yeah. Okay. Okay. And that's the okay.

49:00 Make sense? Yes. Okay. And both of these terms combined... give us our acceleration. We've got the Eulerian perspective, which is how is the traffic pattern itself changing? For example, is it going from green to red if rain came down or something, or if there was an accident, right? Then at a specific spot, that highway would go from green to red on our Google Maps.

49:24 And that's the first term. That's our velocity, the derivative of the velocity field. Yeah, with respect to time, right? With respect to time. Because the field itself is changing. The time derivative. The color is changing on the Google Maps. Right. Right? On the right-hand side, the Google Maps color is staying the same, but I am moving through.

49:41 That portion of the yellow or the red. Yeah. Yeah. And so I'm going to be breaking or accelerating based on that. And effectively a more granular way. There's like a bird's eye and then there's a zoom. Yes. And both of those matter. And that's why it's called actually the Eulerian perspective because that's kind of the bird's eye view.

49:58 And then the Lagrangian perspective is like the nitty gritty. What's happening there? And that's the conceptual trick. Okay. Okay. And. In the Navier-Stokes equations, we're writing Newton's laws for the person in the car, for the car itself, based on the data that's collected by the chopper. The chopper is only finding you.

50:18 The chopper only knows the green and the red and everything like that. It's seeing these aggregate... traffic patterns. But from those aggregate traffic patterns, I could tell you if a car is over there, how fast is it going to have to accelerate or deaccelerate? Does that make sense? Yeah. We're trying to, again, describe the acceleration or deceleration of an individual thing in an incompressible fluid from a vantage point that is only the vector field.

50:52 Yes. Yes. Which the point being the vector field in and of itself does not have enough to describe the acceleration of the individual object until you do all of this math. Until you do all of this. Yeah. Is that the right way to say it back? I think that is the right way to say it. Okay. Exactly. And so now we have an acceleration.

51:09 Right. Right? And all we need for that other part of Newton's laws, right, is F equals MA. We already have the A, the acceleration. Yeah. What is the mass? Well, the mass is just, for a fluid, it's like the density. Yeah. Right? It's just, I could do a... F equals ma per volume. And then just say, okay, density multiplied by acceleration.

51:28 That's my, that part of Newton's laws. That's my ma. Yeah, that's my ma. And that's exactly what we get. We've got the left-hand side of Navier-Stokes equations. Okay. The first Navier-Stokes equation. Right. Because density multiplied by acceleration. Right. Right. Because our acceleration is everything in this larger bracket.

51:46 Yeah. Which is what we just described. Yeah. Which is trying to describe the acceleration of an individual object from the helicopter. Exactly. Okay, got it. So now, again, we're trying to go, we're trying to use Newton's laws as a language to describe stuff that's happening in an incompressible fluid.

52:06 Yes. And we now have half of Newton's force times mass equals acceleration for an incompressible fluid with the description that we've just gone through. Exactly. And now I want to, before we even start talking about the other side of that equation, which is the force, I want to linger on this acceleration term a little bit more, especially that convective term.

52:25 Because that convective term, you might have heard that the Navier-Stokes equations are nonlinear. Yes. Right? Compared to linear differential equations. The Navier-Stokes equation is a nonlinear differential equation. That second term, the convective term there... has everything to do with that. Okay, so let's linger on that for a little bit longer because I want to show you exactly what all the trouble is.

52:46 And just to reiterate here, the convective term that we have on the right was when we were talking about the car in the yellow or in the red moving from the 100 meters per second to the 80 meters per second, describing the acceleration in this slice of the larger system. Yeah, and exactly. And crucially, it had to deal with...

53:05 its own velocity, and the spatial difference in the velocity, right? That's why it's nonlinear, because there's two. There's a compounding effect that is happening. Okay. Okay? Let's take a step back and let's look at, for example, Maxwell's equations in a vacuum. Okay. Okay? These are the equations of electricity and magnetism.

53:24 Okay. They're described, for example, Gauss's Law for Electric Fields... is that the divergence of the electric field is equal to how much charge there is. You know how before I was telling you that fluids are divergence-less? Yes. The divergence is zero. But I showed you those two examples where the vector field was going out or the vector field was coming in.

53:43 And that's because you can have negative charge and positive charge, right? That's Gauss's law for electricity, okay? It just says that del.e, the divergence of the electric field, equals to some charge. Now, crucially, the divergence of stuff is a linear operator. Meaning, if I've got one electric field and another electric field, and I want to find the divergence of the two of the electric fields put together, I can just find the divergence of one, and find the divergence of the other, and I can add them up separately.

54:11 So if I want to combine two electric fields, I just do the math for one of them, I do the math for the other one, and I combine them, and that's totally fine. This is why, for example, if I've got two charges, a negative charge and a positive charge, and I want to find the electric field everywhere, all I do is find the electric field due to one, the electric field due to the other, and I add it up.

54:31 Okay. This is also why telecommunications works with electric fields, right? Like in this room, we've got Wi-Fi, we've got Bluetooth, we've got radio, we've got our phones. We also have the light from the studio, right? All of these are electric fields. But they're all just moving through one another.

54:51 They're like ghosts that are just like moving through one another. And if I can tune my radio to one thing or the other, I can listen to one station or the other. Or I can, like, my Bluetooth isn't randomly, like, interfering with the Wi-Fi. Like, the Wi-Fi is coming from the router. There's a Bluetooth that's probably in my phone.

55:09 But it's not like the Bluetooth signal and the Wi-Fi signal is, like, crashing. right yeah they don't interfere with each other yeah yeah even like um in a radio right in the car like if i'm tuned to 91.5 kusc classical radio um the car is being inundated with all of the radio stations there's also 89.9 coming at me and kiss fm and also the am you know talk radio rush not sponsored we need our check yeah these are just some of my favorites that i listen to in la but like My point is the car is being inundated with all of these radio stations, but my radio can tune in into a specific band.

55:50 And it's not like the KISS FM radio signal is interfering with the KUSC signal, is interfering with Rush Limbaugh. Or he's not... I think he's still on radio. Yeah, he's still on radio. I think he's still on radio. He might be on YouTube now. Yeah, yeah, yeah. Or a satellite. But I think, so part of what you're saying is, like, I don't have to account for the other radio stations.

56:09 Exactly. It doesn't matter what the other radio stations are. And if I want to account for it, all I do is add it. If I want to know what is the electric field here, all I do is add up all of the contributions. It's totally fine. In fluids, we do not have that luxury. Okay. And that is because of that convective term.

56:26 That convective term is nonlinear. And this is what it does if I add up two velocity fields that are separate. Suppose, you know, and before I was adding up E1 and E2, the electric fields. Yes. It's totally fine. Fine. Fine. Suppose I have two velocity fields, right? I got two hoses and I'm like crossing streams or something.

56:43 U1 plus U2. I shouldn't have, you know what? Whatever. Like two things of water. And I add up, right? Like U1 and U2. Suppose I go through the math with those. If it was linear, I would only get the terms in the green. Right? Because I could just do the math for one, U1, and I could do the math for U2, and I'd be fine.

57:06 You do the divergence for U1. Yeah, divergence for U2 multiplied by U2 and U1, and that'd be it. Okay. But because... It's u multiplied by the divergence of itself, right? Or it's multiplied by the gradient, I should say, of itself. Not divergence because divergence has a dot. Anyways, the point is I'm taking the velocity field itself and I'm multiplying by the derivative of itself.

57:29 Because I'm doing that, I have to do the distributive property. You have to do it twice. Yeah, I have to do it twice. And so I'm going to get these cross terms on the right in the red circle. Okay. And those cross terms... are what make this whole thing nonlinear. And that gives us all of the richness of fluids.

57:47 This is why fluids don't just go through one another. They crash. There's rapids, right? When two rivers meet together, they mix and they have turbulent flow. When I was in India last time, we visited Devprayag in the Himalayas, which is the birthplace of the Ganga. where like you have like Bhagirathi coming from one side and Alakananda coming from the other side.

58:10 There are two rivers from the Himalayas that meet in this very holy place in the Himalayas that begins the Ganges River. And you could see the whirlpools forming right at that spot. It was amazing. And the reason for the whirlpools and the reason for all of the chaos in the birthplace of the Ganga is because of those two terms on the right-hand side.

58:31 You know what I mean? Yeah. And so now you can already see that there is trouble with this equation. Right. And I just want to take a quick step back to make sure I'm internalizing what we're saying here. When we talked about Maxwell's equations and we have a positive and a negative charge and we can just add them up, it's because – part of what we're saying is it's because we don't have – That extra level of complexity when we talked about the convective term that's true for fluids.

59:05 Yeah, yeah. In the Maxwell's equation, it's just del dot e, right? It's a single e. Right. Here, there's two versions of the vector field. There's the vector field itself multiplied by the derivative of the vector field. The vector field. Right. And can we just briefly try to connect it back? I'm trying to connect it back to our yellow-red traffic example just to make sure it lands in my brain.

59:28 Because I'm having a hard time finding the leap from – I understood the convective term. We have this idea that the traffic is red in one part, yellow in one part. And we can look at it from the helicopter. And we can look at the whole vector field. Or from red to green? then it is going to accelerate, even though red and green have stayed put.

59:53 And so from that perspective, would it be like the equivalent of saying, if that was it and that was the only level of complexity, it would be similar to Maxwell's equations in that it's not there? Yeah, if all I had to do was take a spatial derivative of red and green and be like, oh, there's red here and green here, and they're 100 meters apart.

1:00:10 So the difference in the velocity is divided by 100 meters. That tells me the spatial derivative of my velocity, and I'm fine. That's fine. That's fine. But crucially, it also has to do with how fast or how slow I was going into the thing. Right? If I'm transitioning from red to green and I'm moving at a snail's pace of like five miles an hour and all of a sudden now I have to get up to highway speed, I got to floor it.

1:00:30 Yes. But if in the red I was already going at 30 and then now I got to get to 60, I don't have to floor it as hard. Right. So there's two contributions. Right. And that's what's giving us this non-linearity. Yes. The fact that there's two contributions. Two contributions that we have to account for as we look at trying to do the mathematics for this.

1:00:50 And that's where we're getting – this is where we're getting having to do it – We have the green, but we have to have the addition of the red because of that second. Because of the fact that, yeah, there's two of them. Right. So I have to do these cross terms. Understood. Yeah. Okay. When I combine velocity fields, when I combine, let's say, a flow in this direction and a flow in this direction, they don't just ghost past one another.

1:01:12 Right. Because they're interacting. Yeah. Yep. Okay. Make sense? Yes. Tracking. Yeah. And so that's... why the Navier-Stokes are so hard. That's one of the reasons. So now let's get back to our Navier-Stokes equations. Finally, we can now start asking, what is the force that is acting on the fluid? So far, we've figured out how to describe the acceleration, and we've figured out how to describe the mass.

1:01:38 Right? Yes. But in order to have an equation of motion, we have to describe the forces that cause this acceleration. And then if we write it out, then perhaps we can solve it. Right? In the same way that for the spring, we could solve mass times the acceleration. In that case, the acceleration was super easy.

1:01:55 It's just the second derivative of the position. And then the force was negative kx. I had an equation of motion that I could solve. In this case, already the acceleration is quite difficult. Right? Right? But now let's ask, what is the force? One obvious thing is pressure. Pressure is force per unit area.

1:02:15 You know, higher pressure means that there's more sort of forces acting on acting on your container. Like, you know, if you've got a high-pressured gas inside of a container, that gas is pushing out on the walls of the container. Compared to a low-pressure gas, it's not pushing out as much. It's just hanging out.

1:02:34 Yeah, exactly. Like when you suck on a straw, what you're really doing is creating low pressure in your mouth, and then the high pressure of the atmosphere pushes down on the liquid and forces it up into your mouth. That's a difference in pressure that you're doing. So crucially, what matters here, what is the force?

1:02:49 on the fluid, it matters what the difference in pressure is, not what the absolute pressure is. There's absolute high, like a thousand, a hundred thousand pounds per square inch, right here, in this room. But I don't feel it, because there's no pressure gradient. There's no difference in pressure. So I don't feel the force.

1:03:10 Now all of a sudden, if outside there was super low pressure for some reason, and we had like a leaf blower in here, and, There would be high pressure in the room, low pressure outside, and we'd feel a wind that's going out. In weather systems, you see this a lot, right? High pressure, the wind goes away, goes away from the high pressure towards the low pressure.

1:03:29 So one of the forces that we will have to worry about is the gradient in the pressure. How the pressure changes from one space to another. And it's negative because it's going from high to low. So the force is always from high to low. Right, right. Okay. Because it's going from a high, it's always flowing from an environment of high pressure to an environment of low pressure.

1:03:47 So you can always have the negative in front of it because it's always going to be. Yeah, it's always going to go from high to low pressure. And so this is a force, right? The whole point right now is we're trying to describe what are the forces on the fluid. In this case, the forces on that velocity field are from negative to from high pressure to low pressure.

1:04:04 You're going to have. velocity is flowing right it's like if you're at a beach and you know those little videos where the surfers create oh yeah yeah the the surfing thing on the from a beach into the ocean yeah and it's it's like the the water pressure is very high and built up at the top and then it wants to flow into the ocean yeah which is a relative area of lower pressure yeah in that context in that context actually you're the next one that i want to talk about is the external forces on my fluid Right.

1:04:36 Because the the like the lagoon or whatever is slightly above the ocean. That's a good point. Right. And so when you create that little tiny thing, gravity is going to force the fluid down that little canyon that you've built. So my example was actually it was not perfectly apt because I was not yet accounting for gravity.

1:04:53 Yes. And the next thing that we're going to do is account for external forces of the fluid. OK. like gravity, right? That makes sense. So on the right-hand side, we've got forces. Yes. The difference in pressure is one such force. Yes. That is going to, you know, like if I were to increase the pressure here and decrease the pressure here, and I started out with some velocity field, those vectors are going to get bigger because there's going to be more fluid going through and the fluid is going to be moving faster.

1:05:21 Similarly, if I introduce gravity now, the velocity is going to get bigger. So that's a kind of force. It's causing an acceleration, right? And so here we've already got somewhat of a Navier-Stokes equation, right? We've got a familiar example is gravity, but honestly that g could be a lot of things.

1:05:40 That g could be the stirring force, right? Right? Like if I put a spoon in a piece of, in a glass and I stir the thing, that's an external force that's acting on my fluid. So the force can be a lot of things. Understood. It's not necessarily just gravity in the context. We just talked about it. Yeah.

1:06:00 We just talked about gravity in that context. But I mean, forces on fluids can be a lot of things. It can be like the turbine in an airplane. That's putting a force on, right? Because the thing is, it's stirring a fluid and making it shoot out in one direction. Yeah, yeah, yeah. So that is about as far as one of our favorite mathematicians on this podcast, Leonard Euler, got.

1:06:22 And I think this is a good point to talk about some of the history behind the Navier-Stokes equation. Because that last equation that we had is one of the Euler equations of fluids. And just to quickly kind of recap, we've started, we're trying to describe... The acceleration of an individual entity within a vector field from only having the context of the vector field.

1:06:56 And we are trying to describe it in the language of Newton's second law. Force equals mass times acceleration. Yeah. And we've built up now for an incompressible fluid. Yeah. All the way up to where Euler arrived in this journey of trying to do so. Yes. He didn't get there on his own, though. Oh, okay.

1:07:19 Ooh. Okay. We said no drama till the end, but... And there is some drama here. There's actually some really weird drama that is very pertinent to today's drama. Yeah. Of OpenAI versus all these mathematicians. So I thought it was kind of interesting to talk about. So Leonard Euler, you know, the great Leonard Euler, we've talked about him a lot.

1:07:37 In the Riemann Zeta episode, we talked about how Leonard Euler invented the Zeta function that then Riemann later extended to create the Riemann Zeta function. function and the Riemann hypothesis, it's a very similar story here. Euler created the Euler equations, and then Navier and Stokes extended those equations to create the Navier-Stokes equations that we know today.

1:07:59 And obviously, you want to name things after the second person who invented everything, because you can't name everything after Euler. But even Euler inventing that first equation, that wasn't on his own. Okay, there is a lot of shenanigans that happened in 1940s Berlin between... 1740s. Sorry, yeah, of course.

1:08:18 Yeah, 1740s Berlin between Leonard Euler. He was working at the Royal Prussian Academy of Scientists. This is the German Academy of Sciences, okay? And if you go to Berlin, you can find a little plaque for where Leonard Euler used to live. So the German Academy of Sciences announces a prize problem about one of the great practical questions in the 1740s.

1:08:44 How much resistance does a body experience as it moves through air or water? Okay. It's a very pressing question because I think around the time is when cannons are being used in warfare. So it's like, you know, we want to know how far a cannon can go. It's their version of the DoD creating, you know, a prize competition to try to understand how to do better warfare.

1:09:09 A guy named Jean Leronde Dupin. D'Alembert. Very good. For those who are long-time listeners. I don't usually do this, but I like D'Alembert. So he submits an essay. It's about fluids. He did not solve the resistance problem completely, but he helped introduce something that is arguably maybe a lot more important, which is this mathematical language of a velocity field to describe a fluid.

1:09:38 He's the guy who invented that concept of U, of the velocity field. You know, all this stuff that we've been talking about, this new language of maybe let's not worry about all of the particles, let's worry about the velocity field of this thing, that was D'Alembert. And, okay, so what's interesting is I want to make a quick zoom in note to note that the prize problem that he set out to answer was about how much resistance does a body experience when it moves through air or water.

1:10:11 And I just want to be, that's the specific question he was trying to answer. In that journey, he did not answer that question. He did not answer the question, but he invented. He gave us the language. He gave us the language for it, right? And his submission didn't win because the Academy of Sciences was like, oh, it's too theoretical.

1:10:28 He also showed that the velocity field has to be divergence-less. meaning that del.u equals zero, like the fact that it's incompressible and things like that. So he already established one of the two Navier-Stokes equations, Dal and Baird. He submits this thing. The Academy decides that it's not going to give out the prize.

1:10:47 No prize was awarded. But here's the thing. One of the people that was on the committee was Leonard Euler. So Euler saw an early submission of D'Alembert's work, and he's reading this thing going, yo, this velocity field is a kind of nice idea. He takes that idea and not long afterward, he presents his own general theory of incompressible fluid motion and that's the equation that we see on the right-hand side.

1:11:13 So on the left-hand side, Dahlembert has already invented the velocity field and he's shown that the velocity field has to have zero divergence, meaning it's incompressible, no sources or sinks. Leonard Euler takes that concept and says, well, I can actually make a Newton's Law version So very similar to Riemann zeta where the Basel problem existed beforehand, right?

1:11:35 Right. The 1 plus 1 over 4 plus 1 over 9 plus 1 over 16, so on and so forth. Leonard Euler saw the Basel problem, solved it, and then presented the zeta function. In this case, he saw D'Alembert's velocity field and he's like, wait, this is a good idea. And he just ran with it and created the derivatives, which are huge, like the...

1:11:57 The derivative term, that acceleration, is massive to create that material derivative. And then also say that, okay, the forces on it are going to be the gradient in the pressure plus whatever external force. And just to say what you just spoke to, that was the convective term that we talked about earlier.

1:12:13 Yeah, yeah. Yes. Okay. And this is so interesting because it's... It's always funny how these things... Basically, we have D'Alembert, who created the vision to think about it this way. And then Euler was like, let me take Newton's framework, D'Alembert's vision, and combine these two things. Yes. And he gets an equation of motion, right?

1:12:42 He gets the first equation of motion for a fluid. This is for an inviscid fluid, as they say. Because these are... as we'll find out, frictionless fluids. Okay? Now, obviously, that can't be everything. And D'Alembert sees this. It's kind of like a mathematical twist that's so perfect it's got to be scripted.

1:13:01 D'Alembert looks at the original competition, and he's like, wait a minute, it was about drag. If I apply Newton's, I mean, sorry, if I apply Euler's equation to the concept of drag, and I ask, what is the drag on a body? I can exactly solve Euler's equations around, let's say, a ball moving through a fluid.

1:13:18 And there's no drag. Like in the next thing we'll see, this is called D'Alembert's paradox. Okay. He uses Euler's equation that's up there and he applies it to a ball, a sphere that is moving through fluid. According to Euler's equations, if you were to solve with all those boundary conditions, the streamlines just go up and then they come back.

1:13:40 There's no difference in pressure from the front to the back. And so there's no drag. But obviously, as we know from our FIFA World Cup episode, where we talked about the drag on a soccer ball, there is a lot of drag on a soccer ball. Just to also note, this was Adidas' peer-reviewed study on their new balls for this year's World Cup.

1:14:03 And they did incredible deep dive studies into the literal drag of how they designed it. It's a great episode. Definitely go watch it. Exactly. And what we can see already is in Real Fluids, the streamlines aren't just meeting up right after. Yeah. Right? There's this region where there's turbulence and all this weird stuff that's happening.

1:14:19 This chaos happening. Right? Right? Euler's equation does not account for that. So basically what we're saying is Euler's equations only accounts for what we see on the left, which is effectively a system that's not actually exists in real context. Exactly. The military can't use it. Exactly, yeah. And so 20 years after the whole competition and everything, Dahlembert says, this is a paradox.

1:14:41 This is not everything. There's got to be more. And this is the part that Navier and Stokes... Help us repair. Okay? They're the ones that come in and they start accounting for viscosity. And so to now kind of quickly recap again, we are trying to build this understanding of what is the construction of the Navier-Stokes equation.

1:15:03 We understand d'Alembert started with this idea that the divergence equals zero. Yeah. uh oiler has come in and used the framework of newton's second law of motion to try to apply it to fluids maybe cheating off of d'alembert's test maybe a little bit maybe yeah um and added for the force side of the equation uh pressure uh and just an external and it's and an external gravity or stirring or whatever whatever it might be yeah and he was like i did it yeah look at me i'm so brilliant yeah however When applied in practice to a ball like we just described, it does not actually describe real world systems.

1:15:45 And what's interesting is the – it's funny because the original prize competition that Euler was a judge on was about resistance and drag. And then he came up with a solution without thinking about resistance and drag. Yeah. Am I missing something? Exactly. Okay. So what he ended up missing was another force that he hadn't accounted for.

1:16:04 Remember, right now we have two forces. We've got the difference in pressure. Yes. And we've got just external forces, whatever they might be, stirring, gravity, whatever. Sure. There is a third force on that side of the equation that is missing, and that has to do with viscosity. Okay. That's the resolution to D'Alembert's paradox.

1:16:21 Okay. So in Euler's ideal fluid, there is no viscosity. This is called an inviscid flow. The fluids can slide past one another. These particles can just slide past one another, and they can slide along the surface of an object without any friction. And the flow divides very smoothly in front of the object, and then it rejoins at the end of the object.

1:16:43 There's no wake. There's no net resistance. There's no draft. And this is beautiful mathematics. It's different than diverge. Like it's different than the problem we were saying where water – form a incompressibility right right yeah this is different it's different this is still incompressible right because the flow is coming in and then it's going out it's just that they when they interact with each other it just rolls yeah they're just they just roll off yeah okay yeah they're just rolling off one another it's not especially useful okay we haven't taken into account friction and we have to take into account something called viscosity viscosity is simply the measure of how thick sticky and resistant to flowing a liquid is.

1:17:25 So we're going from thick with two C's on the left here to thick with three C's on the right. Yes. Okay. Yeah. Vegetable oil, not that thick, right? You drop something, it immediately goes through. Motor oil, also not that thick, but slightly a little bit thicker. Honey, on the other hand, you drop something and it takes forever to get all the way down because there's so much resistance to flow in honey.

1:17:49 The things as they try to move past each other Make it harder. Exactly, yeah. And so for low viscosity, like thin liquids, like oils, the particles are moving past each other with very little friction. But for high viscosity, there's a lot of resistance to that movement. And therein lies the clue in how we can mathematically describe this.

1:18:07 Because we need to have a mathematical language to describe what is viscosity. So let's go back to our traffic analogy. Okay. Okay? Let's consider a multi-laned highway. Like this. So we've got a shoulder that's zero miles per hour, obviously, because you're on the shoulder. You better be stationary.

1:18:24 You don't want to be one of those guys that tries to cut in line and then the cops catch him. So let's look at what exchange of traffic from one lane to another will do because in normal traffic obviously cars are going to change lanes yes so i've color-coded these cars based on their lane and based on their velocity sorry and what i'm realizing earlier you specifically showed us only one lane examples yes Which was fine at the time.

1:18:54 At the time. But now we've got multiple lanes. And now the flow is going to crisscross. In between, which is how real liquids... How real liquids do it, right? Now, what I've done is, in the initial, let's say it's a four-laned highway... The lane that's fastest is going at like 80 miles an hour. And the lane that's slowest is going at 20 miles an hour.

1:19:16 So it goes 80, then 60, then 40, then 20, and then you've got the shoulder. Four lanes, and it's in increments of 20 miles per hour. And then things start shifting around. Okay? You actually wouldn't notice a change in the flow. Because let's take, for example, the lane that's moving at 60 miles an hour.

1:19:36 It's going to inherit some faster cars from the 80 miles per hour lane. And it's going to inherit some slower cars from the 40 miles per hour lane. And the things are going to average out. Right? And so if I were to look at a graph, and that's on the bottom there, if we look at a graph of lane on the x-axis and the speed on the y-axis, it's a line because it goes from 0, 20, 40, 60, 80.

1:19:59 And later on, it's still going to be a line. There's been no change, meaning that the velocity field has not changed, right? It's still big up top, small down below. And the velocity field has not changed from one time to another, meaning there's no force here. Yeah, yeah, yeah. Okay? Even though there's some viscosity, there's no force.

1:20:18 Right, okay. Yes. Right? Yes. There's still like lane changing happening, but there's no force. There's no change in the flow of traffic. Now let's consider a different type of profile. A type of profile where the shoulder is again zero. And I've got the two slowest lanes, both at 20 miles per hour. And then I've got a lane at 30 miles per hour.

1:20:41 But the faster lane is still going at 80. Imagine that. Like the carpool lane is still going at 80. And then the rest is like still very, very slow. And the carpool lane has like a barrier. So there's nothing happening. And then all of a sudden the barrier gets lifted. What's going to happen? Well, the carpool lane is going to slow down a lot.

1:20:58 But also the other lanes are going to speed up because stuff from the carpool lane is going to come in and sort of push that lane forward. This is a bumper car scenario, okay? Not real. But you get what I'm saying? Like before we had this... Curvy profile, which we see on the lower left, where, again, we don't have a line because it's slow, slow, slow, then really fast.

1:21:23 Yes. And with subsequent time, it's going to level out. It's going to level out. Because of the diffusion of cars. Here, there is a force, right? Because I've changed the velocity profile. If I change the velocity profile, that's an acceleration, which means there's a force lurking in here. Right? There's a viscous force that is lurking in here.

1:21:45 And this is what's interesting is, just going back to our previous example, because basically the system was evenly distributed in the initial state... based on how the variables would end up mixing, you had no net effect, no force. No force. In our second example here, it was sort of a not evenly distributed system.

1:22:09 And so when you look over time, and then there's the mixing. Yeah, there's a mixing. There's going to be a change. There's some divergence. Not divergence. Let me not use the word divergence. There's some change. There's a change over time. Over time. Which means that there is a force. And notice, in the first example, there was a straight line, which means I had a positive derivative Right?

1:22:27 Like, the slope is positive, but the second derivative is zero. There's no curvature. In the second version, there's a curvature, right? Which is why when I inherit the fast stuff and I inherit the slow stuff, the fast stuff is way faster than the slow stuff, which is why I'm speeding up. The curvature is the origin of the force.

1:22:49 That's the key. And so that is the form that the viscosity is going to take mathematically. And if we go in the next slide, we'll actually see what the viscosity term looks like. It looks like del squared. Del squared means a second derivative in some sense with respect to space. And there's your viscosity.

1:23:07 The mu, the Greek letter there, that's telling you how viscous it is. Like honey would have a very high value. Water would have a very low value. And that's the force. The force is, again, has to do with the velocity field itself. This is always key. We want to describe everything, except for maybe the external force, as a property of the velocity field itself.

1:23:31 Of whatever the system, as I use in a colloquial sense, as a whole, is a material idea here, which is why we have, these are all derivatives, which we're looking at the system level. Yes, exactly. And it kind of reminds me of when we were talking about the springs, right? The force on the spring had to do with the position.

1:23:48 In this case, the force on the spring has to do with the velocity field itself. Mm-hmm. And on the right hand side, just for those who are engineers and physicists, I want to show you a similarity between the viscosity term and the heat equation, where if you have the heat equation and if you have a temperature profile, as time moves forward, everything gets smoothed out because of thermal diffusion.

1:24:12 The same thing is happening with viscosity. Viscosity is smoothing out the velocity differences, right? And that's what we saw was happening with the traffic analogy. There was a lot of high speed here and low speed here, and viscosity was smoothing it out. and making the flow a bit smoother right i'm starting to see the connection between the words incompressible and smoothness that i keep seeing as it relates to the whole yeah navier stokes piece yeah we're we're starting to see that so the viscous term becomes a smoothing operator in some sense right um and now which feels very natural as a property of trying to measure a liquid because in this context it's it's It's sort of this contiguous thingy when you look at it from the vector field perspective.

1:24:59 Yeah, exactly. And we don't want that contiguous thingy to have like little pockets of really high fluid. I'm sorry, really high velocity. Because if there's a little pocket of really high velocity, well, that velocity is going to spread out. Right. Right? And that's what viscosity does. Right. Right?

1:25:14 That's why in honey, it's really hard to have a small pocket of really high speed honey. Yeah. On the other hand, for water, it's maybe a little bit easier to have a pocket of high-speed water. But then it just depends on what time... Yeah. Anyway. Yeah. Yeah. Exactly. And so now we can solve D'Alembert's paradox because now we have viscosity in mind.

1:25:36 So we can apply this thing called the no-slip condition. This is something that we've looked at with hypersonics and we've looked at in the FIFA episode, where if you've got an object that is moving through a fluid... very close to the object, the fluid is not going to be moving at all. It's going to be stuck to the object.

1:25:55 And there's going to be a boundary layer where it's going to transition between being stuck to the object and moving with the rest of the fluid. And that boundary layer causes the drag because the object is dragging the fluid with it. Right? And this is how we solve D'Alembert's paradox because now if we look at, for example, a golf ball, that is moving in air.

1:26:19 The fluid is sticking to the golf ball. And so... It's creating a wake behind it because the golf ball is moving through the fluid. And as the fluid sticks to it, the part that's right above the golf ball is going to be moving with that sticky layer, but it's also going to be moving kind of with that.

1:26:37 And so you get all of these dynamics happening because of viscosity. There's this transitionary space between the surface of the object that is moving through the fluid and then the rest of space. And in proximity to the surface of the object, there is a transition where the fluid around the object is going to move on some gradient from sticking to the object to being a part of the larger system.

1:27:01 Exactly. Yeah, exactly. And so that's how we get drag. And so this is the resolution. And this is the contribution that Navier and Stokes were trying to capture. Claude-Louis Navier said, in 1822, he presented an equation on viscous fluid motion where he added that viscous term. But he was an engineer as much as a mathematician.

1:27:21 And he tried to explain the internal friction by imagining molecular interactions inside a fluid. This was 1822. So everyone thought he was crazy because there was no such thing as atoms back then. I mean, there was this Greek concept of an atom by democratized or something. But, you know... No one took that seriously.

1:27:38 And it's one of these rare occasions where a good outcome came from a derivation that's based on a questionable premise. Now we know it's not questionable, obviously. And over the following decades, we had Cauchy, Poisson… And others developed the continuum theory of stress and how stress affects bodies.

1:27:58 And in 1845, Stokes, George Gabriel Stokes, derived the viscous equations from a completely different direction from this idea of stress and how you can deform objects. Turns out it's the same form as Navier. And so now that's why we're called the Navier-Stokes equation. equations. They never collaborated, by the way.

1:28:18 Stokes was a teenager when Navier died, but their names are joined because, you know, different physical arguments, but they arrived at these two equations. And now these are the two equations that have been such a headache for mathematicians and physicists all these years. And so as I look at now these two equations, the one at the bottom, which was where we started, is this idea that the divergence always has to be zero for an incompressible fluid.

1:28:45 Yeah. You can't make fluid and you can't destroy fluid. You can't have a fluid emanate from a point in all directions, and you can't have a fluid converge on a point from all directions. That's what the bottom one is effectively saying. Yep. Yep. And then now the top one, which was the more complex one, is we're trying to replicate force on the right equals mass times acceleration on the left.

1:29:11 Yes. In this construction. Exactly. Which was Newton's second law of motion. We're trying to apply it to incompressible fluids. Yes. And in doing so, we constructed the left side of the equation first where we have our mass represented by our volume. Yeah, that's the density. The density, excuse me. And then we then multiply that by everything in the larger brackets, which is our acceleration.

1:29:36 Yeah. which we have the first term being the time derivative of the velocity field. Yeah, that's like rain coming in and changing traffic patterns. Patterns. But we also have to add into that now the convective term, which is how an individual car within that vector field works. is changing its velocity with distance over time.

1:30:03 And so that's the left side. Yeah. And that gives us our mass times acceleration. Yes. Loosely speaking. Maybe there's a few details not quite right. And then on our right… So there's sort of three terms that have complexity on our mass times acceleration on the left. It's really two terms, I would say, right?

1:30:23 Because it's – Sorry, fair. Within acceleration, it has two component parts. Well, I'm nitpicking here. But it's really two terms because it's density multiplied by du dt. Okay. And then it's density multiplied by that second term. Totally fair. You know what I mean? Totally fair. Like density is not its own thing.

1:30:39 That's a fair – No, that's an important distinction. And that's helpful. And then on our right side, where we're trying to define force, we started with the obvious thing, which was pressure. Yeah. And it's negative because high pressure always moves to low pressure. Mm-hmm. Then we moved to – we added our F, which is some general external force.

1:31:03 Yeah, stirring, gravity, whatever you wanted to call it. And that's where Euler ended up. Yeah. And he stopped. And he stopped. And we were missing this concept of drag or resistance. And – Ultimately, both Navier and Stokes, from different directions, ended up on being able to define viscosity as the missing piece, which is, again, talking about there's a boundary layer between the object in the system and then the rest of the medium external environment.

1:31:32 And there's some gradient delta there of change. And so we need to account for the viscosity because that applies a force as we look at all of this. Is that now like an accurate description of what we're looking at? That is the Navier Stokes equation. And we have just built it up from first principles.

1:31:48 As someone who did not do any type of mathematics, it's a testament to your way of taking the story from start to finish. Yeah. And it kind of makes sense though, right? It makes total sense. When you understand the underlying systems. Yeah. I think the hardest part really is getting familiar with the idea of a velocity field.

1:32:06 Yeah. Yeah. You know? Yeah. Because once you have that and you've internalized, oh, it's like a traffic pattern. Then all of the other stuff kind of falls together because you can kind of figure it out. That was the Eulerian perspective was the velocity field. That's the way we described it earlier. Yeah.

1:32:22 Very nice. D'Alembert, dude. D'Alembert. Yeah. And so now we can start asking, at this point, we've got the Navier-Stokes equations. Yes. We know what every term means. Yes. So... Now we can start tackling how to solve them. Right. And because part of what we also define through all of this is this is an extremely complex problem.

1:32:45 problem yes for all of the nuances of each of the underlying terms which also have their own sub uh issues some subcategory issues exactly and so we've got an equation of motion now can we solve it like can we can we find a general like i give you a velocity field now you tell me what the velocity field is later right what is the flow going to look like later and And to just come back to the beginning of the episode, the reason we want to do this is because with Newton's equations that we were talking about, part of it's like then you can use it to make predictions.

1:33:17 Exactly. Yeah. And that's how we do the space stuff and blah, blah, blah. So what we're trying to do is take this equation of motion so that we can – like what is – who cares? Yeah. And the answer is so we can make predictions. We can give you one input, the velocity field, and then you can give me the output consistently that's true every time.

1:33:34 Yeah. That'd be dope. That'd be great. Yeah. Okay, so we have a set of differential equations, and we'd like to have solutions to those differential equations. Now, what do we mean by that? Like, in a normal algebra problem, if I write something like x plus 2 equals 5, I know what x is. x is 3, right?

1:33:51 I can, like, do the thing. In a differential equation, what we're asking is something larger. What is the function that solves the differential equation? In our case, it would be what is the velocity field that solves the differential equation? That solves this differential equation. Let's take a simple example.

1:34:06 Let's take, for example, the derivative of y in terms of t is equal to some r times y. Meaning, I've got some quantity. The change of that quantity with respect to time is proportional to the quantity itself multiplied by, let's say, some rate. That is an exponential, right? The solution to that is e to the power of rt, and this is how people get rich.

1:34:32 This is compound interest, right? You start with some amount of money, and then the rate tells you the interest over time. And the more money you have, the faster your money grows. And this is a very simple differential equation that we can solve very accurately. Right. This is a toy equation and the answer is very nice.

1:34:51 Navier Stokes is less accommodating. Okay. The unknown is not how much money you have, like a single one-dimensional variable. It's a three-dimensional velocity field. Yeah. Right. Yeah. So it's a number. Yeah. at every point in 3D space, but that number has a direction, right? It's a length, and it's got a direction that's also three-dimensional.

1:35:16 So it's quite massive, like this equation. And it's so funny, because now that I understand that, it makes my brain hurt trying to say, we want to solve for this now, given that level of complexity, which is now very clear what that level of complexity is. Exactly. Like the... Fluid is interacting with itself and everything.

1:35:35 So this velocity field has to solve for that entire big thing, right? We've got to find all of the arrows everywhere. Right. In some sense. Yes. Right? For all time. Because the arrows might be changing over time. Over time. And we need to have an expression for all of that. Now, there are certain special situations in which the Navier-Stokes is actually exactly solvable.

1:35:56 Okay, it's not true that the Navier-Stokes is just unsolvable. You can't solve this equation. There are situations, there are certain boundary conditions where you can exactly solve the Navier-Stokes equations. Let's go through one of them. It's called plain co-et, co-et, co-et-ay flow. I never know how to pronounce it.

1:36:16 I think it might be co-et. Is it co-et-ay? Let us know in the comments. Yeah. Yeah. In any case, it looks French. Yeah, it's probably French, which is why I think it's cohete. Here's what's happening. You've got two plates, okay? Those are the Navier-Stokes equations that we're trying to solve. Yes. I've got two plates.

1:36:33 One plate, let's say, is the table, completely stationary. Another infinite plate is moving in the x direction, in just one direction at a certain velocity. And in between two plates, you've got water. You've got some kind of fluid. Okay? What is the velocity profile of the water in between these two plates?

1:36:52 It's very simple, right? There's a stationary plate, there's a plate on top that is moving at a fixed velocity in one direction, and there's fluid in the middle. So what is the velocity field going through? Now we know from the no-slip condition... The fluid that's right next to the table that's stationary is going to be stationary.

1:37:11 And the fluid that's on the plate that's moving is going to be moving with the plate. What happens in the middle? It's actually pretty simple. You solve for the Navier-Stokes equations. And you get this profile, right? Near the bottom, where it's stationary, the velocity is zero. And as you approach the plate in the top, the velocity gradually increases linearly.

1:37:34 And so it's just a function of the height, right? The higher you are, the closer you are to the top plate, the closer you are to the velocity. It's a linear relationship. Boom. That makes total sense. Exactly solvable. That makes total sense. And again, we've sort of created an exactly solvable, simple use case because, you know, by having the bottom plate be stationary.

1:37:56 Yeah. Having the bottom, even if the bottom plate is actually moving in the other direction, it's totally fine. As long as it's the same, like, you know, the same velocity. And there's so much symmetry here. They're only moving it on one plane. Exactly. It's only one coordinate that we have to worry about.

1:38:09 So a bunch of the derivatives over there just go to zero. Like there's so much symmetry and geometry here that it murders all of the difficult terms of that equation. And you've got this steady profile and you're good to go. It's a recurring theme in fluid mechanics, which is if you've got a nice symmetric distribution of stuff, you can solve things.

1:38:29 Fine. You're fine. Yeah. This is called co-wethe flow. You can also do pipe flows. There's a handful of vortices and jets where you can obtain beautiful closed-form solutions based on the symmetry of the situation. And just to ask as a brief side question, part of the other part of why those are so solvable is because we have such a good understanding of geometry as like these other mathematical disciplines.

1:38:55 We can apply some of those learnings and concepts in a way that make these easy or it's not relevant at all. Well, no, certainly. I think, I mean, for that one, it's like pretty easy because you can just take derivatives and they go to zero and whatever, right? But there's other ones, like there's certain vortices and jets where you can manipulate the symmetry to cancel out terms and things like that.

1:39:13 You can use our knowledge of symmetry Right. To make, like, transformations that make these equations much easier to handle. The subtle note I'm just trying to make here is these things don't happen in a vacuum. Like, our general mathematical understanding has applications in subtle, small ways all through.

1:39:34 All through, yeah. All through. Yeah, all through. And so... So that's for a very nice symmetric thing. Formula One cars, not very symmetric. I mean, there's some symmetry, I guess, left and right, right? So sure, you can see that it's symmetric, like you can flip the profile, right? But a Formula One car is insane.

1:39:52 The geometry is extremely complicated. The wheels rotate, the ground moves relative to the car. There's thin boundary layers on the bodywork entirely. Air is... accelerated beneath the floor the vortices are deliberately generated around these waves to cause downforce um the non-linear term does not vanish in the other one the non-linear term actually vanishes just because of the geometry of the thing.

1:40:18 Here, this is the entire problem, right? There's no known formula that you can write out for what a Formula 1 car, the velocity and pressure at every point around a Formula 1 car is going to use. So instead, what engineers do, even in this visualization over here, what they're doing is using a computer to computationally solve the Navier-Stokes equation around this boundary.

1:40:42 And this is called computational fluid dynamics. If there's any aerospace engineers or hydrodynamic engineers in the audience, leave one in the comments and tell us your favorite computational fluid dynamics program. There's a bunch of programs out there. This is one that I just picked up on GitHub. And they always flaunt the number of cells that they can break down their space into.

1:41:07 Yeah. Because, you know, imagine a Formula 1 car in a wind tunnel, right? That's some, like, volume. In order to solve these differential equations, you have to break up that volume into tiny little grid boxes. And then you can solve the Navier-Stokes equations for each of the grid boxes. And then you just can put them all together.

1:41:23 Yeah. You can be like, oh, the velocity is here, but the velocity next to it is something. The velocity to the right and left of it is something. So I can calculate a derivative that way, right? I can be like, oh, the difference divided by the grid spacing is my spatial derivative. Yeah. Right? And I can do this computationally one at a time.

1:41:42 And the more cells I have, the better granularity and the more accuracy I have to the real thing. This makes me think back to our hypersonics episode. where we effectively discuss the same thing. Yeah. And then sort of in addition to all of the challenges we've talked about, when you start going from supersonic to hypersonic, things get even weirder.

1:42:07 Yeah. And we sort of discussed why it gets weirder and how we try to – deal with that with these approximations that we've sort of discussed here. It's again, Navier Stokes is not unsolved in all cases. We found clever ways. Yeah. To like approximate. Approximate. Yeah, the solution, right? Right. And for engineers, this is good enough.

1:42:26 Right. The point, they're like, honestly, like, yeah. Oh, but I didn't solve anything. We've been modeling and having a stroke since before you were born. Yeah, for the Ferrari engineers, it's not good enough because apparently they still can't do it. Like, guys, figure it out. You have Lewis Hamilton and Charles Leclerc.

1:42:44 Like, guys, come, please. I've been waiting for so long. Anyways, the result is not a closed-form solution, okay? It's this enormous table of numbers that tells you what the velocity is at every given point. And you can put any arbitrary shape into it. This is like a meme now in aerodynamics for some reason.

1:43:02 Like, all these aerospace engineers, like, put a cow... a cad model of a cow into their cfd simulations and they're like how aerodynamic is a cow i've seen so many versions of this meme for some reason i don't know why but a cow is a very non-aerodynamic thing that people are trying to figure out you know what is the lift on a cow as it goes through a wind tunnel in any case the idea is any arbitrary shape you can put it into a And you can turn the simulation one after the other and computationally solve the Navier-Stokes equations.

1:43:38 For all practical purposes, that's totally fine. It's an engineering solution. It's not a physics understanding. Yeah. Yeah. And or mathematical understanding. And it's fine that it's an applied engineering solution. It's a spectacular practical achievement. I mean, this is how we get airplanes. This is how we get fighter jets.

1:43:58 This is how we get turbines. This is how we get wind farms for power. But it's not the same thing as... the exact equations, and what do those equations look like, right? What does the solution to this thing look like for any arbitrary boundary condition? A simulation has a finite grid, right? It advances in finite time steps.

1:44:22 It rounds numbers at some point. And so the problem is engineers are pretty happy. Taking Navier-Stokes equations, putting it into a CAD, like putting CAD model in there, using computational fluid dynamics, they are good to go. They'll tell you what the drag is. They'll tell you what the lift is, what the downforce is on a Formula One car and so on and so forth.

1:44:44 And unless you're making hypersonics, it's fine. It's fine. Mathematicians, on the other hand, they're asking, well, what about like the existential problem of like I've got differential equations. Yeah. are the solutions always well-behaved? Okay. That's the question. Here's what I mean by well-behaved.

1:45:06 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 acceleration, and we're talking about the viscosity term on the right, which is a part of our force.

1:45:28 Yes. And these two things do very different things. qualitative transformations to our velocity field. The viscosity term, as I told you, smooths things out, right? If there'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.

1:45:53 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. There's a constant competition between these two terms in the equation.

1:46:18 Because depending on the size of the structure we'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?

1:46:35 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.

1:46:51 Okay, so we're going to talk about this competition in a very... real sense with something called the Taylor Green Vortex. This is a standard test case that is used in computational fluid dynamics because what you can do is actually solve this thing analytically, and then you can solve it in a computer and compare the two.

1:47:09 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 Taylor Green Vortex looks like. This is the initial condition. Okay. On the right-hand side, you're seeing the velocity field at time t equals zero.

1:47:25 So this is what you're starting out with. It's a bunch of sines and cosines for the x velocity. It's a bunch of sines and cosines for the y velocity. And it's zero on the z velocity, meaning all of the particles are only moving in the xy plane. No one's going up and down. Everything is moving side to side.

1:47:43 Flatland. Flatland. In flatland, right? But there's bunches of flatland. Like, it can still be a 3D box, but like... The stuff at the top of the box is still going side to side, and 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 sines and cosines.

1:48:01 So what that amounts to is you have vortices. Okay? You have a... Sines and cosines always means... circles. Okay? If you come out with one thing from this podcast, sines and cosines mean circles. But in this case, what's happening is, you've got these vortices that are circular. Okay? And so all the particles are moving around in circles like this.

1:48:21 Like the ones up top are 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... on this thing. What is going to happen? Well... Let's see what happens. We've got a bunch of math here that I'm going to very quickly go through.

1:48:41 Okay? For those who are not watching, my eyes just expanded to a very large size. Yeah, but it's not that bad. And I'll tell you what to focus on. Okay. Okay? On the left-hand side is our initial conditions. These are our sines and cosines. It's just sine x and cosine y, cosine z. So this tells you that the size of my vortex is 2 pi.

1:49:07 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.

1:49:29 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 sine 2x and a cosine 2x. That's going to percolate into the pressure terms. You're going to get sine 2x and cosine 2x.

1:49:50 And one thing that you should remember is sine x, is a wave. Sine 2x is that same wave squished. 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 the one.

1:50:19 So remember, before I told you that the z component of my velocities was zero. Everything was happening 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 interactions that are happening in the Z direction.

1:50:43 You're going to push up stuff and you're going to push down stuff. And the way you push up and down, those structures are going to be half the size of your original vortex. And this is coming from the squishing. Mm-hmm. It's coming from the squishing and the fact that the squishing is forcing stuff into the third dimension.

1:51:03 So on the left-hand side, you see these vortices. Those blobs sort of represent where the vortices are. You can imagine the orange blob is a vortex going in one direction. The blue blob is a vortex going in the other direction. But all of those vortices are X and Y. Okay? They're revolving in this way.

1:51:19 And they sort of have, for lack of a better term, a consistent, non-disturbed structure. Yeah, and that's time t equals zero. Like, that's what we started with. And when we press play, all of a sudden you start getting these vortices, you're getting these interactions between the vortices up and down.

1:51:36 And the interactions, those structures that are created in that interaction, have a smaller and smaller size because of that convective term. Okay? I see, I see. And if we continue this ad 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.

1:52:01 You're going to get smaller and infinitesimally 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.

1:52:24 And these two things are in competition as we looked at in this example. And at time equals zero, when we have the blobs on the top left, And as time goes by, the left side of the equation is winning. Is winning, yeah. Effectively. Effectively, because it's creating smaller and smaller structure that's leaking into the z direction.

1:52:48 And so our sort of nice discrete blob structures at the top become a more chaotic system as we get these smaller and smaller structures. Yes. But at some point, and the fundamental question is, does this go all the way to infinity in smallness? And do I get smaller and smaller structures? Is the convective term the goat?

1:53:09 And does it always win? 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 for technical audiences, the convective term goes up kind of linearly, but it grows very fast in the beginning. The viscosity term grows like the square.

1:53:33 So at some point it's going to catch up and it's going to diffuse everything and then everything will just sort of be chill. Yeah, that makes sense. So the Taylor Greene 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 And it creates smaller and smaller spatial structure.

1:53:53 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 this versus this. Okay, this versus this. You know what I mean? So eventually, at this meeting point of structure size…

1:54:12 Yes, is when the viscosity is like, okay, that's enough. That's enough. That makes sense. 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. And where it happens depends on the viscosity of the fluid. If the fluid is very, very viscous, it's going to cut it immediately.

1:54:33 But if it's not very viscous, it'll take some time, but it's going to get there. For the Taylor Greene 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.

1:54:51 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 co-wethe flow, and then we showed this Taylor Green vortex. In both of those cases, pretty fine. Everything's smooth.

1:55:09 Everything's normal. Things aren't going crazy. So is this the idea that smoothness always wins? Yes. Like to describe it in another way. The solution is always going to be smooth because of the viscosity. Because of the viscosity. Winning in this race. Yeah. Okay. Okay. Is it always going to win? Is it always going to win?

1:55:27 Yes. We've shown that it can win sometimes. Yes. And for those two times, it's going to win. Sure. 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?

1:55:43 And there's like discretized grids. And so we can't go infinitesimally small. At some point, maybe it's an artifact of the fact that like, you literally can't physically, like there's not enough compute to go down to what? The Planck length or something like that, right? Where it could start falling apart.

1:55:58 Yeah, like, 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 idea. And that actually, it's so funny because now it's, it kind of reminds me a little bit of, was it the Riemann hypothesis episode where we talked about the non- Trivial zeros.

1:56:34 The non-trivial zeros. And it's similarly like, you know, We want to take the question to its ultimate... Yeah, is every single zero on the critical line? On the critical line. Every single one. No, every single one. Because you can still have 99.99. You can even have 100, as we were saying. And that doesn't mean... You can have 100% and still have one off.

1:56:55 And anywhere in all possibility that one is off, it means... And 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 a very mathematical thing to worry about.

1:57:16 Now I understand the mindset. Yeah. Now, crucially, this is now an open problem only in 3D. In 2D, there was a very famous mathematician. She was a Soviet mathematician. Olga Ladijenskaya. Olga Ladijenskaya. She wrote this 1969 book, The Mathematical Theory of Viscous Incompressible Flow, where she showed that in two dimensions, yes, yes, every single initial condition guaranteed viscosity is going to win.

1:57:49 If we're in flatland. Yes. If we're in flatland and we've got like fluids flowing in flatland. And there's no Z up, Z down that we just described. As we saw actually, like even in the Taylor green vortex. Oh, okay. Right. We saw that like the Z dimension was the one that was causing that weirdness because the flow was escaping into the Z dimension and then creating those smaller and smaller structures.

1:58:10 So Olga showed in 1969 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 come up with a smooth solution. Okay. So for 2D, the Navier-Stokes global smoothness is known. It's fine. That's not the question we're asking.

1:58:33 Yeah. For 3D. For 3D is... It is not known. And is this... And I just have a quick question. So when I see the term blow up... Mm-hmm. as it relates to how people describe this. This is what we're... Is this kind of what we're talking about? It's like, does the smoothness... Does viscosity lose? Yes.

1:58:55 Which is effectively... Blow up. Yeah. 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.

1:59:13 Right? It's kind of like, is there a singularity in fluids? Not really. But like. Well, no, but no, no, no. That's literally what they're asking. Is there a singularity? Like, can I create a singularity? That's what they're asking. Okay, fair enough. That's exactly what they're asking. Now, why is it so hard in three dimensions?

1:59:32 Well, we already saw, like, kind of a clue to that with the Taylor Greene 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 sines and cosines 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.

1:59:54 Vorticity is the problem in 3D, okay? Vorticity is the curl, right? For those who have taken calculus, the curl of the vector field, that del operator is back again, but this time we're taking a cross product of the del operator with the velocity field. It's just a way of saying, is the velocity field curling?

2:00:13 Like if I put a pinwheel 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 flatland.

2:00:30 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 about. Like... You know, it's a small vortex, then it becomes a bigger and bigger vortex, right? And that's all it can do in two dimensions.

2:00:49 Because it'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 our... Yeah, let's say a hurricane. It's quite big in space, and it's moving, but the velocity is not that high, okay?

2:01:12 Okay. And now that hurricane, all of that energy gets squished into a tornado. Hmm. What's going to happen? Well, the tornado is going to get very, very tall because 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.

2:01:31 So it's got to go somewhere. It's got to go somewhere. So it's going to go up, right? 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. And so just like, you know...

2:01:47 In 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 kind of reminds me in a different context of our sideways wine bottle and then it getting narrower. And so it increased the acceleration.

2:02:07 Just conceptually, there's a sort of similar thing. Yeah, you're packing stuff in here. But here it's the vorticity that's increasing because of the conservation of angular momentum. Again, 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.

2:02:21 Makes sense. 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 in the Taylor Green vortex, why we're now leaking into the...

2:02:41 Into those higher dimensions, right? Right. And the question is, maybe with this vortex stretching, you can actually create a kind of singularity. Okay? Now, let's get into what we mean by singularity in the first place. We had already done our first toy differential equation, right? Yes. Which is the compound interest differential equation.

2:02:59 Now, this thing does go to infinity. Okay. 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. 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? It just turns out maybe AI is going to end the That's not a real singularity.

2:03:24 A real singularity is an infinite time. I shouldn't say infinite because that sounds like infinite. So I should say before in finite time, can I get to infinity? Yeah. In some constrained amount of time. There we go. Yes. For our listeners out there, that'll be good. In a constrained amount of time, can I get to infinity?

2:03:52 Here's another differential equation that does that. Here, the derivative, my derivative with respect to time, is equal to y squared. So dy dt is equal to y squared. Initial condition, you start at 1 when time is 0. And the solution to that is 1 over 1 minus t. If I plug in t equals 1... I get infinity.

2:04:14 Right? So this is a differential equation that creates a singularity, but the time is not infinity. The time is just one. At one second, I have a singularity. 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 squared.

2:04:34 Yeah. It's the simplest nonlinear differential equation. And already I am getting a singularity, right? The other one was just dy dt equals y. That's a linear equation. I can just add up the terms. But here, the non-linearity is creating that cascade that is creating a singularity before time goes to infinity, at a finite time.

2:04:57 So it's conceivable to extrapolate from this. Yeah. It would be conceivable based on how we've constructed the non-linearity of trying to look at this incompressible fluid and look at a vector field. Yeah, it's like there's a chance. There's a chance because we got a lot of non-linearity going on. Yeah, yeah.

2:05:15 It's just that... There's the viscosity term, right? In this one that I showed you, there's no smoothing. There's no counteracting. It's just going. Right? But in the Navier-Stokes, and that's why it's so interesting. Because the Navier-Stokes has this non-linearity, but it also has this smoothing term.

2:05:32 It has this counterforce, which is the viscosity. And so you're asking, who's going to win? And are they always going to win? So far, it seems viscosity is always won. That makes so much sense now. 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.

2:05:50 Yeah, no, because you can understand – there's good arguments for both sides. One in which viscosity always wins feels true. Yeah, seems legit. Because we've been able to show that. Over and over again. In a variety of contexts as being true. Yeah. But being fundamentally true... Is another thing. Is another thing.

2:06:15 And because of, again, the nonlinear nature of the convective term in this case, that is challenging the viscosity as the counterforce, there's an argument that both of them have a reason that they could win. And in finite time. In finite time. Exactly. And so... It's an interesting problem. I like that.

2:06:36 I have to give it to him, right? It's something that definitely makes you think. And there are other things in physics that have singularities in finite time. I mean, the most famous one would probably be a black hole as a result of the Einstein field equations. The Einstein field equations are also very famously a nonlinear differential equation because you've got these tensors, and these tensors are multiplied by, to themselves and also multiplied to the reciprocal of themselves twice over.

2:07:06 And so you get these squared terms. For example, in a very simple case, 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 what? And so now you get these nonlinear actions on space and time that create the richness of the phenomenon that we see.

2:07:32 And on the right-hand side, I mentioned Lake Geneva. I was actually at the Yerkes 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 Einstein's general relativity because Einstein had visited the Yerkes Observatory many times.

2:07:50 And there on the slab of marble was written out Einstein's field equation. So I just wanted to show that when I received this news, I was looking at a non-linearity. 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 physics 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.

2:08:23 And physicists do not care about this singularity. 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.

2:08:44 And I've got some examples over here. I mean, for example... hypersonics, and supersonics even. Navier-Stokes equations don't apply. Navier-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.

2:09:08 Right. The sound wave is catching up to the sound wave in front of you. And that is going to compress the air. 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.

2:09:33 That's not in the Navier-Stokes equations. In the middle, that's an aquaporin protein. Those things are so small that they let in only water molecules. At a molecular scale, velocity fields don't matter at a molecular scale. Like, what are we talking about? Right, right, right. So again, at the molecular scale, Navier-Stokes equations don't apply.

2:09:54 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 problem, has to do with these two specific equations.

2:10:20 Which are specifically for incompressible fluids. Yes. And so I would even say that the millennium problem has nothing to do with fluids. 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.

2:10:43 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? But it's something that is outside the realm of physics. And I want to make that very clear because there's a lot of hullabaloo. And I think there's a lot of stories out there being like, oh, this is like fluids, right?

2:11:03 Can a fluid achieve singularity? No. 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.

2:11:20 And the fact that the mass of the fluid, like that density term, is going to have a Lorentz factor in there or something because the mass of the fluid is going to infinity the closer I approach the speed of light. I can't go to infinity. So all I'm saying is this is a mathematics question, completely legitimate, but let's not confuse this for something that is physically relevant.

2:11:43 It is not. And 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 singularities in these mathematical constructs that don't apply to the real world, but still have a fundamental value to our mathematical understanding of the world of just math.

2:12:26 Which, again, then can bleed into, like I was mentioning earlier, two ways to think about other problems. Because 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 this framework to fluids?

2:12:46 Yes. And then it created this whole centuries-long journey. Yeah, I mean, 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 things like Stokes' theorem. That's from the Navier-Stokes guy, right?

2:13:04 That's a mathematics theorem, Stokes theorem, that like then, oh, it like works here, right? So definitely something that is super interesting. And Terry Tao actually said that like the Navier-Stokes – Oh, we're calling him Terry now. Look at this guy. Oh, my buddy Terry. We need to edit that out. Hey, yo.

2:13:21 If somebody... All right, Terence Tao. No, no, no. Keep it in. But like Terence Tao, he said the Navier-Stokes equations are one of the simplest, super critical differential equations. So in the theory of differential equations, it's very interesting to think about because you've got this nonlinear term and you've got this competition with the diffusion term, right?

2:13:43 You've got a Laplacian and you've got this like convective thing and they're competing, right? And mathematically, it's a very interesting question to answer. Now, there's been a lot of work since the days of Navier Stokes to try and understand the mathematical properties of this. The first guy I want to talk about is Gene Lire.

2:14:05 He actually proved that fluids always possess something called weak solutions, where the total kinetic energy of the system remains bounded over time. How do you compute kinetic energy? Like, kinetic energy is really just 1.5 mv squared, right? For, like, a normal point-like object, like... You know how Newton's laws can be applied to a point-lock object, f equals ma?

2:14:25 Well, one-half mv squared. For a velocity field, it's a little bit different, right? What we have to do is integrate over a volume, because we've got a bunch of velocities in a volume, and we say the square of all of the velocities in that volume times one-half, that's going to tell you sort of the energy density in that volume, like the kinetic energy density in that volume.

2:14:45 Which would be, in this case, the velocity field. Right. Yeah, yeah, exactly. And so the kinetic energy density can remain finite for these weak solutions. And by weak solutions, these are stuff that don't have this strong smoothness criteria. And I'm going to be honest, at this point, I'm getting into stuff that...

2:15:04 This is the math stuff. This is maybe why I went on a tirade about five minutes earlier about how this is all math. Because maybe there's stuff now that I'm like, okay, I don't really know, right? But... From what I've gathered, what he's done is loosen the idea of smoothness and show that, like, you know, imagine you've got, like, a violent storm system.

2:15:26 He showed that the total energy of that storm can't suddenly go to infinity. Okay? But what he can't rule out is that even though the entire energy density of that storm doesn't go to infinity, you can have small pockets of... where the velocity is still going to infinity. Okay, you can concentrate into an infinitely fast, like, tornado somewhere inside, and you can't rule that out.

2:15:47 So the overall energy is finite, but some local small velocity can be infinite. So already it's starting to show that maybe there could be ways that the convective term wins. Mm-hmm. 1982, Kaffarelli, Cohn, and Nuremberg, they show that if the Navier-Stokes equations for a velocity field work, then you have partial regularity.

2:16:12 What that means is, if you have singularities, if they exist at all, They can't be like everywhere. Okay. You can't have giant chunks of fluid moving at infinite speed. But what you could do is have infinitesimally small point-like singularities, kind of like at the center of a black hole. And that works.

2:16:33 Okay? And so the system has some fundamental limitation where it can only happen at particular scales. Particular scales. At very small infinitesimal scales. This is why, again, it's like a mathematical problem, right? It's not... It's infinitesimally small scales, and it's happening only at particular places.

2:16:50 It's not like a continuous singularity, right? It's at the center of a black hole. It's a single spot. Okay, so that's them saying, okay, partial regularity. In other words, you've got these decades of work that is placing a very tight fence around any possible singularity, but we haven't proved that that fence has nothing in it.

2:17:11 There could still be a singularity. It just has to be constrained to these things. And so the box in which... The convective term wins. We've now created a very small box. Yes. It can still win. It could still win. Against viscosity, but it's a very small box. Very small box, right? And this brings up to 2000.

2:17:32 This is when the Clay Mathematics Institute announces the Millennium Prize problems in mathematics. These are seven very famous mathematics problems that were formerly Hilbert's problems. Um, that, you know, a bunch of them got solved over in the 20th century, some by John Nash, of course, um, and others.

2:17:51 And finally, we have, um, seven out of those that were picked as like the big ones. The point car rate conjecture has been solved already by Gregory Perlman in 2003. Um, And that was huge. He never accepted the million dollars because he said that someone else deserved it. He's kind of like this weird guy.

2:18:12 I'm going to be honest. He's a weird Russian guy, brilliant mathematician, but like lives kind of like in his mom's basement kind of thing. Yeah. Anyways, brilliant mathematician. So here's the problem statement. And this is a problem statement by Charles. I want to make one quick note here, which is for those viewing who can see the list, I want you to remember this list, right?

2:18:36 Because this list is going to come into relevance when we talk about the race between the frontier labs. And this is the gold medal that they're all chasing after. Yeah. is and the infinity stone and these are the infinity stones and they are coming after all of these maybe not yang mills and maybe not p versus mp yeah but all the other ones all the other ones reman hypothesis they came after it um or at least something that is related related um navier stokes is the one that we're at right now um yang mills i mean we we did yang mills for an episode last year.

2:19:09 That's one of my favorite episodes. Check it out if you haven't. And so here is the problem statement. So each of these problems have to be well-defined, right? Like when the Clay Mathematics Institute is like, this is the problem, the problem has to be well-defined mathematically. So Charles Fefferman, field medalist, Also a student of Elias Stein at Princeton University, the guy who wrote the complex analysis textbook and my professor of complex analysis also.

2:19:36 So he's a brother, I guess, an academic brother of Terence Tao. Because they both had the same PhD advisor, also field medalist. I just want to note for those who might not know, field medalist, meaning that they've won this, the most prestigious mathematics award that is available. For those who are under 40.

2:19:55 For those who are under 40. Yes. I would say the most prestigious mathematics award that people don't really know about is the Apple Prize. We talked about this. Which is, I think, like the Nobel Prize of mathematics. Fair enough. People keep saying it's the Fields Medal. I really don't think it is. Argue about it in the comments.

2:20:10 So Charles Fefferman puts out this statement of what it is that we are trying to prove, okay? Okay. We want the existence and smoothness of Navier-Stokes on R3. That means in three dimensions. Two dimensions, we've already figured it out. Olga said, it's smooth. We're good to go. Okay. A and B say existence and smoothness of Navier-Stokes in three dimensions.

2:20:38 C and D say the breakdown of Navier-Stokes in three dimensions. And just to quickly touch on this. So we're saying A and B means viscosity wins. Yes. And C and D means the convective term wins. Exactly. Yes. Okay. So A and B say it's always going to be smooth. C and D say it's... going to lose. In the small fence that we've already defined, the convective term ultimately wins and we get a very discrete small singularity somewhere.

2:21:10 Exactly. And so in A and B, that's a proof of global regularity, and that would mean you need to identify some kind of mechanism that prevents this infinite concentration at these very small scales, right? In practical terms, one would like some kind of quantity or some kind of scale critical quantity that's controlled for all time.

2:21:29 And it's some structural argument saying that whatever vortex stretching is happening that we were talking about, it's going to lose out to viscosity, okay? A proof for the breakdown would be the opposite. You just have to construct a genuine Navier-Stokes flow. You have to construct some kind of initial condition in which the concentration reinforces itself.

2:21:48 That convective term keeps making smaller and smaller things. And all of that pressure and nonlinear cancellations, after all of that, the mathematics becomes demonstrably something like a singularity. Okay. To make an analogy, because this is sort of how people always frame these things when they talk about AI's progress as it relates to these things, which a year ago people didn't think it could compete in the Math Olympiad and now it's solving C&D of a millennium problem.

2:22:18 But putting that aside, it almost seems, based on this construction we've just described, that it's kind of akin to a counterexample. Yes, 100%. Versus sort of the global solution. Mm-hmm. Not minimizing in any way because it's still a big deal. But people will nuance and nitpick on that point. It's like, great.

2:22:39 You found one. You're looking at this large search space and you found something that has blow up in finite time. Well, it's an incredibly small search space. Yeah. Right? And everyone's been looking. Looking at it. Right? But I think you raise a very big point. And I think we're going to come to that in the next few examples that I'm showing you.

2:22:59 Because in 2000, this is when the Clay Mathematics Institute came up with it, right? In 2016, big players are involving themselves in this. Because they smell the blood in the water. That there is a counterexample hanging out somewhere. Okay? More and more analysis is going into showing that perhaps there are situations where you get singularities.

2:23:26 So Terence Tao, he creates this averaged version of the Navier-Stokes equations. And he proved that the standard energy conservation laws are mathematically too weak to prevent a singularity. So he created this like averaged version of the Navier-Stokes equations. It's kind of like a blurred out version.

2:23:45 Maybe you can think about that. It's like a stunt double equation. of the fluid equations. And from the outside, this thing obeys all of the exact same conservation laws that the real Navier-Stokes equations does. But tau proved that this stunt double definitively explodes in finite time. Okay? So this averaged version can create singularities, which means that the principles that you would have used to prove that it wasn't possible are...

2:24:13 are also not going to work for the real thing. If it was global, you wouldn't be able to have an average stunt double that blew up. Yes, exactly. So there's a chance now that at least the original arguments that people were using, saying, oh, energy conservation and things like that, that might not work anymore.

2:24:31 Okay? So A and B are already not looking great. Right? Maybe there is... A singularity that's lurking somewhere. Okay? And for decades, mathematicians tried to trigger this blow-up using something like a fractal shape, like a whirlpool that's shrinking into smaller and smaller whirlpools and so on. This usually failed.

2:24:52 Because it required an infinite amount of energy. And crucially, you can't do that. One of the things that Pfefferman said is, you can't just stir it infinitely fast and be like, oh, I got infinity. That's not the interesting question we're trying to ask. Okay, so you supplied an infinite amount of energy and you got an infinite amount of velocity.

2:25:10 Great. No, it's supplying a finite amount of energy to get an infinite amount of velocity because the nonlinear convective term is concentrating energy. into smaller and smaller structures. So both finite energy and finite time are both relevant in the context of this. Yes, 100%. And so Diego Cordova and Luis Martinez Zorora and Fan Zhang in 2023 and 2025, this is very recent, They changed the paradigm by abandoning this whole single shape thing and relying on something called a layer cascade.

2:25:43 It's kind of like a bunch of nested gears or, you know, those Russian dolls where you have like a big doll and then smaller doll and smaller doll and smaller dolls. That's what they were constructing. Instead of a single vortex that they do this vortex stretching in, they've got slow moving outer vortex and then a medium inner vortex.

2:26:02 and then a smaller inner vortex that spins faster because of the medium vortex. And the medium vortex is kind of like a parlay, or like, not a parlay, like an intermediary between the outer and the inner. And it acts upon an even smaller layer, and so on and so forth. And you can design this geometry so that the feedback between these gears doesn't disrupt the big gears.

2:26:25 And the feedback between the smaller gears... goes up, up, and up, and you can get unbounded velocity. Okay? And if you have an infinite number of gears, then perhaps you can have an infinite velocity. And so the mechanism they called a dynamical amplification across scales. And they used this to prove finite time blowup for the unforced 3D Euler equations.

2:26:49 Remember the Euler equations without viscosity? Without viscosity. Yeah. They showed that without the smoothing term, you could actually have finite blow up. The pressure won't do as much because the pressure also does a little bit of smoothing. Because it was both the pressure and the external force were the only two of the three on the force part of the equation.

2:27:05 Yeah. And so this is unforced. So no force. You just set up an initial. Ah, okay. Okay. Yeah. No force. There's no stirring. There's no force. You just set up an initial condition, and the scales converge in such a way that you get an infinite blowup. Now, this is huge. They're chipping away at each of the variables on the right side of the equation.

2:27:27 Yes. Wow. The problem was, so that was for the unforced 3D Euler equations, right? But obviously we're interested in the ones that have viscosity. So there's this one type of equation called the hypodissipative Navier-Stokes equations. It's a variant that utilizes this thing called a fraction Laplacian instead of the actual Laplacian.

2:27:52 The del squared u that we had for viscosity, the triangle squared u, that's a Laplacian. This thing has a fractional Laplacian. Fractional Laplacian. And they showed that in this case, we can use the same sort of paradigm and get blow up. The only problem was that they required a forcing term that was rough.

2:28:15 Meaning it's not smooth. Like, the forcing term itself has tiny little bits of infinity. Like, the way you're stirring it has, like, little jolts of infinite stuff. You're pumping infinity into your stirring. And so... We're getting closer, though, because the infinity is small enough that you can maybe figure out a way to smooth it and still preserve the dynamics.

2:28:40 And so I just want to pause really quick because what we're sort of saying is human mathematicians... have now taken this Navier Stokes Millennium Prize, looking at finite blow-up, and then they started with looking at taking out the smoothness and taking out all the force. We got some solutions, which are effectively the Euler equations without the F.

2:29:11 And then we've now started adding in, back in the viscosity. But that weird version. But the viscosity wasn't smooth. It was rough. And so when people see this smoothness and roughness, part of what we're saying is the viscosity term has some special sauce that doesn't quite get you to the natural smoothness that arises in the equations with viscosity un- Yeah, the raw viscosity.

2:29:40 The raw viscosity. But it's still important because these are fundamental. No, and this is what even OpenAI's thing is built on. Okay? It's like, there's a reason why the Navier Stokes is the first of the six infinity stones to fall. Which we just talked about. It's because we got so close to... as human beings it's like a racetrack and someone was on the lap the fourth lap of whatever the 800 meter yeah and uh you know open eye just shows up grabs a baton and finishes yeah yeah yeah Not minimizing.

2:30:14 Not minimizing, of course. But I'm just saying, a lot of human input has gone into this, right? So Cordoba and Martinez-Zorora, they established this layer-by-layer scale amplification. It's a huge deal because everyone can sense that we're getting close. Okay? And shortly before the OpenAI announcement, mathematicians Tristan Buckmaster and Levant Alpoghe, they published preprints that elevated this methodology and, to accommodate for smooth forcing of Euler, meaning you've got the Euler equations, but now with a smooth force.

2:30:44 The Boyzonesque equation, which is something that's like kind of similar to the Navier-Stokes equations. And this is where the sort of drama starts, but I'm going to save that for later because a few days after that, OpenAI announces that they have solved it. Here we go. Here we go. On the Navier Stokes Millennium Prize problem.

2:31:05 They announce it on Twitter. They have it on their website. And the actual paper is something like 165 pages long. The theorem proved by OpenAI constructs a smooth, compactly supported external force that drives a fluid starting from rest into a singularity. That's what they're doing. They're solving C and D. Right.