Note: I wrote this years ago, to be precise on 7 January 2007 — so some of the links might be out of wack. At that point I still lived in LA and was a historian. But I thought since I haven’t been posting all of my current fabulousness, I would at least give you some of past fabulousness.
This quest was spurned on by a friend who was interested in learning more about the Navier-Stokes (N-S) equations. (They’re a system of equations, which is why they are referred to in the plural. All the equations together, describing the system in 3-linear dimensions and 1-time dimension, define how fluid flows.)
I learned about them in 18.354, a class devoted to the study of fluid dynamics at MIT. What are the N-S equations, you ask? As stated, the equations describe the way fluid flows – but fluid means more than just things like water, but almost anything from honey (very viscous) to a gas (well, a gas moving at speeds much less than the speed of sound). The only limitation is that the system has to abide by something called the “continuum hypothesis”. One website nicely puts it like this: “The basis for much of classical mechanics is that the media under consideration is a continuum. Crudely speaking, matter is taken to occupy every point of the space of interest, regardless of how closely we examine the material… it is well known that the standard macroscopic representation yields highly accurate predictions of the behavior of solids and fluids.”
So the N-S equations can answer some pretty cool questions about everyday life. Why does an airplane fly? The answer lies in how air flows around its wings. How long does it take for a stirred cup of coffee to become still? The answer lies in the effect of the cup on the coffee. (Believe it or not, the velocity of the coffee very very close to the edge of the cup — called a “boundary layer” — is actually zero. The coffee doesn’t move. And this layer, over time, eventually affects the rest of the coffee spinning until the coffee is totally still. Of course, to be totally precise, you need to take into account the effect of the bottom of the cup too.)
It’s hard to explain what makes the equations so neat. First is that even though they look complicated (see Wikipedia), they are actually pretty easy to derive from first principles (read: from scratch). Second is that they apply to so many phenomena — and much experimental work that has been done confirms it. Third is that they are still pretty mysterious. I’ll get back to that soon.
The N-S equations straddle the boundary between the pure and the applied. To be more accurate, perhaps, they do a good job of demolishing the myth that there is a “pure” and an “applied.” They are used as tools in a number of real-world problems. But at the same time, they represent a challenging problem of pure mathematics. This is what I meant by “mysterious.” They aren’t quite as well understood as mathematicians would like. Right now, the Clay Mathematics Institute has offered a $1 million prize to the first person to make some real headway into understanding the N-S equations: it is a millennium problem.
For the laypeople, not us, the Institute describes the problem as such:
“Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.”
That’s a bunch of fluff. The way the problem is posed by the Clay Math Institute for mathematicians, it appears insanely complicated. But I am going to try to explain the problem for a population who wants more than fluff and less than incomprehensible jargon.
The N-S equations are something called “Partial Differential Equations” (often referred to as PDEs). You might remember seeing differential equations in calculus (e.g. dx/dt=2x). Well, a partial differential equation is like a regular differential equation you’ve seen, but with more variables (e.g. du/dx+du/dy=3). When you first learn to solve them, in your intro PDE class, you are given a hodge-podge of tricks to solve a limited set of those equations. But this isn’t bad teaching; it’s the nature of the beast.
Let me explain. Remember in calculus when you were given an integral, and you had to figure out how to simplify the problem in order to integrate? You couldn’t integrate everything put in front of you… even if you wanted to, you wouldn’t have been able to integrate (x*sin(x)*5^x)^x by hand, because there’s no good way to do that short of putting it in a computer and churning out an answer. So you could only solve by hand certain, easy integrals. The same thing goes for PDEs… they are really tough to solve by hand… sometimes they can be solved by computer… and sometimes there isn’t even a solution! (How could there not be a solution? Sometimes, no solution exists for a problem. For example, the equations “x+3y=7” and “3x+9y=4” can’t give you a solution for x and y. No x and y exist that satisfy both equations.)
So in an intro PDE class, you learn to solve only certain “classes” of PDEs that have solutions. Sometimes you’re lucky and you can do them by hand. Othertimes you can’t and you have to put them in a computer. But, as I said, sometimes there isn’t even a solution.
What the Millennium problem asks a mathematician to prove is “merely” that the N-S equations have a solution (this is called proving the “existence of a solution”).  What does this mean?
So say you’re given a fluid system (imagine, say, a really really big cube of water, so large that for our purposes, it is so huge that it fills the universe… in other words, all of space is filled with this fluid) and you are given the following pieces of information:
1. the fluid’s viscosity (in this case, the viscosity of the water; remember viscosity represents the internal frictional forces of the fluid), and
2. the initial conditions of the system (the velocity of the water at every point in the universe at a certain time t=0)… [The initial conditions you are given are “smooth”… continue reading to find out what this means.]
So you have this giant cube with liquid in it, and you know how the liquid is moving at the beginning. You let the liquid continue to move around, defined by the N-S equations which describes fluid flow. Liquid with a velocity pushing upwards, for example, will displace other liquid which will displace other liquid, etc., and the whole system is churning.
EXISTENCE: The millennium problem says that you have to be able to prove that a solution exists to the N-S equations. You don’t need to find the solution, but you need to prove that it exists. What does “a solution to the N-S equations” mean? What it means is that by solving the N-S equations, you can (1) give the pressure of the fluid at any point in the universe, and any time in the future, and also (2) give the velocity of the fluid at any point in the universe, and any time in the future. These two things (the pressure and the velocity) define the system; if you can find both of these, then you have the solution to the N-S equations.
But recall that the problem is abstract. So to solve the problem, you can’t merely say that for a single particular system, you can show a solution exists. It would be pretty easy to show, for example, that a universe filled with water which is at rest at time t=0 (the initial condition of the fluid is still) will never change. So you can say that a solution to this system exists. But you haven’t solved the millennium problem. What makes this problem hard is that you have to say for all systems, solutions exists; in other words, you want to say that for a universe filled with a fluid with ANY viscosity, given ANY initial conditions, a solution exists. That’s what makes the problem hard.
SMOOTHNESS: There is one thing I left out, but now I can add it in. The Millennium problem doesn’t just ask that you show the existence of a solution, but also that the solution is smooth. In math, “smooth” has a particular definition, but what you need to know is that in this problem, the desire for a “smooth” solution comes out of a physical concern. The system, at every point in time, must have a finite energy. (For those who care, mathematically, this is calculated by taking the integral of the square of the speed of all the points of the system, and showing that it is less than infinity.) 
COUNTEREXAMPLE: Of course, one easy way to solve the problem is to prove the opposite. I think mathematicians generally are fairly confident that there is a proof that can show the existence and smoothness of a solution to the N-S equations. But if you can come up with just a single system with smooth initial conditions, a particular viscosity, and smooth external forces acting on it (like gravity), and prove that that system DOESN’T have a solution, then you’ve also solved the problem. Because you’ve shown that no matter how hard mathematicians try, they can’t find a proof to the problem, because you’ve find a counterexample.
 To make the problem easier, the Millennium problem people even said that you don’t need to consider ANY external forces on the system (so, for example, in the universe, you don’t need to have any gravity). In the most general version of the N-S equations, these are incorporated.
 Recall, however, that the initial conditions have to be “smooth.” This fact should make it easier to show that the solution will be “smooth.”
PS. I’m aware that I probably got some of this wrong. Plus there’s the added difficulty of being 100% truthful mathematically while using words without just writing the math out, which pretty much defeats the purpose of me trying to do this. Feel free to correct.
UPDATE: I finally found a really nice explanation of the problem on this blog which attempts to explain a proposed (but now shown wrong) solution. Read up to but not including the paragraph beginning “So, how does Penny Smith’s analysis approximate this by a set of hyperbolic equations?”