Navier-Stokes Millennium Prize Problem Solved?
Back in 2000, the Clay Mathematics Institute put bounties on seven of the world’s most difficult mathematical problems. Some of these problems have been around for over a century, with many a brilliant mathematician racking their brains over them. If you manage to solve one of them, there’s a 1 million dollar prize waiting for you. To date, only one of the Millennium Problems has been solved. Now, however, there’s a claim from a Professor of Mathematics from Kazakhstan, who believes he has solved the Navier-Stokes problem.
The Navier-Stokes Equations
The equations we’re talking about are particularly nasty ones that describe the motion of liquids and gases. They have very important applications in modeling the weather, ocean currents and aerodynamics. The equations are named after the physicists who first came up with them back in the 19th century (Navier and Stokes.)
However, even though the Navier-Stokes equations are over a century old, they’re still tough nuts to crack. Solving them is tricky and, most interestingly to the mathematicians, no one has yet been able to prove that solutions even exist in all circumstances. That’s the problem to solve if you want that Millennium Prize: prove that the Navier-Stokes equations always have a solution and that this solution is always smooth (no annoying singularities or discontinuities.)
So why is this such a difficult problem? What makes the Navier-Stokes equations so problematic? Or, from a more physical perspective, what is it about the movement of water or air that makes it tricky to describe? Much can be said on the topic (a very interesting discussion can be found here) but to keep things simple, the answer is turbulence.
For a lot of physical systems, things become simpler if you zoom in on them. The smaller the feature of the system, or the shorter the time interval in which you look at it, the less interesting it tends to become. A race car going around a track can be very exciting. But if you film it with a high speed camera and only look at its movement for a fraction of a real-time second, it will just be going in a straight line.
Not water or air, though. If you’ve ever blown out a candle and observed the smoke, you’ll know what I mean. (See Fig. 1.) On a large scale, the smoke just rises up into the air and slowly vanishes. But if you look more closely, you’ll see that the smoke is performing all kinds of chaotic motion: it’s highly turbulent.
It is this turbulence that makes our lives so difficult. Not only is it chaotic (making it difficult to predict the weather, for example) but the fact that it gets worse on smaller scales also means that a lot of mathematical tools that are commonly used to solve these kinds of equations simply don’t work.
So, can we cross this one off the list? It will probably be a while before we know for sure. Prof. Mukhtarbay Otelbaev, who produced the 100-page(!) article proposing a solution to the Millennium Prize Problem, wrote in Russian. (Even if it were in English, though, it would probably go way over my head, so I won’t be able to tell you.) A project has been started to translate it and some (critical) discussion is underway here but we will have to wait until the Clay Institute has reviewed Prof. Otelbaev’s work before it becomes official. This could take a few years (for comparison, there were about seven years between the moment Grigori Perelman first wrote down his solution to the Poincaré Conjection Millennium Prize Problem – the only one that’s been solved – and his actually being offered the award.)
So, we will have to wait and see. We can’t get too excited yet, but if Otelbaev’s solution holds up against review, an old and very difficult problem will have finally been resolved.