Navier-Stokes in 3D

It's not exactly the Nature Trail to Hell (also in 3D), but this is pretty important mathematically. Today I saw a talk from Yakov Sinai, one of the preeminent living mathematicians in dynamical systems and mathematical physics, on his work on blow-up of solutions to the Navier-Stokes equations in three-dimensional Euclidean space. A sufficiently complete solution (actually phrased as "substantial progress" on the official website) to existence and uniqueness of Navier-Stokes in 3D is worth $1 million from the Clay Mathematics Institute, as this is one of the millenimum prize problems.

Here is what is written on the website:

"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."


The Navier-Stokes equations are partial differential equations describing fluids*, and its existence/uniqueness issues become much more difficult in 3D than in 2D. (The 2D version of this millenium problem is solved, although that's not to say there aren't still tons of things to learn about the 2D system... This basically results from there being an inequality in the 2D case whose validity doesn't carry over to 3D.). The prize can be awarded either for an existence/uniqueness theorem or for a result about lack of existence/uniqueness. (For the precise description of the problem, go here. Today's speaker was discussing solutions that blow up (become infinite) in finite time. If I read him correctly, he seems to think he's on the right track for the millenium problem prize, although that's quite a grandiose claim if that's what he actually meant. Given all the stuff he's accomplished in his career (and the fact that he's now in his early 70s), it certainly is reasonable for him to tackle such a problem. It's not like he has to work on shorter-term projects, so he might as well go for the gold. Understanding these issues would aid in the understanding of turbulent dynamics, although putting on my applied hat, we need to know the dynamics of the solutions and not just their blow-up properties. (The prize is for a pure math problem, but engineers using the Navier-Stokes equations to study turbulence are going to be interested in much different things.)

Also, my postdoc advisor at Georgia Tech (Leonid Bunimovich) was a Ph.D. student of Sinai's. (These are the two main people with mathematical billiards named after them.)


* For a typical fluid mechanics problems, one will start with Navier-Stokes and then go to a simpler equation appropriate in the regime of interest and then one will study the "simpler" equation, which usually itself isn't so simple. Also, there are studies of fluids from a statistical mechanics perspective, etc. Anyway, I'm not a fluid mechanist, and I'm not trying to be precise about things here.