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now I am trying to understand what happens within a 10 inch by 10 inch by 10 inch piece of a hurricane. That may sound depressing, the idea of studying such a little speck of a big nasty storm, but the good news is that I’m making progress! To be of use to you, we will have to get much better at understanding my little 1,000-cubic-inch box, and then eventually move up to boxes 1,000 cubic miles in size. It will be awhile.
    In the meantime, the National Hurricane Center makes predictions that essentially assume a hurricane is a solid cork floating in a swirling but well-behaved soup of water-loaded air. How well do these cork predictions work? Not well at all. Hurricanes aren’t corks. They are not floating in a well-behaved soup. The soup is a mess, as is the hurricane, which is not surprising, since trying to make a distinction between the hurricane and what surrounds it is never precise.
    I wish to describe and predict where a fluid particle inside my 1,000-cubic-inch box inside a hurricane will go. So everything in that bit of geek texting above called the Navier-Stokes equation relates to either the nature of the fluid or how fast it’s moving. ρ is how dense the fluid is,
p
is its pressure,
v
is its velocity, and
t
is time. The equation states that if you want to know how the velocities of a fluid change with time, you need to keep track of how the fluid’s pressure changes with space and the stresses,
T
, and external forces,
f
, on that fluid.
    It is routine for upper-class undergraduates who are concerned with fluid dynamics to derive the Navier-Stokes equation, known for more than 150 years, on their own. It is simple to do this, actually. You don’t even have to know anything about fluid dynamics. Just start out assuming Newton is right and apply Newton’s fundamental laws to fluids. Voilà. But please don’t misunderstand and think that just because it is easy to do, this work is trivial. It took more than 150 years to go from Newton to Navier-Stokes, and the work involved the best and the brightest mathematicians of Europe, including Leonhardt Euler.
    I am happy to use the Navier-Stokes equation. But I am not a mathematician. I’m a user of Navier-Stokes, not an inspector of its correctness. David Hilbert was, however, an inspector. In 1900, he announced to the world that there were twenty-three major problems in mathematics that awaited solutions. One of those problems, number six, expresses the need to prove the fundamental correctness of using equations like Navier-Stokes to describe the physical behavior of materials. Then, in 2000, one hundred years after Hilbert, a group of mathematicians examined the future of mathematics again. Many of Hilbert’s problems had in fact been solved at least partially over the interim years. But Hilbert’s sixth remained a complete enigma. In a nutshell, the new committee dramatically reduced the scope of Hilbert’s original problem to something more manageable than all equations used to describe physical processes. They chose just one equation—perhaps the most important and certainly one of the most baffling—Navier-Stokes.
    The fact is that when your geeky college niece or nephew or my students derive this equation from Newton’s laws (with the help of Euler’s work), they are making assumptions about the behavior of fluids that are so naïve as to be ridiculous.
    Think of a hurricane, the water and air violently going every which way with a mixture of order and chaos. We call this mixture of movement, this wildly erratic dance of fluids, turbulence. You know this word from air travel, and it never means anything good. But to me, it means something beautiful. Without it, understanding hurricanes would be boring. Mathematicians’ worries about this equation would be the height of neurosis.
    Add turbulence and mathematicians’ concerns are quite sane. The committee of 2000 thought they were so appropriate that they offered $1 million, a Millennium Prize, to