Sunday, October 08, 2006

An immortal smooth solution withdrawn

The widely discussed possible proof of a $1 million Clay Institute problem called
by Penny Smith has unfortunately turned out to be mortal and falsifiable, and it was withdrawn today due [to] a serious flaw. Well, the article in Nature written just 1 month after Prof. Smith started to work on the problem was arguably a little bit too fast, much like many other recent articles that search for really fast answers to really difficult questions.
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The terminology "immortal" refers to solutions that exist for all times in the future given some reasonable boundary conditions. It is non-trivial to prove the existence of a regular solution of the Navier-Stokes equations because the equations admit complex phenomena collectively described as turbulence - behavior that can get out of control.

Figure 1: Some literature about Navier-Stokes equations...

Nevertheless, I think that Prof. Smith has nothing to be ashamed of: serious thinking sometimes requires a trash can. Also, my guess is that this is not an infinitely difficult problem and everyone experienced with rigorous proofs in the field of differential equations should try to attack it even though you will probably have to develop some new methods, just like Prof. Smith has bravely tried.

And yes, my guess is that her yes/no answer is right: a smooth solution exists anywhere. In this sense, hydrodynamics is more well-behaved than general relativity. General relativity is beautiful but we often hear inappropriate religious comments that it is "special" and has mysterious properties that make its UV behavior better than the infidels could ever imagine, and so forth. ;-)

When you actually look what's going on, you see that the UV behavior of pure general relativity is much worse than the behavior of renormalizable field theories including Yang-Mills and its ability to create problems such as singularities exceeds all other systems of differential equations including the turbulent Navier-Stokes equations.

As mentioned above, two days before the paper was withdrawn, Nature celebrated it. A few hours before it was withdrawn, it was already used by some other nerds to prove that "the last claim of sexists fell" and the debate was over: everything about the male-female cognitive differences is caused by culture. ;-) Well, not yet.

Thanks to Charles. See also an encouraging sentence of Clifford Johnson.

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