You touch an essential point: There is a big difference between
100 and 10^100 = googol. If you do not make this distinction
in a quantitative analysis, it is not an analysis. I am surprised
to see that some (pure) mathematicans do not make this distinction,
and I wonder what mathematical tradition it can reflect.

Claes

A more practical example: suppose you have a computer program with array references like a[x*y + z]. It is provably impossible in general for a compiler to tell without running the program whether there will ever be a subscript-out-of-range error. Suppose instead the references are like a[37*y++19*z+3*w]–that is, you never multiply two variables together in a subscript, you can only add them or multiply by constants. Then compile time checking is possible (this is called Presberger arithmetic). The running time formula is a tower of iterated exponentials in the size of the expressions, which is absolutely intractable in the worst case, but it’s a remarkable theoretical result that it’s decidable at all. And it turns out that for most real practical cases, the worst case exponential tower is avoided. And even without that, the decidability is of interest to pure mathematicians. Maybe they’re not of interest to anyone else, but pure mathematicians went into that field precisely because they’re into that sort of thing. It’s not up to anyone else to talk them out of it.

Who knows, maybe global stability will involve growth functions like that. If so, the Clay math people want to know about it. In a physical sense it’s as bad as a singularity but from a pure math point of view it’s not the same at all. And as Terence and others have said, reformulations of the problem may be very interesting from a physics perspective, but the pure math question posed as a Clay problem is very specific.

You are right that an unstable numerical scheme for a stable problem
can give artifacts. But our computational Euler solution is validated
by a posteriori output error estimation by Euler residuals multiplied
by stability factors obtained by solving dual linearized problems.
A computed Euler solution is thus a representative solution and
not an artifact. Since the computed solution shows blowup and
is representative, there is blowup.

Correspondence (with e.g. Terence) on the blowup problem is published on
http://www.nada.kth.se/~jhoffman/pmwiki/pmwiki.php?n=Forum.Clay

Claes

The issue is that the NS equation is highly numerically unstable, so simulations showing blow up tell us very little; the blow up may be an artifact of the simulation rather than an indication that NS actually blows up.

[different anonymous from 7:20pm above]

This discussion is important. I hope e.g. Terence will take part.
You express a common misconception, addressed in the article, that
computational evidence is not mathematical evidence.
The central concepts are wellposedness and turbulence, and in this
context a computational solution is as much a solution as anything.
Please read the article with an open mind. Looking forward to
your comments.

Claes

I don’t think this paper is called “resolution”… it is just some numerical evidence. For incompressible Euler, finite time blow-up or not, has been a long time controversial question. Both sides have numerical “evidence”…

For incompressible Navier-Stokes, physicists tend to believe no finite time blow-up.