Via NEW: a serious-looking claim by Penny Smith to have solved the millenium prize problem on the Navier-Stokes equation and a web page explaining it. I don't understand this stuff very well at all but I'd be interested in finding more reactions from people who do.
ETA2: Withdrawn due to a "serious flaw" (as of 8 Oct). Too bad, it was exciting while it lasted.
Very interesting, indeed -- and thanks for the links; I wouldn't have seen it otherwise.
Despite having nearly finished a Ph.D. in computationally solving the Navier-Stokes equations, I can't say that I understand it any better than you do. Though there's one piece that might be of interest:
The basic idea of considering the solution of the actual equations to be a limit of a set of hyperbolic-equation solutions is an old trick on the computational front; some of the simpler methods amount to just taking one of the a hyperbolic equation and considering it "close enough". And, actually, one of the methods that I've used works by a process that's not too different from computing the convergence of a series of such solutions at each timestep. I haven't translated the notation to tell whether the set of hyperbolic problems that she's using are the same as the ones that are used computationally or not, though; the representation is rather different.
Everything beyond that in the paper is pretty much out of my field of knowledge.
Thanks! That's definitely more than I understand of this stuff. I know what a differential equation is, have some idea what it means for it to have a solution in this sort of context, and that's about it.
Glad it was useful, then!
My other impression of the paper, on second look, is that it was relatively hastily written -- or, at least, rather hastily proofread. Aside from the frequency of grammatical typos in the text, I noticed an significant error in one of the equations (which has since been corrected).
Is this sort of haste typical in mathematics? (I suppose it would be, in a case where one is wanting to claim priority for a million-dollar prize.)
And the fact that you found that useful inspired me to do a bit more of a writeup of those things, in my otherwise-long-dormant technical blog here. It turns out that the hyperbolic equations she uses are slightly different from the ones I used in computational approximations, but not a lot.
Cool — it'll take me a while to get through what you wrote, but I added it to my list of "ETA" links.
I just received this note from a friend of mine who is more than a fairly decent numerical analyst.
"I was discussing her papers on 2nd order hyperbolic equations with (another numerical analyst). We decided that those results are pretty amazing by themselves. She has only withdrawn the extenison to quasilinear systems. I want to get those papers on the linear systems and see what they say about matrix approximations. Both <Numerical Analyst friend> and I are very surprised that these results escaped the experts in the field."
So maybe it doesn't get the Clay Prize, but I wouldn't throw the baby with the bathwater.