[lolipop1]

So first, the equations seems to be mostly used to approximate and model a lot of stuff. Very precise, I know.

Second, here's wikipedia take on the Millenium problem: Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularity (smoothness). These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example.

Does the article directly relate to the Millenium Prize problems? I can't conclude anything but it seems like just a small part of the whole. It think people are excited because there was a lot of talk about the possible solution to the P equals or not NP problem.

Maths at those levels are esoteric to most people and as such, presenting papers like that is pretty useless in mainstream media. If one could find a good analysis of the conclusions and possibilities, it might be a lot more useful.

[golwengaud]

It looks like the author is attacking statement C from the official problem description (http://www.claymath.org/millennium/Navier-Stokes_Equations/n...). That statement asks for "a smooth, divergence-free vector field u0 (x) on R3 and a smooth f(x, t) on R3 x [0, ∞), satisfying [some physical-reasonableness conditions] for which there exist no solutions (p, u) of the [the Navier-Stokes equation, a physical reasonableness condition, and a sort of sanity condition]." In essence, you're finding a counterexample for uniqueness.

The author claims to have found exactly such a combination of initial-value field u0(x) and force function f(x), solving the millenium problem. I can't really judge his claim yet, as I haven't read the whole paper (and I'm probably not qualified to say even once I have), but slackenemy presents some good reasons to be rather dubious elsewhere in the thread. Also, applying Scott Aaronson's "Eight Signs A Claimed P≠NP Proof Is Wrong" (http://scottaaronson.com/blog/?p=458) isn't pretty.