Counter-Example to the Navier-Stokes Millennium Problem Found | Hacker News Reader

Counter-Example to the Navier-Stokes Millennium Problem Found

ejde.math.txstate.edu

31 pointsjperras7 comments

I spent half an hour composing meaningful description of insignificance of the work (without judging if it is indeed a result, attacked "uniqueness" is more of a technicality than anything much of a big deal) only to find out author also published stabs at some other Clay problems: http://arxiv.org/abs/0809.4935 (PNP), http://arxiv.org/abs/0806.2361 (RH)… at which point I have nothing more to say.

Flagged.

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.

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.

what

oh, this might help

"Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations."

http://www.claymath.org/millennium/Navier-Stokes_Equations/

that doesn't seem to setup an assertion for which a counter-example could be shown.

My understanding is that the Navier-Stokes problem has to do with showing that smooth solutions always exist for the three dimensional system of equations, so a counter-example would be a set of initial conditions for which no smooth solution exists. On Wikipedia, the problem is described as:

"Prove or give a counter-example of the following statement:

In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations."

http://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smo...

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