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Another Millenium Problem May Have Been Solved
Posted by
Zonk
on Sat Oct 07, 2006 03:32 AM
from the we-all-miss-our-loved-ones-and-gas-equations dept.
from the we-all-miss-our-loved-ones-and-gas-equations dept.
S3D writes "After recent verification of the proof of the Poincaré conjecture, another of the Clay Institute's Millenium Problems may have been solved. This new solution is for Navier-Stokes equations under physically reasonable conditions. Navier-Stocks equations describe the motion of fluid substances such as liquids and gases. Penny Smith has posted an Arxiv paper entitled 'Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System' which may prove the existence of such solutions."
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Hm. (Score:5, Funny)
LOUD ones.
Re:Hm. (Score:5, Funny)
Parent
Neat indeed (Score:3, Interesting)
Note: Not considering P vs. NP as it is quite possibly unprovable.
Re: (Score:3, Informative)
Just because we can't prove it doesn't mean it's unprovable.
Godel's incompleteness theorems [wikipedia.org]
Re: (Score:3, Interesting)
Re:Neat indeed (Score:5, Insightful)
Not necessarily -- it is conceivable that there exists a poly-time algorithm for an NP-complete problem, but there is no proof (within ZFC, say) that it is correct. The physical truth is certain -- but what we can know about the physical truth is limited.
Now, I'm with you in believing that that's extraordinarily improbable, but math doesn't always respect what we consider to be likely.
In my opinion (as a complexity theory grad student), the "maybe P=NP is independent" speculation is bunk. There are genuine, interesting results talking about the limits of how we can resolve P vs. NP, but none of them come anywhere near logical independence, and giving up on a field-defining problem after 30-odd years is just very odd considering how long the really major open problems often take to solve. I believe the solution exists, and I hope it is found soon, but I will be unsurprised if it takes another 100 years or so while we get a better handle on what computation really means.
Parent
Re: (Score:3, Informative)
Not really. Actually solving Navier-Stokes for concretely given boundary conditions is very much an applied math problem, maybe the most important one of them all, and it is done with computers and algorithms from numerical analysis. But the paper we're discussing here is pure math: she proves that for a certain class of boundary conditions a solution must exist, without saying what it looks like or how to get it. It's of pure intellectua
Quite impressive (Score:5, Informative)
Re:Quite impressive (Score:5, Informative)
Parent
Re:Quite impressive (Score:5, Informative)
it's about the existence of a solution for certain boundary / initial conditions of the NSEs. This is still a very big deal because you can now expect correct results when doing numerical calculations. By the way you probably meant FEM (Finite Element Method), not "fractional element methods". FEM is rarely, if not at all used for solving the NSEs, you'd rather use Finite Volume Methods (applicable for structured and unstructured grids, as are FEM).
Parent
Re:Quite impressive (Score:4, Funny)
Parent
Re: (Score:3, Funny)
but rather the PROOF of THE EXISTENCE OF A FORMAL SOLUTION. You still have to find it,
either analytically or (most probably) numerically.
Bottom line: about this a mathematician gets horny, an engineer says SO WHAT!!!
Ciao
Whuh? (Score:5, Funny)
Someone had better tell the Formula One teams (Score:3, Interesting)
Re:Someone had better tell the Formula One teams (Score:4, Informative)
Parent
I solve 3 millennium problems before breakfast (Score:3, Insightful)
Wait for the peer review to begin. I've not seen anyone familiar with the field say anything about the paper yet, only then does it gain credibility.
FatPhil
The toughest millenium problem of all... (Score:4, Funny)
Re:The toughest millenium problem of all... (Score:5, Funny)
A millennium is mille + annus: a thousand years.
A millenium is mille + anus: a thousand assholes.
If you get it wrong, you're anal; if you get it right, you're annual.
Parent
What is the geometry? (Score:3, Informative)
It is a big deal for the mathematicians. That is all
The N-S Eqn has been "solved" in 2D using Velocity Potential, Stream Function approach. But in 3D stream function does not exist and the method does not extend. But in practice the only problem that is really "solved" even in 2D was was this driven cavity problem, a box with a moving wall.
Take the much more simple to solve for a hundred years, the Heat Equation. Analytical solutions exist for simple domains like a semi infinite plate or a box with Dirichlet boundaries. But in practice ANSYS sells numerical solutions to Heat Equations and the industry has been buying millions dollars worth every year. Similarly FLUENT (Recently acquired by ANSYS) does not have to worry its market has fallen out of the bottom. For real life geometries we will be using numerical solutions of NS Eqn for the foreseeable future.
Further though I could not see any geometry restrictions in the paper, it appears as though they have just proved solutions exist, and not actually solved it. Depending on the assumptions made and terms neglected, engineers may be able to build better turbulence ing out of this.
Caveat: Though I started out in CFD I have not read CFD papers for some 12 years. and frankly I dont understand much of the math in this paper.
Re: (Score:3)
I wouldn't go so far as to say it is only interesting for mathematicians. Fluid dynamics and Navier-Stokes especially, is what, for example, many 3D engines use to simulate water by now. Granted, they use simplified equations, usually only taking the surface into consideration, but any breakthrough in the theory their models are based on might have implications for those models as well. I'd say let's wait until a) those new findings have been properly p
An important step (Score:5, Informative)
Back to the paper... While I am not a mathematician, the paper appears kind of rough to me - lots of punctuation errors, commas in the wrong place, unclosed parehtneses... I suspect this paper has not been fully through the peer review process. I don't know how the mathematicians do it, but I would say this paper is a draft (not discrediting the work - I am not quallfied to judge it - but it looks rough).
Re: (Score:3)
Withdrawn (Score:4, Informative)
Re: (Score:3, Interesting)
http://en.wikipedia.org/wiki/Catastrophe_theory [wikipedia.org]
Re: (Score:3, Interesting)
One of the things that I understood was a real problem with NS is that not only were there no existence proofs, but there were no uniqueness proofs. Does nayone know if the uniqueness question has been answered?