Another Millenium Problem May Have Been Solved

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."

Pretty nifty stuff

If this turns out to be true, then it's a pretty big deal. I remember studying this kinda stuff a few years ago... suffice to say that it really makes my head hurt, even now. Having had a quick look at the article, it does promise to be a very interesting

Hm.

I have no idea what any of that means, but rest assured that by the time this thread ends I will have developed ironclad opinions on the subject.

LOUD ones.

Re:Hm.

I have a truly remarkable proof that will convince you, but the dang lameness filter is getting in the way.

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Never before have I laughed this hard at a slash post.

You sir have pwned me.

pr0n

This new solution is for Navier-Stokes equations under physically reasonable conditions. Navier-Stocks equations describe the motion of fluid substances such as liquids

who needs a description of the motion of fluid substances? I want video, perferably i

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I believe you are thinking of a different Stokes [suzannestokes.com].

Smoother rendition possible...

I bet if I put on a pimp hat and read it while drinking a glass of Courvoisier, I could make it "The Even Smoother Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System".

Don't player hate, player appreciate baby.

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Courvoisier is anything but smooth. Try a proper cognac like Delamain or Hennessy (assuming X.O or older of each brand, of course).

Neat indeed

As a math major I may say the this is impressive: after understanding the significance and complexity of the problem seeing a solution has been found is really exciting. Although I'm looking forward to see something done about the most significant of the Millennium Problems (IMO and from the pure maths POV) -- the Riemann hypothesis [wikipedia.org].

Note: Not considering P vs. NP as it is quite possibly unprovable.

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How could it be unprovable?

It certainly couldn't ever be proven unprovable, like some things can be, since proving it unprovable would also prove there was no way to implement a conversion P = NP, and, therefore, P != NP.

Just because we can't prove it does

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How could it be unprovable?

Just because we can't prove it doesn't mean it's unprovable.

Godel's incompleteness theorems [wikipedia.org]

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That is true. However, note that unlike Godel's incompleteness theorem, P = NP has direct and obvious connections to the real world. We're not choosing between competing logical theories that exist in a vacuum. P = NP allows us to do certain interesting th

Re:Neat indeed

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.

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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.

Yes, that's conceivable but seems unlikely. A more likely scenario (and in fact my money is on it) is that we

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It can be independent of the current accepted axioms.

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That's true. However, P = NP is interesting because it has practical uses. So, while there is obviously a mathematical meaning to it, the part that I am personally interested in is "is it doable on modern computer hardware".

So even if it does turn out to b

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"is it doable on modern computer hardware"

The question P=NP is thoroughly uninteresting when restricted to existing computers. Every existing computer has a finite and bounded amount of memory and storage, and hence a finite set of internal states, and i

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See, from my POV (I'm an engineer), an analytic solution to Navier-Stokes would be far more important. It would mea a huge advance for our understanding of aerodynamics (among other fluid-flow problems).

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This Navier-Stokes thing seems to be more of an applied-math problem

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 co

Quite impressive

As a mechanical engineer, I have some idea of what this means.. Fluid dynamics is a fairly pervasive subject which goes into the design of airplanes, irrigation canals, industrial machinery, turbines and a lot of other places. The solution of the navier stokes' equation in three dimensions is quite fabulous, since without such a mathematical tool it's not possible to estimate how a fluid will flow in three dimensions.. Till now, we typically use either special conditions (ex. along a turbine blade, constant pressure) or fractional element methods (think of fluid as lots of tiny balls) or physical modelling for such problems. To put some perspective, it's about as cool as being able to determine the movement of n planets simultaneously attracting each other gravitationally.. quite tough!

Re:Quite impressive

That is not "the solution" of the Navier-Stocks system - they could be solved only numerically (fractional element methods or other discretization), but this is the next best thing - proof of the existance of such solution. From the practical point of view that mean, if you have correct physical starting conditions and working numerical method you will get correct result after calculation. Until now, you couldn't have been sure if you will get physyically reasonable result of numerical calculations, even if starting conditions would be correct.

Re:Quite impressive

Correct,
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).

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Your analogy is completely misleading.

You take a quartic equation [nerdparadise.com] and choose to call "exact" what is called "a solution by radicals".

Yes, a solution by radicals can be hard to find even when it turns out to exist. (Indeed quartics weren't solved by rad

Re:Quite impressive

I agree. Fluid dynamics is very fascinating. Since I'm not so smart I've devoted my limited abilities to trying to understand the things we put conventional fluids into so that we can transmit them.

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Information is a fluid now? You kids today with all your newfangled inventions...

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Actually the whole thing IS NOT about FINDING SOLUTION of the Navier-Stokes equations,
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

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Actually, engineers care about this a lot. Inexperienced people take a formula and assume it works under all conditions. However, when you take a [good] numerical analysis course, you'll do more than just learn how to use a formula. You'll spend time doing

Whuh?

Man, I haven't had a date in like 4 years, and even *I'm* not nerdy enough to know why this matters...

Someone had better tell the Formula One teams

While I know they perform many, many computer simulations, I think aerodynamics is still regarded as one of the "black arts" in the field. Wind tunnels are still used extensively (it's often about who can build the better wind tunnel, never mind car). Maybe complete solutions of fluid movement will mean some odd-looking cars in 2007!

Re:Someone had better tell the Formula One teams

Not really. This proof of the existance of the solution won't substatially affect the real-world application of fluid dynamics (including aerodynamics) for quite a ling time (maybe within my lifetime, probably not). Numerical and real simulation will still guide the principal advances at the full assembly level. Nonetheless, this is a pretty cool event. I remember studying N-S in undergrad. Still makes the hair on the backof my neck stand up is apprehension. (tensor math and pdes both make me ill).

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Using the Finite Element Method (FEM) will give you very good results. I've worked with Comsol and Floworks simulations designing a variety of things - but mostly cooling loops. This is where the problem lies - these simulations are very computer intensive

I solve 3 millennium problems before breakfast

Well, at least contributors to arXiv between them seem to. (The 'GM' section in mathematics has been dubbed by some serious mathematicians "garbage machine", for example.)

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

blink blink !

the article could have very well been in french or latin

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I know some French, some Latin, and more math than either and have used the NS equation in my work (including nuerical slutions to subsets of the 3D problem). However ths would take me at least a couple of years of work to understand.

One of the things that

The toughest millenium problem of all...

...is getting people to spell it "millennium". Cracking that one would be a million dollars of anybody's money...

Re:The toughest millenium problem of all...

Just remember:

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.

What is the geometry?

Abstract of this post

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.

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It is a big deal for the mathematicians. That is all

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

An important step

As a previous commenter stated, this is a mathematical proof that such a solution exists. You cannot explicitly solve the Navier Stokes equations as written. If you could, my job would be much easier (I model thunderstorms at very high resolution on massively parallel supercomputers). The Navier Stokes equations, along with some other conservation laws, and some physical parameterizations, can be "closed" such that you can approximate a solution using numerical tehcniques, given an initial state and boundary conditions. It is not easy. From a practical standpoint, dealing with massively parallel computers is not much fun. I've spent the past couple of months debugging my own stupid coding errors, competing with hundreds of other scientists running their models, and finding ways to manage the terabytes of data these models produce when they do run succesfully.

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).

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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).

Not that I think you

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[On arXiv,] there are dozens of papers there that claim to have solved the Goldbach conjecture, or the Riemann hypothesis, or proven that the real numbers are countable, etc.

The difference is that the authors of these papers have no track record of gettin

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The arXiv (pronounced "archive") is a preprint server where people post their papers. Sometimes the papers are awaiting publication, sometimes they aren't going to be published, and sometimes they are just rough things like lecture notes that people just

a fitting tribute

it's about time someone named an institute after Clay Aiken [google.com]

Going to need to follow how this reviews

A generic solution of Navier-Stokes under any kind of realistic conditions is huge. I'm sure it will still be necessary to discretize for most aircraft or boats. I'm also certain the solution is "strange" (as in "highly sensitive to boundary conditions")

What the World Needs

I'm holding out for a solution to the Navel-Strokes system.

An arxiv article does not a headline make

Hmmm... the arxiv, of course, has a bit of a 'reputation'. They'll take anything, and more power to them for being willing to do so. However, it does tend to mean that if one's a non-specialist, the cranks can look awfully convincing. Without, obviously

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Check the last link in the summary. The author is a highly-respected mathematician in the field and this follows previous work that has been peer-reviewed. That doesn't mean it is *right*, but that does make it newsworthy.

Withdrawn

Well, I guess peer review has already taken its toll. The paper has been withdrawn from the arXiv due to "serious flaws."

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Sorry, that is Catastrophe theory:

http://en.wikipedia.org/wiki/Catastrophe_theory [wikipedia.org]

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Hey, go easy on them. They've only had seven years to learn how to spell the word. Don't rush them! I mean, look how long it took them to get a CSS-based layout for this site, and CSS is only three letters!