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."

Comments

Pretty nifty stuff

[Morlark]

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 read, at the very least.

Hm.

[ZombieRoboNinja]

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.

[Anonymous Coward]

You sir, are my hero. . .

Re:Hm.

[arun_s]

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

Re:Hm.

[plantman-the-womb-st]

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

You sir have pwned me.

pr0n

[macadamia_harold]

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 in slow-motion and from multiple angles.

Re:pr0n

[Ignominious Cow Herd]

I believe you are thinking of a different Stokes.

Smoother rendition possible...

[nixkuroi]

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.

Re:Smoother rendition possible...

[Prune]

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

Re:Smoother rendition possible...

[Anonymous Coward]

Whoooosh!

Re:Smoother rendition possible...

[bodan]

sqr("Woooosh!");

Re:Smoother rendition possible...

[chawly]

That would have to be a tin-foil hat, I think.

Neat indeed

[Zx-man]

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.

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

Re:Neat indeed

[ZorbaTHut]

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 doesn't mean it's unprovable.

Re:Neat indeed

[Famatra]

How could it be unprovable?

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

Godel's incompleteness theorems

Re:Neat indeed

[jonadab]

Indeed. When it became evident that the Poincaire conjecture is not going to be considered an open problem much longer, my thoughts too were of the Reimann Hypothesis. Throughout most of the twentieth century those were widely considered to be the two biggest open problems in pure math, and it'd be really cool to see them both solved in our generation.

This Navier-Stokes thing seems to be more of an applied-math problem, and although I'm sure it's important, it's just not as exciting to me as the more abstract theoretical stuff of pure math.

Re:Neat indeed

[rbarreira]

It can be independent of the current accepted axioms.

Re:Neat indeed

[ZorbaTHut]

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 be independent of current accepted axioms, which I will admit I'm skeptical about, I feel that knowing that allows us to immediately add an axiom to make that view of math approach "physical computer hardware". Or, alternatively, to define a subset of all NP problems that include things like Hamiltonian path.

Re:Neat indeed

[ZorbaTHut]

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 things on computers. If it turns out we can prove we'll never be able to do those, that is the same thing as saying it is impossible.

Re:Neat indeed

[Garse Janacek]

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.

Re:Neat indeed

[rogerdr]

I'd say that this can be considered a 4D generalization of complex analysis, if true. In that case, it could go far beyond applied math into Relativity.

Re:Neat indeed

[Anonymous Coward]

And why NOT tack on another axiom (like the axiom of choice)? Having trouble with your proofs? Make up a corny axiom that magically "solves" them for you, reality be damned! ZFCX, where X is the Axiom of "Damn My Dissertation Needs Some Help."

Re:Neat indeed

[be-fan]

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).

Re:Neat indeed

[AxelBoldt]

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 can eventually prove that ZFC can neither prove nor disprove P=NP, and in that case we don't know whether your scenario above is correct, or if on the contrary no such algorithm exists but ZFC is simply too weak to establish that.

Re:Neat indeed

[AxelBoldt]

"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 is therefore a finite state machine. Everything a finite state machine does can be done in linear time.

Re:Neat indeed

[AxelBoldt]

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 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 intellectual interest and won't help the engineers in any immediate way.

Re:Neat indeed

[gkhan1]

You are fundamentally misunderstanding the incompleteness theorem. The incompletness theorem states that in a powerful axiomatic system, there are statements that are true, yet they cannot be proven using the system. That is, it is possible for there to be a polynomial algorithm for an NP-complete problem, but we cannot prove that it is polynomial-time. Conversely, it could be true that there is infact no such algorithm (ie NP != P), but we cannot prove this fact, even though it is still true. Pick up Gödel, Escher, Bach next time you pass through a bookstore, it's really quite awesome.

However, is it probable that that P vs NP is undecidable? No, it is not. P vs NP is very likely not undeciable, we just haven't been able to decide it yet ;) That doesn't mean it's not impossible.

Lastly, why do you think algorithms are any different from abstract mathematics? Just because algorithms have an actual use in the real world doesn't mean that it is fundmentally different from pure mathematics. Fundamentally, they are the same thing, it's just that in our feeble human minds they seem different. But they aren't, they're part of the same abstract logical universe.

Re:Neat indeed

[gkhan1]

It should probably pointed out for all people that hasn't studied advanced mathematics that ZFC is Zermelo-Frankel set theory, the most common way to define the foundation of mathematics. See article for some axiomatic set theory fun!

Re:Neat indeed

[gkhan1]

That should be Zermelo-Fraenkel set theory. Apologies to Fraenkel.

Re:Neat indeed

[Pseudonym]

There have been some problems (e.g. the weak pigeonhole principle) where it's been shown that any proof must be intractably large. There is some evidence http://citeseer.ist.psu.edu/cache/papers/cs/27779/ http:zSzzSzwww.wisdom.weizmann.ac.ilzSz~ranrazzSzp ublicationszSzPchina.pdf/raz02np.pdfthat this is true of P!=NP.

If you think about this, there's a certain amount of poetic justice. NP-hard problems are solvable in principle, just not in practice. And the conjecture that P!=NP may be true in principle, but not provable in practice.

Re:Neat indeed

[Retric]

The are several things that a finite state machine can do faster than linear time and several that will take longer than linear time.

AKA find index of Y in a sorted set X vs sort set X.

Re:Neat indeed

[fluffy99]

You could certainly prove that the equation is wrong or incomplete.

Re:Neat indeed

[chawly]

"allows us to do certain interesting things on computers" - while streched out on top of them, I bet.

Re:Neat indeed

[YU Nicks NE Way]

Well, actually, ZFC is Zermelo-Fraenkel with the Axiom of Choice. Although AC is almost certainly "true", there's a lot of exceedingly interesting mathematics you can do without it, or even with various strong forms of its negation.

Re:Neat indeed

[AxelBoldt]

Yes, some things a finite-state machine does don't require linear time (e.g. checking whether a given input string is empty or not); nothing a finite-machine does ever requires more than linear time. Sorting cannot be done with a finite-state machine.

Re:Neat indeed

[noigmn]

I don't think the Riemann Hypothesis will have that much value. It's just a big challenge. We know it holds to at least squillions, and could guess that it is most likely right without proving it.

P vs. NP on the other hand would be more than quite significant. It would turn the world upside down, inside out, and potentially make our mathematical abilities unstoppable. The day someone does an RSA challenge number by hand or in their head will be a grand day indeed.

Re:Neat indeed

[azipsun]

In general, computers have some form of external removable storage be it floppy, tape drive, or USB key. So, with just a little human intervention, you can have unbounded storage on an existing computer. I'll admit that it isn't particularly practical and you still have the problem that there are only a finite number of floppies in the universe, but the notion that the P=NP question is uninteresting to existing computers is also ridiculous.

Re:Neat indeed

[jd]

Navier-Stokes turns chaotic is you sneeze the wrong way, which makes me think that no analytic solutions exist for the problems likely to be of interest. There is a flip-side to that. If an analytic, computable solution does indeed exist to the completely generalized Navier-Stokes, then it goes far beyond any "applied" solution (such as the solution of compressible or incompressible flows, say in aerodynamics, climate modelling or even canal maintenance!) - this would smash into the heart of chaos theory and fractal geometry. If it is possible, by any means whatsoever, to transform or convert some example of a chaotic system into a non-chaotic system and then transform/convert the results back to obtain the solution to the original problem, then a general solution may exist to handle chaotic functions directly, rather than using potentially infinitely-looping iterative methods. Assuming everything up to this point is correct, then if you meander down this twisted path of dubious reasoning, you'd eventually conclude calculus would have to be a special case for a much more general approach which could analytically solve a much broader range of problems.

Much more likely, the result of the published work will prove that certain classes of Navier-Stokes problem that have not previously been reducible can now be reduced to something that is solvable, but will not be able to do much with the flows aircraft contend with, where even the best algorithms in use (which are horribly complicated) are absurdly over-simplified simulations based around oversized meshes that has a passing resemblance to a physical system at the level of an operator but certainly not to your average air molecule.

(Aircraft are designed using CFD, but most designs are then tested by sticking on bits of tinfoil, placing it in a wind tunnel, and recording the rattling sounds, and/or using smoke. Even once all that is done, the consensus is that test pilots will report results wildly different from anything predicted and as the DeHavilland Comet proved, even test pilot reports don't tell you that much about what an aircraft will do under "real" conditions. If this new work can improve any/all of this process, it will be a gigantic leap forwards, but I'm inclined to think that whilst it might tell you more about very specific cases, it's unlikely to help much in the general case.)

I would add one other thing, though. CFD software can sell upwards of a few tens of thousands of dollars. Well, since that's for a per-year license, I guess that should really be "rent", not sell. Someone comes along with a simpler, quicker, easier solution - doesn't matter if it only applies in certain cases - it could utterly crush companies who rely solely on selling this kind of uber-expensive software. The reason they can charge so much is that it's a hard problem with very few people who can both understand it and write good code that's fast enough to be worth using. Someone makes CFD easy and more accurate at the same time, who is going to bother with code that needs a dedicated mainframe to get anything done that week?

Re:Neat indeed

[AxelBoldt]

there are only a finite number of floppies in the universe

That's precisely the point. If you are willing to imagine that you have an unlimited supply of floppy disks, then P=NP becomes an interesting mathematical question. If you stick to the real world with a finite supply of floppy disks, every algorithm is linear and P=NP is uninteresting.

Re:Neat indeed

[gkhan1]

Yes, the axiom of choice would be the Z. Who was it that said that great quote, "The axiom of choice is obviously true, the well-ordering principle is obviously false, and who the hell knows about Zorn's lemma!"?

(the joke being, all three of those things are mathematically equivalent, if you accept the axiom of choice, you accept the other two).

Re:Neat indeed

[YU Nicks NE Way]

Actually, everything but reflection and choice would be Z. Reflection is F, and AC is neither.

(And we won't even get into the correct attribution for CH/GCH.)

Re:Neat indeed

[Anonymous Coward]

The Riemann Hypothesis has a lot of consequences in number theory. Some of which are directly related to cryptography.

Re:Neat indeed

[jonadab]

> See, from my POV (I'm an engineer), an analytic solution to Navier-Stokes
> would be far more important.

I don't deny that applied math can be _important_. I only said that I find pure math more _interesting_. Plenty of things in this world are more interesting than important, or vice versa.

Re:Neat indeed

[Retric]

You don't understand what a finite state machine is. Read http://en.wikipedia.org/wiki/Finite_state_machine

Then look at this a finite state machine which takes a binary number and counts down.

Aka
11111 =
11110 =
11101 =
11100 =
11011 = ...
00000

Starting with a set of N binary digits counting down with this finite state machine takes at worst (N^2)-1 steps.

Re:Neat indeed

[noigmn]

Real ones or perceived ones? ie. do we know any proof of the Riemann Hypothesis will be useful in these ways, or do we just have a feeling that it might give insight into it?

Re:Neat indeed

[AxelBoldt]

I think you mean (2^N)-1 steps. And no, there is no such finite-state machine. As Wikipedia correctly explains, a finite-state machine is fed an input string letter by letter from start to finish, and in each step, based on currently read letter and current internal state, the machine changes its internal state (and, optionally, outputs a result letter). Once the string is read in full, the computation is over. Thus the computation always takes linear time.

Re:Neat indeed

[rpresser]

There are dozens, if not thousands, of actual results already in use today that assume that the RH is true. If the RH could be proven false, it would have definite implications. It would make and break mathematician's careers, if nothing else.

There are few results that depend on RH being false, however. The methods developed to actually prove RH will probably have more use as general methods than the simple fact of RH's truth will have.

In my opinion. I am not much of anything.

Re:Neat indeed

[Verteiron]

You are all insane.

Quite impressive

[adityamalik]

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

[S3D]

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

[pipingguy]

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.

Re:Quite impressive

[vogon jeltz]

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).

Re:Quite impressive

[legrimpeur]

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 a mathematician gets horny, an engineer says SO WHAT!!!

Ciao

Re:Quite impressive

[Anonymous Coward]

You're a milkman?

Re:Quite impressive

[OmnipotentEntity]

IANA Expert in Fluid Dynamics; however, though an exact solution exists, this doesn't mean that it'll be easy to find even if we have a method or formula to solve it exactly.

Here's an example. Two board, one 3m and one 2m are laying crisscross in an alley, with one end in each corner of the alley, and laying the other end on the opposite wall.

Their intersection is exactly 1m from the ground, how wide is the alley?

This problem is very easy to find a numeric solution, but suprisingly difficult to find the exact solution (by hand). But we already know the method to find it, it's simple algebra.

Just because an exact solution exists doesn't mean it's pratical or it will be used, if the approximation is good enough, or much faster. As I understand it though, Fluid Dynamics is a chaotic system. So an exact solution is probably preferrable unless the calculation disparity is wide enough. Just thought I'd point out the obvious because no one else was doing it.

Re:Quite impressive

[WhoBeDaPlaya]

It's kinda like trying to find an exact analytical solution for something as innocuous as the current distribution on a current carrying conductor for more accurate impedance calculations. Major PITA.

Re:Quite impressive

[This is outrageous!]

Your analogy is completely misleading.

You take a quartic equation 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 radicals until Ferrari in 1540.)

But the question whether a solution by radicals exists has nothing to do with whether a solution (period) exists. Indeed polynomial of higher degree have the latter (Gauss' fundamental theorem of algebra) but not always the former (Abel's quintic counterexample).

Penny Smith's achievement is analogous to Gauss's and not *at all* to Ferrari's.

Fortunately the former, not the latter, is what's needed to guarantee that numerical methods (e.g. Newton's, in the analogy) converge to an actual solution rather than nonsense.

Re:Quite impressive

[pyite]

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 a lot of real analysis that you don't necessarily enjoy doing (I didn't) but the point is clear. A lot of times we really, really, care about existence of solutions. At times, we even care about the uniqueness of such solutions. Or, how about convergence of the series we're approximating a function with? Or maybe, does this Fourier series really work on this set of data that has a bunch of discontinuities in it?

Here is what happens when such problems aren't approached with proper rigor. In short, we read (regarding an offshore oil drilling platform):

The post accident investigation traced the error to inaccurate finite element approximation of the linear elastic model of the tricell (using the popular finite element program NASTRAN). The shear stresses were underestimated by 47%, leading to insufficient design. In particular, certain concrete walls were not thick enough. More careful finite element analysis, made after the accident, predicted that failure would occur with this design at a depth of 62m, which matches well with the actual occurrence at 65m.

So, yes, engineers VERY MUCH care about these things.

Whuh?

[LiquidEdge]

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

Re:Whuh?

[The Cydonian]

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

May be if you dropped watching so much porno and picked up a math book instead... :-)

Re:Whuh?

[Anonymous Coward]

Dude, WTF? 4 years? Get out there, be a MAN, and score some pussy!

Re:Ouch...

[tsq]

Female AND good at math? What else could a /.er ask for?

Someone had better tell the Formula One teams

[Simulacrus]

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

[Overzeetop]

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).

Re:Someone had better tell the Formula One teams

[quanminoan]

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 and even a simple simulation such as a cooling loop through copper (one bend) can take over a day to converge to a solution (and i would make all sorts of assumptions to cut the time down, like perfectly smooth walls). A desktop computer wouldn't even be able to handle a more realistic simulation of the same loop. So the problem isn't with our knowledge of teh equations or the algorithms, it's a lack of available computer power. It turns out it's easier to build a wind tunnel than a supercomputer.

Re:Ouch...

[richy freeway]

A free lifetime WoW subscription?

I solve 3 millennium problems before breakfast

[fatphil]

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

Spell check

[MrYotsuya]

I wish that there were spell checks on whatever is submitted. "Millenium" indeed.

blink blink !

[phreakv6]

the article could have very well been in french or latin

Re:blink blink !

[Anonymous Coward]

I know french and some latin. I can say with confidence that it is neither language and I didn't understand anything either...

Re:blink blink !

[Anonymous Coward]

It's all Greek to me.

Re:blink blink !

[the eric conspiracy]

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 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?

It's the mathematics of ....

[PermanentMarker]

it's about the mathematics of farting
hahahaha


oops (sorry)

Re:It's the mathematics of ....

[thomasa]

Sorry, that is Catastrophe theory:

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

meh

[Anonymous Coward]

I have had a date in like.... ever and I don't understand this... ok.. ok at this point I must Concede that I am more a loser and less of a nerd... but still...

'kinell.

[Anonymous Coward]

At least spellcheck the ARTICLE TITLES.

It's "millennium".

Millenium?

[Threni]

I thought it was "Millennium"? Certainly that's how the linked-to website spells it.

Re:Millenium?

[eric.t.f.bat]

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!

The toughest millenium problem of all...

[john-da-luthrun]

...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...

[Frater 219]

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.

The toughest mathematical and computational probs

[Sem_D_D]

The fliud dynamics is an area, which is purported to be on the current edge of supercomputer capabilities, along with nuclear weapons simulation, weather forecasting and GO http://en.wikipedia.org/wiki/Go_board_game/ . Chess, on the other hand, is no longer part of this exclusive club, as the comparison says http://en.wikipedia.org/wiki/Go_board_game#Numeric al_estimates.
I suspect, some fancy hardware and breakthrough programming was needed to assist the geniuses, that managed to pull this one out.
That is a good sign of the advances in this outer-limits areas.

What is the geometry?

[140Mandak262Jamuna]

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.

Re:What is the geometry?

[Anonymous Coward]

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.

That's OK - this is slashdot.

Most commenters won't have even read the article, let get as far as failing to understand it!

Re:What is the geometry?

[Hakubi_Washu]

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 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 peer-reviewed and b) more computer scientists, engineers, and other theory-appliers have had a few months of thinking about it.

An important step

[Orp]

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).

Re:An important step

[Anonymous Coward]

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 are making an attack on mathematicians here, but I just want to comment on this for anyone that might construe it as such.

Mathematicians do subject papers to full peer review before being published in any reputable journal, but the arXiv is not a journal in any sense of the word. It's a sever that holds preprints --- literally ANYONE can put ANY paper on it. 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.

Likely this paper has not been peer reviewed at all. Take it with a pound of salt.

Re:An important step

[jschrod]

[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 getting articles accepted in the best math journals, also have no track record of previous ground breaking new work in math, and have not caused a stir in the community that is as positive as it is this time. Penny Smith has.

Re:An important step

[Ibag]

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 figure others might appreciate. As such, it is fairly unsurprising that the paper would be a rough draft. That doesn't mean that the ideas aren't all there, but it most likely hasn't been peer reviewed in any broad sense yet. Putting it on the arXiv is perhaps a first step in getting peers to look it over.

Re:An important step

[egork]

I wonder, if what they wanted to say was "this is more then proved"?
Proof: This is less than proved in the brilliant paper of [H]. QED.

Re:An important step

[TheoMurpse]

my job would be much easier (I model thunderstorms at very high resolution on massively parallel supercomputers)

Oh, you're a weatherman?

a fitting tribute

[seven of five]

it's about time someone named an institute after Clay Aiken

Going to need to follow how this reviews

[StressGuy]

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") in many cases. Still, this is a major breakthrough if it verifies.

What the World Needs

[Baby Duck]

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

yeah, right

[Anonymous Coward]

Sorry, but this doesn't pass the sniff test, just on the basis of the title.

"Immortal" ?
"three space dimensions"?

Now, if the title was "Time-invariant solution of the Navier-Stokes equation in three dimensions", okay.

But the choice of words for the title indicates kook-infestation.

Re:yeah, right

[AxelBoldt]

You're wrong. "Immortal" in this context is standard terminology for a solution that exists for all times. Your choice of "time-invariant" is not the same. "Three space dimensions" is much clearer than your "three dimensions", because the latter could be mistaken for one time and two space dimensions, a much simpler scenario.

A series of tubes?

[tepples]

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.

You mean the series of tubes that make up the Internet?

Re:A series of tubes?

[pipingguy]

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

Engineer here.. and can I say WOW!

[PWNT]

not that WOW, but the surprise! If there do exist solutions to the navier-stokes equations be prepared for something huge. These equations play a dominating role in anything where something is moving in the presence of many other moving particles (think water, gas, landslides etc) The significance of this finding cannot be emphasized enough, if it proves to be true. Current solutions revolve around using computers to actually simulate the ENTIRE section of the system you want, then you test for conditions at specific points. Also, only sufficiently large systems will give an answer, as you try to get a finer and finer solution to a zoomed in region, the data and simulation available either gets too small to work with, or you get so many non-linear equations to simultaneously solve that it takes up too much RAM and memory. Second: The existance of a solution to these groups of NON-LINEAR equations may provide an answer to how non-linear equations react to starting conditions. I am personally very excited about this, and hope it pans out properly.

An arxiv article does not a headline make

[Wooster_UK]

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, wishing to ascribe that appellation to the good Associate Professor, I would note that this paper carries some of the hallmarks: an extremely dodgy abstract, poor punctuation (as described above in comments), ropey spelling, dubious use of English (whassiss "immortal"?) and poor LaTeX skills.

As I say, far be it from me to call "crank", but I'd wait for this to appear in a peer-reviewed journal and get responses. I suspect the Millennium (sp!) Prize committee may well be doing likewise.

Re:An arxiv article does not a headline make

[flawedconceptions]

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.

Re:An arxiv article does not a headline make

[Wooster_UK]

Hokay, missed that. All the same, I'd wait for final verification before breaking out any bubbly (and then proceedng to analyse its egress from bottle to glass).

Shoehorning in my field

[mnemonic_]

Can I just say "As a mathematician/engineer/chef," to get modded up now?

Re:Shoehorning in my field

[Krakhan]

Hmm, a Mathematician chef would probably be able to make one hell of a pie at least. :)

You know what's more impressive?

[Anonymous Coward]

Your stupidity.

Re:Ouch...

[Anonymous Coward]

Your ridiculous anti-intellectualism is a disgrace to geekdom.

I am not amused.

[Millennium]

You are correct: the letters l and n both appear twice. But then, this is Slashdot; correct spelling may not be a reasonable expectation.

What millennium?

[Anonymous Coward]

Since these "millennium problems" aren't problems that have plagued mankind for an entire millennium, and they probably won't even last a whole millennium until they're solved... what exactly is the point of calling them "millennium problems"?

Re:Last claim of sexists falls

[Anonymous Coward]

The plural of anecdote is not data?

The example of Penny Smith, I'm afraid, cannot by itself disprove the general statistical assertion of greater variance for males. For that, one needs large scale group studies. :)

Why the constant Slashdot self-hate? +5 funny

[SaberTaylor]

Did I miss the big event when nerdism became negative instead of positive?

another explanation

[David Eppstein]

Brooks Moses has some additional explanation which I found helpful: http://notes.dpdx.net/2006/10/06/penny-smiths-proo f-on-the-navier-stokes-equations/

Cancel the survival gear!

[AndroidCat]

I was disappointed when there was no mention of a new Y2K-type problem.

...Waitaminute.

[Anonymous Coward]

Holy shit holy shit holy shit holy shit holy shit what? NS solution? Jesus Christ. Awesome!

Re:Last claim of sexists falls

[Anonymous Coward]

Great!

Let's send Penny down to Miami, and have her tweak the NHC's computer modelling programs,
so that we'll know exactly where the next hurricane will strike and with what wind speeds
and rainfall rates, etc.

Either that, or have her clone herself, and work the problem in parallel, and then teach
scads of other young minds how to do it.

Re:Last claim of sexists falls

[Anonymous Coward]

As the slashdotter above me stated...

So there is all that bullshit talk about being men better than women because they tend to have a greater variance (hence having the best and worst results) and you just now disproved it because this one women solved a Millennium Prize?

Poor Marie Sklodowska Curie (who won two Nobel Prizes and was one of the few people to do it)... why didn't you use her for you counter-example? At least her work is already proved, unlike the work from this lady.

Statistical significance comes from the size of the studied population. Otherwise, based on this:
"A B-17 ball turret gunner, Magee had no choice but to jump out of a disabled, spinning-out-of-control bomber from about 22,000 feet.
A drop of more than four miles. Without a parachute. And Magee miraculously lived." (taken from here)
jumping of a comercial airplane in trouble would be less risky than waiting for it to try to emergency land (as we know people die on plane crashes and apparently free-falling people do not).

So, next time will you jump of the airplane? Thought so...

Penny Smith's usenet posts!

[programmeratarms]

I read her very entertaining posts for many years, until she suddenly quit killing time on Usenet.

FEM is used plenty for solving Navier-Stokes

[Richard Mills]

Why do you think that FEM is rarely used to solve the Navier-Stokes equations? A quick Google search will indicate otherwise.

The choice of method for solving the equations does seem to vary quite a bit between disciplines. Engineers tend to love FEM, while, say, atmospheric modelers seem to prefer finite-volume or finite-difference approaches.

As long as we get ...

[Anonymous Coward]

quiet & more powerful leaf blowers

You'll have to tame your enthusiasm

[nonlnear]

Even if the result ends up holding, it's not time to close the book on simulation methods yet. The result shows (or claims to show) existence and uniqueness. It does this through an indirect method that is not constructive. (i.e. It doesn't tell you what the solution is, only that it's out there somewhere.)

If this holds up, the methods used are doubtless going to lead to better approximations and possibly - after a lot more research - to constructive methods. It's going to be exciting to see what happens to the field of fluids if this result holds, but it's not time to close the book on NV. In fact, this might open up some new subfields of PDE theory.

It would actually be very surprising to me if there ever were constructive solution methods that are significantly faster than simulation. It's a complexity thing. (That's just me musing in a highly speculative manner, though.) IMHO the real power of a constructive solution method would primarilly be for analysing other theoretical characteristics of solutions - not necessarilly for fast algorithms. (Like how if you want to compute a numerical solution to a potential problem, it's easier to do a numerical simulation of a stressed membrane and let it reach equilibrium than to numerically integrate a Green's function. The Green's function gives a theoretically constructive method that lets you determine properties of the solutions, but if you want to actually copute the thing, it's not the best way to go.)

MOD PARENT UP

[Btarlinian]

This is not offtopic. Penny Smith is the author of the paper in the article.

Re:MOD PARENT UP

[Ignis Flatus]

my bad. this post should undo it.

wtf

[Anonymous Coward]

wtf

Withdrawn

[mathcam]

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

Re:Ouch...

[Anonymous Coward]

Female AND good at math? What else could a /.er ask for?

Given this picture I'd ask for the lights to be turned off...

Paper is withdrawn

[excalibur313]

I was trying to find a copy of the paper on the preprint server and it now says that it has been withdrawn for serious errors. It's too bad.

If it was right...

[PDExperiment626]

As a Ph.D. student in partial differential equations, I was very excited to hear about the possibility of NS being cracked finally. I was even more shocked to see that the techniques used in this paper to prove the existence were 'oldschool'. Quite literally, the core of the technique (perron's method) has been around since the early-mid 1900s. The regularity extension pulls on difference quotient methods, another classical technique from decades past. It is widely thought that to prove existence of NS with classical regularity, new techniques would need to be invented; if this proof is salvagable (could be a big IF), it would be a massive lesson in humility for the field... this proof should have been seen a long time ago, if it is true. I would be very interested to see how some of the mathematicians in the area would be reacting to this proof if it is salvaged. I know the paper looks very rough with the lax grammer and punctuation; but from what I can see, this isn't a crack-pot claim. Really, the most surprising part of this proof is the comparison principle used that was supposedly published in 2002 by the same professor. The rest of the techniques have basis in classical PDE theory. I am not sure what exactly the error in the proof is (NS is not my area, and I haven't had time to look into it thoroughly); but it's indicated that the error is in a paper cited in this one. So, the proof may be salvageable, it may not be. If the problem is with the comparision principle... that might be the final nail in the coffin for this line of attack (which is a strong possibility). I admit I feel for Dr. Smith, it looks like she is a reputable researcher; and I think making a NS proof claim might hurt that. Even eccentrics like Dr. Perelman (guy who did the final bit to prove Poincare) had the prudence to avoid claiming he had solved the Poincare conjecture explicitly with his work; on the contrary he presented his proof and idea in the most specific way possible. In short, he understood that if caught 'crying wolf' on a millennium problem is a serious blow to one's reputation in the maths community. Of course, it is hard to be subtle when claiming a classical solution to NS; Perelman's proof completed Hamilton's very long and technical program using Ricci flow... this proof stands on it's own. Even so, the paper was clearly rushed to arXiv with little or no editing; and this reflects poorly upon the researcher. I think rushing the preprint was just a mistake of a researcher who got too excited about a result; and I think this might cost her reputation quite a bit.