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

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.

[arun_s]

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

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

[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.

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

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...

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).

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

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.

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).

Withdrawn

[mathcam]

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 The paper has been withdrawn from the arXiv due to "serious flaws."