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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."
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!
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.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
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).
Is it just my observation, or are there way too many stupid people in the world?
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).
A squid eating dough in a polyethylene bag is fast and bulbous, got me?
How could it be unprovable?
Just because we can't prove it doesn't mean it's unprovable.
Godel's incompleteness theorems
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.
Well, I guess peer review has already taken its toll. The paper has been withdrawn from the arXiv due to "serious flaws."