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."

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

[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).

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: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).

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: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 [wikipedia.org]

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 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: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.

Withdrawn

[mathcam]

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 The paper has been withdrawn from the arXiv due to "serious flaws."