A few months ago, I wrote about the Poincare conjecture, and the fact that it appeared to finally have been solved by a reclusive russian mathematician named Grisha Perelman. Now there's news that *another* classic problem may have been solved. This time, it's the Navier-Stokes equation, apparently solved by [Professor Penny Smith](http://comet.lehman.cuny.edu/sormani/others/smith.html) of Lehigh University. She's published the steps leading up to her solution in top peer-reviewed journals, and a [preprint of the final paper is now available via arxiv](http://arxiv.org/abs/math/0609740). There's also a pretty good detailed description of the solution on [Christina Sormani's website](http://comet.lehman.cuny.edu/sormani/others/SmithNavierStokes.html).
The Navier-Stokes equations form a classic problem that I actually know a bit more about, although I have to admit that the proof of the solution is beyond my ability to understand. Why should you care? Aside from the fact that it's a famous problem with a million dollar reward posted by the Clay Institute for a solution, it's *useful*. Unlike the Poincare conjecture, the reason why we care about solving the Navier-Stokes equations isn't just theoretical. If Professor Smith's solution and proof do turn out to be correct, it would be a really incredible accomplishment, with direct, immediate, practical implications.
Navier-Stokes isn't really *an* equation, but a family of equations that's used to solve fluid dynamics problems. The idea is to describe precisely the forces in a moving fluid. Suppose, for example, you want to design a wing for a new airplane. To figure out how air will flow around the wing, how much lift it will generate, and how much drag it will produce, you need to be able to precisely model the behavior of air flowing around it. That's a very typical example of what the Navier-Stokes equations are used for.
The equations themselves are in some sense sort of straightforward. In all physical interactions, there are a collection of fundamental conserved properties: mass, momentum, angular momentum, energy. You can express the velocity and pressure in terms of *how* the fluid motion preserves those conserved properties. The equations themselves, when all of the dimensions of all of the variables are worked out, turn incredibly messy. But the basic concept is just conserving the fundamentals.
The problem with Navier-Stokes is that they're a group of extremely difficult differential equations. They don't actually tell us *what* the values of the variables are; they talk about the relationships between rates of change. But until now, we haven't been able to actually *solve* that differential equation in a way that gives us a useful closed-form equation.
Since we don't have that closed-form solution, there's been a huge amount of work in a field called computational fluid dynamics (CFD) at computing simulations of fluid flow using a very complicated iterative approach to finding an answer to the differential equations. (That's why I know a little bit about them: in grad school, most of my advisors other students were working on automatic parallelization of scientific code, most of which was CFD.) To simplify to an almost ridiculous degree, the way that we do CFD is roughly equivalent to the "rectangle approximation" approach to computing the area under a curve.
As a refresher: suppose you want to find the area under the curve y=x2
from 1 to 3. If you don't know the integral of x2dx, then one way of
doing it is to take the section of the curve from 1 to 3, and break the x over that range into a bunch of segments. For each segment, draw a vertical line from the x-axis to the curve, and then a horizontal line from the intersection with the curve over to the vertical line from the next segment. So you end up with a collection of rectangles like the following:
Calculating the sum of the areas of those rectangles gives an approximation of the area under the curve. If you reduce the size of the segments of the x-axis, thus *increasing* the number of rectangles, you'll get a *better* approximation. The smaller the segment size gets, the more rectangles you fit under the curve, the better the approximation becomes; the limit as the width of the rectangles approaches zero is the precise area of the curve.
The way that CFD algorithms compute solutions for the Navier-Stokes equations is similar to this - except that instead of dividing the x axis into small segments, we divide just about *everything* into small segments. We divide space into tiny little cubes; we divide time into tiny little intervals; and then we use the differential equations to figure out an *approximation* of the change in each tiny little patch of space over each tiny little interval of time, and add those up.
The iterative CFD approach is very complex, and it requires a huge amount of computation to get a good approximation. This means that computing things like weather, or airflow around a prototype wing is *hugely* expensive, and takes quite a bit of time. If we didn't have to do the massive iterative process, we'd be able to compute better results much faster.
That's the real implication of this. Everything from weather prediction to the design of cooling fans for computers can be improved using the better CFD algorithms we'll be able to develop using the solution to the Navier-Stokes equations.
It's a big deal, and extremely exciting. I really hope she got it right; and all signs so far are that she did.
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Two Clay solutions hit the headlines in (practically) the time it takes me to blink. Awesome.
(Holds breath for proof of Riemann hypothesis. . . .)
Sorry, stupid question: Does "immortal solution" mean closed-form solution? Or something else?
"If Professor Smith's solution and proof do turn out to be correct, it would be a really incredible accomplishment, with direct, immediate, practical implications."
uh, yea, like solving all of geophysical fluid dynamics. I got my phud in physical oceanography with minors in atm science and climatology. N-S is more or less the root of everything for the entire field.
Capt:
Alas, I'm out of my depth. I'm not sure what an "immortal" solution means. From what I've been reading about it, she's got a closed form solution for a version of the equation with some set of constraints - but the constraints are very minimal; all of the "real-world" applications of fluid dynamics, up to and including relativistic fluid dynamics, all fit the constraints of her solution.
Kevin:
Amazing, isn't it? My experience with this kind of stuff is aerodynamic and atmospheric CFD; this looks to be the basis of an incredible leap forward for *everything* that uses fluid dynamics. If it proves out, it could be the most important mathematical discovery of our lifetimes in terms of impact.
I think, but am not sure, that an immortal solution means a function of coordinates+time that is defined for all possible time values.
Solutions for differential equations tend to sometimes be defined only on a small interval, exploding to infinity at singular points. Seems she claims that her solution is defined everywhere.
wow...
Exciting times for math! Fermat, Poincaré, Navier-Stokes...! What's next? Goldbach? Riemann? P/NP? Any bet?
Besides it being interesting times, it may be interesting to speculate on reasons... why is math SO successful these days? And, funny enough, it happens just when some people were predicting stagnation, or at least a dearth of big analytic results, due to the extreme complexity of the frontiers (and a shift to "experimental" math, math-in-computers).
IMMORTAL SOLUTION sounds like it ought to be a death metal band, really.
I think you overestimate the practical importance -- knowing that a solution exists is very different from knowing what the solution is. And we already knew "on physical grounds" that solutions exist....
anon:
She didn't just prove that there was a solution. She *found* a solution a proved it correct. So if she's right, then we *do* know the solution.
The explanation is quite lucid enough to convince me that I don't understand the topic. ;-) Nevertheless, I must ask: what about turbulence, and the associated chaotic behavior?
Mark:
I'm looking for some hint of this in her paper. I can't make sense of anything below the abstract, so I'm just going to go by the abstract: why does the abstract make the weaker claim of existence of solution rather than the stronger claim of method of solution? I know the prize is for a proof of existence, but that shouldn't affect how she writes her paper.
I'll have to read the paper again, her 5 preceeding papers, and Heywood's paper (the 6 important references), but I don't think that this proof is constructive. Best I can tell, it's an existence theorem for a form of the NS equations, but doesn't really offer a method.
Since short-time solutions were already known, her goal is to prove that there are smooth solutions for all future times (i.e., immortal, t bigger than 0). Basically what she does is convert a system of 1st and 2nd order nonlinear partial differential equations into a system of 1st order nonlinear PDEs, and then, carrying it one step further, make an assymptotic expansion which can be smoothly matched to the short-time solution to extend it to all time. Pretty slick if you ask me.
The solution that she proves the existence to and the goal of the Clay prize is in all of R^3, that is to say that there are no boundary conditions. So, the real world applications of this are even fewer since I know of only one interesting problem in fluid mechanics that can reasonably be posed in an infinite spatail domain (or equivalently, generally speaking, in a spatially periodic one). No airfoil problems are going to be solved with this technique!
All of that is, of course, not to diminish this work. It's simply amazing that with 6 papers published or to be published this year, she's proved one of the holy grails of mathematical physics. Maybe she'll have a look and see what she can do about the Yang-Mills mass gap by the end of the month! :)
This is an existence proof that I cannot see having any applied consequences at all. The question is whether or not NS forms singulariites in finite time. Most of us who are not pure mathematicians assume that the answer is no -- that a little bit of viscosity is sufficient to maintain smoothness. This proof will confirm this intuition, which will be nice, but i do not see this as affecting how anyone using NS goes about their business.
I am astonished. When my eyes read the headline in my RSS administrator I just wanted to cry. I never thought the Navier-Stokes equations would be solved in my lifetime. Now, I could die in peace with a proof for the Riemann Hypothesis .
Well, an arxiv paper means that the author is endorsed and that the paper passes an overview to be fit for arxiv publication. But of course it looks promising.
This is not my cup of tea, but the Clay Institute challenge were to place a proof on any of four alternatives (existense and smoothness or breakdown, in R^3 or R^3/Z^3). Smith claim the first with solutions in R^3x[0, oo], and seems to call that immortal as ParanoidMarwin says.
At least originally, it seems her (or rather Perron's) comparison principle used both upper and lower envelopes to box the solutions in. Smooth! :-)
Theoretically, since Clay asked for solutions with initial conditions, there is no need for boundary conditions. Though I suspect billb is correct, it could be awfully hard to construct the correct initial conditions, and you rather want the initial boundary anyway which is possible for some other hyperbolic DE's. Furthermore Smith doesn't discuss any examples what I can see, so her constructions from initial conditions may be hard to impossible to realise.
Perelman et al used Ricci and related flows, which is as I understand it based on the parabolic heat equation's tendency to smooth solutions out. Here instead the hyperbolic NS was solved. Feels nice to see work with such classic stuff as DE's.
"What's next? Goldbach?"
Funny you should say that, perhaps there is some new progress on the twin primes conjecture. There is a new result on small gaps between primes ( http://in-theory.blogspot.com/2006/10/primes-are-random-except-when-the… ).
Okay, I'm making a wild guess that P = NP is proven in 2010, the Riemann hypothesis around 2015, and the Goldbach conjecture not by 2020.
There is, however, a much simper and therefore more elegant solution to the 3-space periodic Navier-Stokes posted on arXiv 2Oct2006 by David Purvance
Let us hope this leads to more women taking up a career or advanced studies in Mathematics and the Sciences. And discredit cranks who trot out trashy hypotheses such as the gender caused difference in intelligence.
Imagine how much more exciting Math is going to be for the next generation.
Great work Dr. Smith!!
This is sort of off-topic, but I'm just wondering: what tools do you use to produce the graphics in your posts?
It's seems impossible that she constructs explicit solutions. She probably proved that solutions extend for all future times i.e. that they do not develop singularities.
So, as others have pointed out, this probably has *no* practical interest.
As a side note, a theoretical problem in QFD is that you can't (in general) prove that your solution will converge to the true solution as your grid sizes go to zero.
As a second side note: why are so many people enamored with Goldbach conjecture? It is not interesting and has no impact on any field of math. It's only redeeming point is that it is easy to understand. Hardy talk shortly about it in his (excellent) "Mathematician's Apology"
That was just a beautifully written post on a very exciting topic. How well I remember being tortured by Navier-Stokes equations in fluid mechanics...how sweet to add Penny Smith to my pantheon of goddesses!
Larry Summers, eat your heart out. You can make all the pompous pronouncements you want, but women will still be kick-ass mathematicians anyway. In SPITE OF all the roadblocks put in their way.
MarcCC: Did you do a post on the Summers thing? Seems like Zuska could stand to read it! ;)
Yes, like Kovalevskaya. The Cauchy-Kovalevskaya theorem was the first especially beautiful proof I read. (Read: complex for me, but I still got the gist of it, and the central idea is simple.) But I'm partial to Noether, since Noether's theorem is such a beatiful result.
UN statistics says about 5 % of the earths resources is owned by women. When it becomes up to 50 % in a society for a few generations, when we can start comparing differences for real.
Mark, there's really nothing in math that is of *no* practical importance. We might not currently have any idea what importance it may hold, but much of what was once 'pure' math has become applied once someone found something to apply it to. It just needs to exist, and it will find a use.
At the very least, the fact that it's such an extremely difficult problem means that mathematics itself is advanced in the search for solutions by it. So, even if the result itself is practically useless, the work leading up to the result will likely be of great importance.
And hey, weirder things have proven useful. Iirc, partition theory has a use in fluid dynamics as well. It's just counting how many ways one can write a number as a sum of positive integers! That just plain sounds silly, but there you go.
Yes, IIRC Lubos Motl uses to mention that the fact that 1 + 2 + 3 + ... = -1/12 is immensely valuable in quantum theory. :-) (AFAIK it is a result, probably by studying poles, from the Riemann function and becomes natural when renormalising string theories.)
In fact I was just listing further famous open questions; however, I think that its interest may be defended. Xanthir makes the general point of the frequent long-term usefulness of "pure" math; I'd add that in the case of Goldbach, the simplicity of the statement and the fundamental objects involved (natural numbers, and their "atoms", the primes) suggest that any proof would teach us something deep about numbers. Atoms both through addition and multiplication, and a proof connecting that... Wouldn't be so bad!
Sometimes the proof itself is useful, even if the theorem isn't. Fermat's Last Theorem is not useful to anyone; the tools in algebraic number theory that were developed in order to prove it are not only immensely useful in mathematics but also at times applicable to the sciences.
@Torbjörn Larsson:
The bit about 1 + 2 + 3 + 4 + ... = -1/12 comes from taking the analytic continuation of the Riemann zeta function and evaluating it at -1. If you look at the common form of the zeta function, i.e., the sum of reciprocals of successive integers raised to the power s, it sure looks like ζ(-1) = 1 + 2 + 3 + 4 + ...
However, that definition only converges if the real part of s > 1, so you have to break out the analytic continuation, and then life just gets weird.
Why would this arithmetic trickery matter? That's a long and technical story which can be found in Chapter 12, I believe, of Zwiebach's First Course in String Theory. In abbreviated form, you have to put an infinite number of non-commuting operators in the proper ordering, which gives an infinite series of corrections to a Hamiltonian. . . Eh, well, it isn't exactly pretty.
See also this homework assignment (PDF) from John Baez and Derek Wise's quantum gravity seminar.
Thanks, Blake!
"The bit about 1 + 2 + 3 + 4 + ... = -1/12 comes from taking the analytic continuation of the Riemann zeta function and evaluating it at -1."
Yes, Riemann zeta, I should have been more careful.
A similar thing happens when calculating the force due to the Casimir effect, in which (I believe) you have to sum the effects of an infinite number of photon modes. The magic phrase to feed a search engine appears to be "zeta function regularization".
It looks like Professor Smith has withdrawn her paper from the arXiv. The only comment is "This paper is being withdrawn by the author due [to, sic] a serious flaw."
I think the Goldbach Conjecture is appealing to the general public because it is simple to understand (i.e., I can explain it to my mother) and it has endured for so long.
Formulated by Clay Mathematics Institute the sixth Millennium Problems about existence and smoothness of solutions of the Navier - Stokes equations periodically was discussed at numerous forums (http://grani.ru/Society/Science/m.112524.html). On recognition of some commentators the complete presentation of problem's solution can demand about thousand pages for mathematical formulas (http://lib.mexmat.ru/forum/viewtopic.php?t=4289). The author of Official Problem Description-Charles Fefferman has set the task about demonstration of existence and smoothness of the solution, instead of solution's obtaining. However, the Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
More in detail on a site http://continuum-paradoxes.narod.ru the link "Russian pages", "Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods (in Russian)".
Yours faithfully, Alexandr Kozachok
Formulated by Clay Mathematics Institute the sixth Millennium Problems about existence and smoothness of solutions of the Navier - Stokes equations periodically was discussed at numerous forums (http://grani.ru/Society/Science/m.112524.html). On recognition of some commentators the complete presentation of problem's solution can demand about thousand pages for mathematical formulas (http://lib.mexmat.ru/forum/viewtopic.php?t=4289). The author of Official Problem Description-Charles Fefferman has set the task about demonstration of existence and smoothness of the solution, instead of solution's obtaining. However, the Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
More in detail on a site http://continuum-paradoxes.narod.ru the link "Russian pages", "Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods (in Russian)".
Yours faithfully, Alexandr Kozachok
This paper is being withdrawn by the author due a serious flaw.