The Navier-Stokes equations describe how a fluid flows. They are derived by applying Newton's laws of motion to the flow of an incompressible fluid, and adding in a term that accounts for energy lost through the liquid equivalent of friction, viscosity.

In 2000, to add some glitz to number-crunching, the Clay Mathematics Institute in America offered seven Millennium Prizes of $1 million each for the solutions to seven major problems in mathematics, including the Navier-Stokes equations.

"This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding." Seven Millennium Prize Problem.

"Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations." Seven Millennium Prize Problem.

Claude Louis Marie Henri Navier, (1785-1836), was a French engineer and physicist who specialized in mechanics.

Sir George Gabriel Stokes, (1819-1903) was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics, optics, and mathematical physics.

Prof. Penny Smith, a mathematician at Lehigh University, has posted a paper on the arXiv that purports to solve one of the Clay Foundation Millenium problems, the one about the Navier-Stokes Equation. Her solution has not yet been verified. Smith withdrew the paper from arXiv because it contained a "serious flaw."

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Flawed solution to famed math problem spurs cyber soap opera.

October 27, 2006. Source: seedmagazine.com by Stephen Ornes

Mathematics is a field rarely associated with human drama, but earlier this month, a soap opera-worthy narrative unfolded in the wooly world of conjectures, theorems and lemmata.

It all started when a mathematician tackled one of math's most enduring open problems—one that happened to be worth $1 million—in a paper she posted online. The journal Nature quickly published a story on its web site; news of a great mathematical breakthrough began to spread.

But less than two weeks after she posted the paper, the author learned that she had made an error and withdrew her work. In another era—as recently as, say, 10 years ago—that would have been the end of the story.

"Before blogs, here's the way this normally would have shaken down. [She] would have put the paper out, people would have heard about it and downloaded it," said Peter Woit, a physicist and mathematician at Columbia University. "Other people see it over a few days or weeks. There would have been a much slower exchange taking place, privately."

Oh, but times have changed. In the world of instant communication and public access to sophisticated research, this small story blossomed into a veritable cyber-drama. The narrative at "Not Even Wrong," Woit's blog, escalated quickly. Within a week, it had become a revealing chronicle of scientific hope, human disappointment, and the perils of undertaking the often messy enterprises of science and math in the age of the blog.

It began on Oct. 5, when Woit started a new thread titled "Navier-Stokes Equations Progress?" A Lehigh University mathematician, Penny Smith, had posted a paper online that tackled a famous unsolved problem. In the abstract of her paper, Smith claimed to have proven the existence of a solution to the Navier-Stokes equations. (Mathematicians are often just as concerned with proving the existence of a solution, as they are with finding the solution itself.) The equations describe the behavior of liquids and gases in motion, and a solution would give scientists insight into the chaotic natural phenomenon of turbulence.

The Navier-Stokes problem, which has lingered unsolved for more than 150 years, was designated a Millennium Prize problem by the Clay Mathematics Institute in Cambridge, Mass; anyone who solves it will win $1 million from the Clay Institute.

Smith posted her paper at arXiv.org, an online preprint server owned by Cornell University where physicists and mathematicians often place their work before it has gone through peer review or been published in a journal. The arXiv (pronounced "archive") serves as an online billboard where scientists can share ideas and research, but lately it seems to have taken on a new importance.

In 2002 and 2003, the reclusive Russian mathematician Grigory Perelman used the arXiv to post three ground-breaking papers that contained a proof of another million-dollar Millennium Prize problem: the Poincaré Conjecture. Even though Perelman's work had not appeared in a refereed journal, he was named a winner of this year's Fields Medal, math's highest honor. (Perelman declined to accept the award.)

People flocked to Woit's blog, eager to discuss Smith's paper. Giddy admirers congratulated Smith while cautioning each other against getting too excited until the work had been verified.

"Since [Perelman's] proof came out of nowhere, maybe people were thinking, 'oh it's going to happen again," Woit said of Smith's proof. "I was amazed at how much attention it got."

But doubts about Smith's work began to surface; uncertainty swelled until the morning of Oct. 8, when Smith withdrew the paper from arXiv because it contained a "serious flaw."

The drama, however, didn't stop. Readers of Woit's blog, some of them physicists and mathematicians themselves, proved to have much more to say, especially to Smith, who declined to comment for this article. Some reminded her that advancement in mathematics requires risks and failures, while others criticized her work. Some lobbed personal insults.


Though the flare-up was not an isolated incident—flawed proofs show up regularly on arXiv—the blog discussion it ignited suggests that the protocols of mathematical research may have to change in the face of growing technology. It might also give mathematicians of the future a strong incentive to be hyper-meticulous about their work.

"Penny did not do anything wrong by posting her paper on arXiv.org; I just wish she had been more careful about assessing the correctness of her work before she posted the paper," said Deane Yang, a mathematician at Brooklyn, New York's Polytechnic University, via email. "But in the end, no matter how hard we try to catch our own mistakes, we're all capable of doing the same thing she did."

Perhaps mathematicians will have to reconsider posting their work on arXiv. On the one hand, the site can disseminate information quickly within a wide mathematical audience. This worked to Perelman's advantage; news of his breakthrough spread quickly through cyberspace, and any interested mathematician—or layperson—could download history in the making. Of course, at the time, his papers were new and subject to scrutiny, and for three years mathematicians scoured his equations for signs of errors.

On the other hand, however, this same public scrutiny can subject a researcher who submits a flawed paper to public grief. As scholars often spend years preparing a single paper or result, the swift appearance of a grievous fault can be devastating.

"This incident serves as part of our learning process on how to use the Internet and blogs to communicate and discuss ideas more effectively," Yang said. "Blogs have made these discussions much more public, drawing both appropriate and inappropriate comments from a broad range of people. This makes a lot of mathematicians very uncomfortable."

Smith's own posts to Woit's blog reveal that she began to suffer under the weight of the experience: "ArXiv is supposed to be a preprint file—not a journal or a newspaper," she wrote. "This has hurt me a lot."

With the discussion showing few signs of quieting down after nearly a week, Woit decided to shut down the forum on Smith's proof and send everyone home. "This endless discussion has become both unpleasant and tedious," he wrote. "No one seems to have anything else to say about mathematics."
 

Latest news: This paper is being withdrawn by the author due a serious flaw.

Sun, 8 Oct 2006. Source: paper on the arXiv

Has famous math problem been solved, and in only a month?

Has $1 million math problem been cracked using novel techniques?

October 6, 2006. Source: nature.com by Jenny Hogan

A buzz is building that one of mathematics' greatest unsolved problems may have fallen.

Blogs and online discussion groups are spreading news of a paper posted to an online preprint server on 26 September. This paper, authored by Penny Smith of Lehigh University in Bethlehem, Pennsylvania, purports to contain an "immortal smooth solution of the three-space-dimensional Navier-Stokes system".

If the paper proves correct, Smith can lay claim to $1 million in prize money from the Clay Mathematics Institute, based in Cambridge, Massachusetts. In 2000, the institute listed the Navier-Stokes problem among its seven Millennium Prize Problems.

What mathematicians want to know is whether these equations always behave themselves, or their solutions sometimes diverge — which would amount to physical impossibilities, such as fluid mass disappearing. Smith claims to show that solutions to the equations never diverge.

Experts in the field say that it is too early to make a call on whether the paper is correct, but they are beginning to comb through the work. Among those looking at the paper is Charles Fefferman, a mathematician at Princeton University, New Jersey, who wrote the description of the Navier-Stokes problem for the Clay Mathematics Institute. "It would be a spectacular achievement of the highest order if it turned out to be right," he says.

The Navier-Stokes equations serve not only as the basis of a challenge to mathematicians but also underpin many practical exercises in physics and engineering, such as the design of chemical plants. Any mathematics developed to tackle the equations could also be put to use in computer simulations or might provide new insights into the nature of complex phenomena such as turbulence.
 

The challenge

Smith says that she was prompted to tackle the problem by a colleague, beginning work on it only one month ago.

Her expertise lies in solving differential equations, and she says that she has developed new mathematical tools to do so. She gave a lecture on how these tools could be applied to another set of equations. At the end of the lecture, she says, "somebody asked me why don't I work on one of the Clay problems. So I looked around for one to do with differential equations".

She spotted that the Navier-Stokes equations could be rewritten in the form of the differential equations that she knows how to solve. The method works by setting upper and lower bounds to the solution, then squeezing them together to show that they converge. The paper that she posted online has also been submitted to the Journal of Mathematical Analysis and Applications1, she says.

"I'm pretty confident that my result is right, or I would never have submitted it anywhere," says Smith. "Of course, when there's so much attention being paid it does activate every piece of insecurity one has ever had." This anxiety has led her to revise the paper a number of times since posting it to the arXiv preprint server — but the changes were to correct typographical errors, rather than anything mathematically significant, she says.
 

The proof

There have been previous claims of solutions to the Navier-Stokes problem, says Fefferman. He recalls seeing maybe half-a-dozen such papers over the past few years, most of which he discovered to have obviously fatal errors within a matter of hours.

He expects that assessing Smith's work will take much longer than that. Although the paper itself is only nine pages long, it relies heavily on her earlier publications, so Fefferman will have to trawl through those too. The earlier papers are in peer-reviewed journals or are listed as "to appear" in such journals. "That increases the probability that they're right, but for something this important I wouldn't trust that," says Fefferman.

Smith used to attend seminars, she says, with Grigory Perelman, the Russian mathematician who is believed to have solved another of the Millennium Prize Problems: the proof of the Poincaré Conjecture. He recently turned down the most prestigious prize in mathematics, a Fields medal (see 'Maths 'Nobel' prize declined by Russian recluse') ; it is rumoured he would not accept a Clay million if it was offered to him.

Like Perelman, Smith says that she is motivated by the mathematics, not the money. She hopes to serve as a role model for women in mathematics. "On the other hand, I certainly want the prize," she says.

James Carlson, president of the Clay Mathematics Institute, says he is seeking opinions on the paper, but adds, "It is far too early to say whether it is correct or not." To win the prize, Smith's work will have to withstand two years of scrutiny after appearing in a peer-reviewed journal.

References
Smith P. J. Math. Anal. Appl., preprint at http://arxiv.org/abs/math/0609740  (2006).
Latest news: This paper is being withdrawn by the author due a serious flaw. Sun, 8 Oct 2006