On October 2, Nature published this news brief about a claim of a solution to the Navier-Stokes equations:
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.
The headline of the article asked: “Has Famous Maths Problem Been solved, and in Only a Month?”
The answer, it seems, is no.
In September 2006, Penny Smith of Lehigh University posted a paper which appeared to have proven the Millenium Problem on Navier Stokes. She then retracted the paper in October 2006 when an error was found in a prior published paper which is described below. Here is a description of her research that was posted in September 2006 to assist in the verification process. At the Navier-Stokes mathforum, people may feel free to ask further questions about Navier-Stokes rather than this particular attempt.
My understanding of the situation is that while Smith’s initial attempt at a solution was clearly incorrect, the errors may not be fatal to her enterprise. Only time will tell if her basic method can be made to work.
The Navier-Stokes equations, incidentally, refer to a certain set of partial differential equations that arise in the study of fluid mechanics. Roughly, you imagine a fluid with a particular viscosity flowing under the influence of a force that varies with time. The problem is to determine the motion of a speck of dust moving along with the fluid. If you make certain reasonable assumptions about the nature of the fluid and the force acting on it, and invoke some elementary ideas from physics and calculus, you can derive a set of PDE’s that describe the motion of the speck of dust.
The equations themselves are not especially difficult to derive. Not today, anyway, when the necessary ideas from physics and mathematics are routinely taught to college freshman. But when people like Euler and Bernoulli, and then later Navier and Stokes, first attacked this problem, their equations represented a significant accomplishment.
Alas, writing the equations down is easier than solving them. Finding functions that satisfy them is quite another. Nature seems to have little trouble solving them. I mean, fluids do flow, after all. But humans are finding the problem a bit trickier. Wikipedia has a typically informative essay on the subject.