Nature: Navier-Stokes solved?

From Nature, there is word that another famous math problem may have been solved.

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

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.

Two posts here, Intuition and space maps and Poincaré and proof positive, describe a little about another towering math problem, the Poincaré conjecture, which also may have been solved recently.

I'm intensely interested in the problem of knowledge. This is probably a massive overreach, but given the complaints leveled against string theory (where is the physical evidence?) and the suggestion that science increasingly rests on information (an idea too firmly wedged in my brain), I wonder: is rationalism opening a knowledge gap on empiricism? What does it mean to know when knowledge thrives in our heads but is inaccessible to our fingertips? And related: is language the summit of cognition?

Wayne