Good News Everyone! I learned just now from Good Math, Bad Math there may well be a successful proof of yet another of the great Millennium problems. This one pertains to solutions of the Navier-Stokes equations which are of central importance in fluid dynamics, for example. (The figure to the left -click for larger- is from a computer simulation, by Greg Ashford, of the airflow around an aircraft wing (in cross section) and the foil that it uses to change the overall shape of the wing for maneouvering. I got it from a site linked here.)
Penelope Smith, at Lehigh University has presented a preprint with her proof, which uses a set of techniques that she has been developing for a number of years. Christina Sormani has a website which describes aspects of the proof, and Smith’s website is here. Read also the commentary at Good Math, Bad Math
As usual, there will be a long process of independent checking and re-checking her results before the mathematics community declares this problem solved (should her solution pass those tets). So there is a bit of a wait to follow. Don’t be under any illusions about the importance of this problem. It is an incredibly difficult system of equations to solve, while at the same time they arise as governing the physics of so many situations (pretty much anything inolving the flow of fluids: water, air, etc), so they show up in fields as diverse as the study of the weather, the structure of the interior of a star, the design of more aerodynamic (and hence more fuel-efficient) cars, etc.
I can’t help but wonder if the Clay Institute, which will have do dole out another million dollars in such a short time, had budgeted for such a rapid rate of mathematical progress…