The" Clay Mathematics Institute":http://en.wikipedia.org/wiki/Clay_Mathematics_Institute in America has offered 7 $US1,000,000 prizes (the "Millennium Prizes":http://en.wikipedia.org/wiki/Millennium_Prize_Problems) for the solutions to each of 7 unsolved mathematical questions, one of which is proof of existence and smoothness of the "Navier-Stokes Equations":http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness, also called the Equation of Continuity. These equations describe the 3-dimensional behaviour of any fluid in (almost) any circumstances, like air passing across an aircraft's wing, or "air vortices in water":http://en.wikipedia.org/wiki/Bubble_ring. A Kazakhstan university professor. Mukhtarbay Otelbayev. has published a paper that seems to have solved the equations, but it's in Russian, and most Western mathematicians with the specialist knowledge needed to understand the mathematics can't understand the Russian. "http://www.newscientist.com/article/dn24915-kazakh-mathematician-may-have-solved-1-million-puzzle.html":http://www.newscientist.com/article/dn24915-kazakh-mathematician-may-have-solved-1-million-puzzle.html

Navier-Stokes equations used to be my bread and butter. I like to quote a person who said “If they were any easier than we wouldn’t have a job because anyone cold solve them. If they were any more difficult then they would be impossible to solve.”

> the Navier-Stokes existence and smoothness problem

I read a book about this 30 or so years ago. It was really pushing my limits to try to understand it. I do not recommend it for the mathematically illiterate.

Now that I’ve had a look at the paper, the maths isn’t that bad at all. There’s nothing there that is intrinsically difficult.

I note that he’s assumed periodic boundary conditions. That will satisfy the conditions for the Clay institute prize, but in most of the cases I’ve had to solve the boundary conditions aren’t periodic.