The Navier-Stokes equations describe how water flows in turbulent situations.
A huge mathematical breakthrough might have just been made, but a language barrier is slowing things down.
New Scientist reports that Kazakh mathematician Mukhtarbay Otelbayev may have solved an extremely difficult and useful mathematics problem: the Navier-Stokes equations.
This is one of the Clay Mathematics Institute Millennium problems — six unsolved problems (and one solved problem) that are both of deep theoretical interest and have many useful applications. Finding a solution to one of these problems that stands up to strict mathematical scrutiny carries a prize of one million dollars.
Unfortunately, in addition to the normal difficulties in verifying a complicated new proof, Otelbayev's paper is currently only published in Russian, making things a little harder for the international mathematical community.
The Navier-Stokes equations are a set of differential equations. Differential equations describe a quantity in terms of how it is changing throughout time and space. In many situations in physics and economics, it is more intuitive to describe mathematically how a quantity is changing than to directly write out an expression for the quantity itself.
But when you solve a differential equation you do just that — you find a formula to describe the actual quantity at any particular time or place, based on the differential equations describing how the quantity changes.
Solving the Navier-Stokes differential equations is key to understanding fluid dynamics — how smoke moves off of a fire, how water flows through a pipe, or how air glides over a car driving down a highway.
The equations describe how the flow of a fluid changes at different times and places. They are based on Newton's second law of motion — the statement that the amount of force exerted on an object equals the rate at which the object's momentum is changing — applied to the special case of a moving fluid.
Despite the relative straightforwardness of the equations, and their many applications, mathematicians do not know, in general, how to solve them. That is, given a fluid, we do not, in all cases, know how to find a mathematical expression that describes the patterns of flow in that fluid.
Mathematicians and physicists are especially confounded by the behavior of turbulent fluids, like the smoke spreading out chaotically on the table in the GIF to the right.
While we can use powerful computers to come up with approximate simulations of fluid behavior, without exact solutions to the Navier-Stokes equations, our theoretical understanding of these kinds of processes remains very limited.
Mathematicians do not even know whether or not these solutions exist, and this mystery, combined with the many applications of these equations to physics, is why this is such an important problem.
Time and the tireless efforts of many translators and mathematicians will tell if Otelbayev has truly solved this intriguing problem.
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