In the news this week, a Kazakh mathematician has hit the headlines* for having supposedly submitted a solution to the Navier-Stokes existence and Smoothness problem, one of the Clay Mathematics Institute’s 7 Millennium Prize problems. This is big news, as these problems represent the 7 biggest unsolved problems in mathematics, and only one has been solved so far. We may have to wait some time to determine if his answer is correct, as he had the gall to write it in Russian and not English (which he doesn’t speak). Each of these problems carries a $1 000 000 prize.
The “plain English” phrasing of the problem is (I’ll put the “official” wording at the end):
Given initial pressure and speed, can we always predict the behaviour of a fluid over time?
Fluid behaviour is defined by the Navier-Stokes equations, which are below:
(this is actually three equations rolled into one for convenience’s sake).
I think we studied a simplified version of this in one dimension as part of the “Motion and Flow” course we did in third year. In any case, what the equation is describing is how fluid is flowing in particular places at particular times- it’s how we can predict weather, flow in pipes, and air flow around a wing.
The problem is that for certain initial conditions, we don’t know if the equations yield “nice” answers (in the language of the original problem, the solutions have to be “smooth” and “globally defined”), which means that we get turbulence, which will probably bring back memories of bumpy plane rides. Turbulence is mathematically problematic, because it’s hard to pin down (in official terms, it’s chaotic).
You’ll win your million dollars by proving that you can find a solution that doesn’t involve turbulence, no matter what state your fluid is currently in, or if you can show that there is a state that it’s impossible not to get a non-turbulent answer for.
The interest in this is almost entirely theoretical, which means that it’s unlikely that if it’s proven you’ll see an end to bumpy aeroplane flights, but then Mathematicians and the “real world” have always been strange bedfellows…
Below is the full statement of the problem (I’ll explain it afterwards!):
Prove or give a counter-example of the following statement:
In three space dimensions and time, given an initial velocity field (i), there exists a vector velocity (ii) and a scalar pressure (iii) field, which are both smooth and globally defined (iv), that solve the Navier–Stokes equations.
So to explain some of this terminology:
i) and ii) A velocity field is an equation that tells you how fast and at what direction a fluid is flowing at any given point at a given time. An initial velocity field is what the equation looks like at the start of the given time period. It is a vector because it has a size and a direction.
iii) A pressure field is an equation that tells you how much pressure is being experienced at a given location at a given time. It is a scalar because pressure doesn’t have a direction associated with it.
iv) Finally, “smooth and globally defined” means that the relevant velocity and pressure fields don’t have any gaps or jumps in them, and that they give an answer irrespective of what values you put in (an example would be that the graph of sin (x) is smooth and globally defined, while the graph of tan (x) is neither, as it gives no answer for 90 degrees.)
* in relative terms, of course!