Solving one of the most important mathematical problems

Their ability to describe motion of liquids and gases makes them useful in a variety of applications. Yet there’s no one who fully understands the Navier-Stokes equations. Large parts of their nature remain a mystery. But that’s something that Denis Gaidashev intends to change.

The existence of solutions of Navier-Stokes equations is considered as one of the most important unsolved mathematical problems of this millenium. Ever since the equations first appeared in the nineteenth century, they have helped us to understand various physical phenomena involving movement of gases and liquids. Biomedical researchers calculate blood flow with them, meteorologists model weather with them and engineers use them to predict how oil will move through pipes. This system of equations has countless applications, and yet no one has managed to show that it has smooth global solutions.

How is it possible to apply equations that aren’t solved mathematically? It isn’t as strange as it may sound. There are ways of simulating solutions by a computer, but only up to a certain limit. Although these numerical calculations may be perfectly sufficient for solving particular physical problems, they tell us nothing about how the equations act everywhere and at all times. In order to understand their complete behaviour, we need a rigorous mathematical proof.

What makes the Navier-Stokes equations so difficult to solve?

- They have a term that is non-linear. The problem is that, given some initial conditions, we don’t know if the equation system has a solution at all times. Could it be so that there is a time at which the solutions become unbounded?

Translated to physics, the equation system describes flow of a liquid. Flow depends on the pressure and the velocity of the liquid, both of which vary over time. The initial condition is the velocity and the pressure that the liquid has at an initial point, i.e. when time is equal to zero. So, the question is: Will the flow always be smooth, or will there be a point in time when it explodes? In other words, is there an initial velocity for which the energy will be unlimited?

Denis Gaidashev says that many mathematicians have tried to solve the problem, but still, no one has managed to reach all the way to the answer. Even though many of those attempts have lead to calculations that could not be completed, some of them have interesting aspects that could be helpful in finding a solution.

- One of the most prominent mathematicians right now, Yakov Sinai, has proposed a new method, using something called renormalization. Renormalization means that we introduce an operator that relates solutions for a later point in time to a solution for a previous point. Sinai and his collaborators haven’t managed to solve the problem analytically, it was again very difficult. Now, what we are suggesting is to try to use computer aided proofs, as we do in our research group, CAPA, says Denis Gaidashev.

Does it mean that the part processed by the computer would be impossible for a human to calculate?

- Exactly. Some calculations would take a great amount of time, or they are simply impossible for a human to do. But today, there are techniques for programming proofs on a computer. What we plan to do is to implement a renormalization scheme on the computer and make it perform the difficult calculations for us.

How long will it take?

- Maybe two to four years.

Thanks to a new investment in Swedish mathematical research, Denis Gaidashev has received funds to hire a postdoctoral researcher to help him tackle the problem. So far, he has no particular person in mind, but it must be someone who can understand both renormalization and computer proofs.

Denis Gaidashev and the new post doctor will begin the search by making experiments with numerical simulations. If this gives results, it will be an indication that the problem can be solved in the way that he has intended.

- The simulations will operate within the framework of techniques that Yakov Sinai and his collaborators have proposed. We will not only take the system of equations and put it into a computer and see what it gives. No, we’ll have to program the renormalization operators and implement the whole technique.

Why did you choose this problem?

- Maybe because I have experience in renormalization and computer aided proofs. And I think that our group, CAPA, has a rather privileged opportunity to solve the problem in this way. There aren’t many specialists in the world who have the knowledge of both techniques, but we do.

What will it mean for mathematics if you manage to solve the problem?

- First, I’d like to say that it's pretty optimistic to say that we can solve it. But if we do, I expect that some would rather like to see an analytical evidence, and maybe they won’t think that a computer aided proof could tell us much about the mathematical structure. On the other hand, just to answer the question if there are initial conditions for which energy is unlimited in finite time would be a success.

Denis Gaidashev thinks the answer to that question is yes. What it will mean for other disciplines if he succeeds is hard to predict. Hopefully, we’ll see in a few years.

Alma Kirlic

December, 2016