In 2000, the Clay Mathematics Institute published a list of seven open problems, each of which will earn the person who solves it a million-dollar prize [1]. One of these problems concerns the existence of “reasonable” solutions (we will explain exactly what this means later) to a set of highly important equations known as the Navier–Stokes equations.

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The Navier–Stokes equations describe the behavior of fluids. A fluid is any substance that undergoes continuous deformation under a force acting tangentially to its surface (also called shear stress). If this sounds somewhat confusing, think of a fluid as a material that takes the shape of its container. For example, a gas is (usually) a fluid, and a liquid is of course a fluid, but so are plasma, molten lava, galaxy clusters, and even continental glaciers can be described as fluids (they flow very, very slowly). 

Now that we have explained what falls under the definition of a fluid, we can understand the immense importance of the Navier–Stokes equations. Every fluid in the world can be described by these equations. Whether we want to calculate the trajectories of ballistic missiles in the atmosphere, simulate blood flow through vasculature, or forecast the weather, the Navier–Stokes equations allow us to calculate what the flow field in the system that interests us will look like.

The Navier–Stokes equations work so well because they are based on two very fundamental laws of physics: the law of conservation of mass and Newton’s second law of motion. For fluids, the law of conservation of mass leads to an equation called the “continuity equation”, which essentially states that if we want to introduce an incompressible fluid (such as water) into a given volume, we must remove an equal amount of fluid from that volume, or “punch” a hole in it through which the fluid can escape. Newton’s second law states that the rate of change of a fluid’s momentum is equal to the sum of the forces acting on it, and it leads to three additional equations (one for each spatial direction) that describe a competition between the two most important terms in flow: the advection terms and the viscosity terms.

Imagine a motorboat traveling across a calm lake. In the region near the boat’s engine, the advection terms dominate, representing the transport of momentum in the fluid (think of them as kinetic energy), and the water churns into vortices of many different sizes. But if we turn off the engine and wait a while, the water’s surface will calm down and all the vortices will disappear. Where did all that kinetic energy go? This is where the viscosity terms come in: they represent the “internal friction” within the fluid, and they are responsible for converting the system’s kinetic energy into heat. The viscosity terms dominate the flow field in cases of very slow flows, highly viscous liquids (such as honey), and very small system dimensions (such as flow in microchannels or capillaries).

Remember the vortices of many different sizes created by the boat’s engine? Their different sizes are the product of a fascinating phenomenon in fluid flow called the energy cascade. In this phenomenon, the kinetic energy of large vortices is transferred to increasingly smaller vortices, until the vortices become so small that viscosity takes over and converts all the kinetic energy into heat. In the video below, you can see a beautiful demonstration of the energy cascade in turbulent flow.

One might even think that because of this action of the viscosity terms, any flow field we choose, no matter how strong, will decay after enough time and the flow will stop. Or, more generally, we would expect that even in the presence of an external energy source (such as the engine in the boat example), the fluid’s kinetic energy will remain bounded and the flow field will not “blow up” (that is, it will not take on infinite values).

There is just one problem... We do not know how to prove either of these claims. In fact, we do not know whether, for every “reasonable” initial condition, the Navier–Stokes equations will yield a sensible solution—namely, one that does not blow up in finite time.

The Clay Institute’s Millennium Problem concerns the Navier–Stokes equations for an incompressible Newtonian fluid (like most of the fluids we know), which is the most common form of these equations. Its official formulation is: “Given an initial velocity field in three spatial dimensions and one time dimension, prove or disprove by counterexample that there exist smooth (that is, continuous solutions that do not change abruptly), everywhere-defined velocity and pressure fields that solve the Navier–Stokes equations”. So far, the following partial results have been proven:

1. In two spatial dimensions and one time dimension, the problem was solved as early as the 1960s, and the answer is positive: smooth, everywhere-defined solutions always exist.
2. In three dimensions, the answer is positive provided that the initial velocity field is very small.
3. Given any initial velocity field in three dimensions, it can be shown that smooth solutions exist up to some finite time (but we do not know what happens after that time).
4. In 2016, Terence Tao, a Fields Medal-winning mathematician, showed a method for constructing solutions to an “averaged” version of the Navier–Stokes equations that blow up in finite time. These averaged equations can be thought of as describing the flow field at low resolution, so that not all the fine details are visible.

Although Tao’s proof applies to equations that are “almost” the Navier–Stokes equations, the way the solution is constructed is in principle also possible for the unaveraged equations. It is possible that someone will eventually find a non-smooth solution to the Navier–Stokes equations based on this idea. Then again, perhaps not.

So far, the Navier–Stokes equations have passed every test and experiment successfully. Yet despite countless attempts at proof (and disproof) by mathematicians and scientists around the world, the existence and smoothness problem for the Navier–Stokes equations remains one of the greatest and most important challenges in mathematics. Solving this problem is closely tied to our ability—or rather, our inability—to understand what Richard Feynman called “the most important unsolved problem of classical physics”: turbulent flow. A one-million-dollar prize may encourage more people to work on the problem, but just between us, there are probably easier ways to earn a million dollars.

English editing: Elee Shimshoni

Reference:

1. The Clay Mathematics Institute Millennium Problems