The Navier-Stokes existence and smoothness problem has been one of the Clay Institute's Millennium Prize Problems since 2000, with a $1 million bounty available for a proof or disproof of an associated conjecture. The technical statement of this conjecture is that the three-dimensional Navier-Stokes equations for incompressible flow admit globally defined solutions which are smooth, given that the initial conditions are themselves smooth and satisfy certain decay conditions. In practical terms, the problem is analogous to the physical statement that if we have some fluid in a container and initially its velocity field is "well behaved," then if we wait for any finite amount of time and look at the container once more, the velocity field will still exist and be well behaved in the same technical sense.

This statement might seem obvious: How can a fluid have an ill-defined velocity after a finite amount of time in which we leave it inside a container if its initial velocity was well-defined? However, the claim turns out to be surprisingly difficult to establish for the Navier-Stokes equations in three dimensions. In this essay I'll explain the origin of this difficulty.

The Navier-Stokes equations

The Navier-Stokes equations for incompressible flow are typically stated in terms of a smooth velocity field $u(x, t): \mathbb R^3 \times [0, \infty) \to \mathbb R^3$ and a smooth pressure field $p(x, t) : \mathbb R^3 \times [0, \infty) \to \mathbb R$.

Here is one way of justifying the Navier-Stokes equations: Suppose that there is an infinitesimal particle in the fluid whose position at time $t$ is given by $y(t)$. Since the fluid is incompressible we assume there is the same density of fluid everywhere in $\mathbb R^3$, and so Newton's second law reduces to the statement that force is proportional to acceleration. If we normalize so that the fluid density is equal to $1$, we get

$$F = \frac{d u(y(t), t)}{dt} = (y'(t) \cdot \nabla) u(y(t), t) + \frac{\partial u(y(t), t)}{\partial t} = (u \cdot \nabla) u + \frac{\partial u}{\partial t}$$

where in the last equality I omit the position and time coordinates for brevity. This equality is simply an expression of Newton's second law in our specific context, but without specifying the force $F$ it doesn't tell us much about what our fluid will do. In the Navier-Stokes equation, we consider two contributions to $F$: pressure forces and friction forces. The pressure contribution is simple, as the force is simply the negative of the gradient of pressure at a point, $- \nabla p$. The friction contribution, also called viscosity in the context of fluid dynamics, requires a further assumption about how friction works in our context. Navier-Stokes assumes that the friction force between two adjacent flows is proportional to their relative velocity with a proportionality constant $\nu \gt 0$ called the kinematic viscosity. If we make this assumption and do some vector calculus, we get that the viscosity contribution to $F$ is given by $\nu \nabla^2 u$, where $\nabla^2$ is an operator known as the Laplacian. We end up with the equation

$$(u \cdot \nabla) u + \frac{\partial u}{\partial t} = -\nabla p + \nu \nabla^2 u$$

The final ingredient we need is specifying what determines the pressure $p$, as right now we have one equation in the two variables $u$ and $p$. We can close this system by explicitly requiring that the flow defined by $u$ is incompressible—in other words, that there's no net flow of mass into or out from any region in space. By the divergence theorem, this is equivalent to the equation $\nabla \cdot u = 0$.

Here we have the Navier-Stokes equations for incompressible flow in all their glory:

$$(u \cdot \nabla) u + \frac{\partial u}{\partial t} = -\nabla p + \nu \nabla^2 u$$ $$\nabla \cdot u = 0$$

In the first equation, the first term $(u \cdot \nabla) u$ is called the transport term. Remember this name since this term is going to cause significant problems for us later on.

The ultraviolet catastrophe

The original ultraviolet catastrophe was a distressing conclusion reached by physicists who applied the methods of statistical mechanics to analyze the intensity and spectrum of light emitted by hot objects following Maxwell's remarkable discovery that visible light was a kind of electromagnetic wave.

Simply put, the equipartition theorem of statistical mechanics states that all quadratic degrees of freedom in a system must contain the same amount of energy in a statistical equilibrium. Electromagnetic waves have a quadratic degree of freedom corresponding to each individual frequency. These two claims combine to produce the result that almost all of the energy contained in an electromagnetic wave emitted by a hot object must be in very high—that is, ultraviolet—frequencies.

This basic problem also comes up in the Navier-Stokes equations. The transport term is nonlinear, and nonlinearity tends to make statistical equilibrium a good approximation for the behavior of a system in general, though there are exceptions to this general rule. When the ultraviolet catastrophe phenomenon occurs in fluid dynamics, we call it turbulence. Turbulence is the process by which the nonlinear tendency of the Navier-Stokes equations asserts itself to ensure that energy flows from low-frequency or high-length scales—of which there are few—to high-frequency or small-length scales—of which there are many. Think of large vortices gradually devolving into smaller ones ad nauseum as the initial large scale and simple motion of the fluid is broken up into an enormous number of messy and small-scale components.

Turbulence is unpredictable and difficult to understand in many ways due to the nonlinearity of the underlying transport dynamics. However, the viscosity term balances out the nonlinearity of the transport term at very high frequencies because parts of the fluid that are moving too fast dissipate energy due to friction and so eventually slow down. This fact allows us to get a good statistical description of turbulence. Kolmogorov used this method in 1941 to obtain a power law description of turbulence which states that in the presence of viscosity, the energy at some frequency scale $f$ will be proportional to $f^{-5/3}$. In particular, since $f^{-5/3}$ is an integrable function, the presence of viscosity has eliminated the ultraviolet catastrophe.

Global regularity

Unfortunately, Kolmogorov's argument is statistical in nature and therefore while it works for generic initial conditions, it's not strong enough to rule out the ultraviolet catastrophe for all initial conditions. In particular, we can imagine that some exceptional solution of the Navier-Stokes equations has the property that all of the energy in the fluid ends up being concentrated in a small, rapidly spinning ball. Resolving the existence and smoothness problem for Navier-Stokes then comes down to showing that such a situation—in which the radius of the ball shrinks to zero in a finite time as the velocity diverges to infinity—can't happen.

To understand why this is hard, remember that the problematic term in the Navier-Stokes equations is the transport term. In other words, when the viscosity term is much bigger than the transport term, the viscosity forces the velocities together, and over time the nonlinear dynamics die down as the fluid settles into a steady state. By simple dimensional analysis, if we have a ball of radius $R$ spinning with a velocity of $V$ at the surface, the viscosity term is proportional to $V/R^2$ while the transport term is proportional to $V^2/R$. In particular, we expect the viscosity to be dominant when $V/R^2 \gg V^2/R$ or $1/R \gg V$.

On the other hand, the kinetic energy of such a ball is proportional to $MV^2$ where $M$ is its mass, which is proportional to $R^N$ where $N$ is the number of dimensions in space by the assumption of incompressibility or constant density. We therefore know that $R^N V^2$ is bounded by the initial energy of the fluid $E$, since the transport effects conserve energy and friction can only drain energy from the system, not contribute additional energy into it.

In this situation, if we had $N = 2$, the worst case scenario $V/R^2$ and $V^2/R$ would be roughly comparable in magnitude. Here we say that the partial differential equation is critical: That is, the Navier-Stokes equations in $N = 2$ spatial dimensions are critical. It turns out this fact is sufficient to solve the existence and smoothness problem in two dimensions.

However, when $N = 3$, $1/R$ can only be as big as $V^{2/3}$ times a constant, which is a far cry from the condition $1/R \gg V$ that we had hoped for in order for viscosity to win the battle with the transport term. In other words, nothing about energy conservation and the size of the viscosity effects seems to preclude the velocity field diverging to infinity in finite time. Another way of stating this is that the Navier-Stokes equations in $N = 3$ dimensions are supercritical, and this is why the existence and smoothness problem in three dimensions has proven to be so difficult to settle.

Should we expect existence and smoothness to hold?

I think the Navier-Stokes existence and smoothness problem is unique among all of the Millennium Prize Problems in that it is more likely to be resolved in the negative rather than in the affirmative. In particular, I believe it is more likely than not that the Navier-Stokes equations can, indeed, blow up in finite time.

In a 2016 paper, Terence Tao constructed a finite time blowup solution to an appropriately averaged version of the Navier-Stokes equations in which he ignored inconvenient frequency interactions implied by the transport term in the equation. While this doesn't prove the conjecture of existence and smoothness false due to the very strong simplification assumptions made in the argument, in my view it is suggestive evidence that all the problem might require is the specification of sufficiently clever initial conditions to achieve the elusive finite time divergence.

The fact that in the real world we don't observe finite time divergences in velocity is not a good argument against their existence in theory. There are many reasons for this but here is one that is often overlooked: The friction between fluid flows in the real world is not proportional to their relative velocity, especially when the relative velocity is large. There are higher order contributions to friction which would manifest themselves as hyperviscosity terms in the equations—friction terms that are especially strong in the pathological situations we imagined earlier and that would be strong enough to make the equations critical or subcritical even in three dimensions.

From this perspective, the existence and smoothness problem for Navier-Stokes is not of much practical relevance for engineering or computational fluid dynamics applications, but as a question about the limits of supercritical behavior in partial differential equations and about the reach of turbulent effects, it nevertheless has theoretical significance, justifying its inclusion as a Millenium Prize Problem.

Forecasts

In my view, how long we expect to wait until a resolution of the existence and smoothness problem depends on whether the existence and smoothness conjecture is true or not. I think the chance of the conjecture being true is about 35%—which I mention explicitly since while it is possible to infer it from my forecasts on the two questions below, the interface only displays my 50% confidence interval.

If the conjecture is true, then I see no particular reason to expect it to be proven anytime soon, because in my judgment there has been no progress whatsoever toward a method which would prove that the conjecture holds. In this case, my expected timeline would be in line with the base rate for Millennium Prize Problems so far, and more pessimistic than Laplace's rule.

On the other hand, if the conjecture is false, then I think we can be more optimistic about a possible solution given that just in the last decade there has been progress toward a better understanding of the precise transport dynamics in Navier-Stokes and how they might be exploited by carefully chosen initial conditions. While we're still far from a solution, I think in this scenario we should be more hopeful.

These considerations led me to operationalize the forecasts for this question in the following ways:

A further question may be asked about an issue I've neglected to mention so far: While the specific definition I gave in this essay was over an infinite three-dimensional space $\mathbb R^3$, we can also define the Navier-Stokes equations over other spaces. The most frequently studied alternative space is the three-dimensional torus $(\mathbb R/\mathbb Z)^3$, which is convenient because it's compact and therefore we don't need to worry about technical growth conditions on the solutions as we did before. Solving the equations on a torus amounts to looking only for periodic solutions of the original equations over $\mathbb R^3$.

While unlikely, it is possible that the global regularity properties of the Navier-Stokes equations end up depending on whether the universe over which they are defined is compact or not. For instance, if certain mechanisms for achieving a finite time blowup of the equations require an infinite amount of space, the compact torus wouldn't allow the mechanism to function while the noncompact $\mathbb R^3$ might. Will this turn out to be true? You can find the relevant question below.

Conclusion

The Navier-Stokes existence and smoothness conjecture may not have direct applications or consequences relevant for practical contexts in which engineers must cope with turbulence, but it is nevertheless an important theoretical question whose resolution would give valuable information about the limits of turbulent effects in situations where they aren't dampened by friction. I think the conjecture is somewhat more likely to be false than true, and if it is indeed false I believe there's cause to be optimistic about the problem being settled in the next few decades.