The Navier-Stokes Situation

03 Aug, 2026

Consider a lake flowing.

If I gave you the velocity and pressure at every single point in the lake, would you be able to predict how the lake flows over time?

Seems like a reasonable question. We know the physics. Water is not magic. It has mass, momentum, pressure and viscosity. Newton has been dead for almost 300 years. Surely we have this figured out.

This question points to the Navier-Stokes equations, a set of equations that can describe essentially any fluid you can think of, from water to air to honey. They are some of the most important partial differential equations in physics and engineering. We use them to forecast weather, model airplanes and rockets, predict water currents, design turbines, understand combustion and generally prevent fast-moving fluids from ruining our day.

There is just one small problem. We don't actually know whether they always behave themselves.

There is a $1 million prize for proving (not anymore) that sufficiently nice three-dimensional initial conditions remain nice forever, or showing that somewhere, somehow, the equations can blow themselves up. We fly airplanes designed using equations whose most basic mathematical regularity problem is still open.

So how did we get here?

Water is just Newton with extra steps

For what follows, assume the fluid is Newtonian and incompressible.

Newtonian means that viscosity does not itself depend on how quickly we shear the fluid. Ketchup is the classic counterexample. Hit the bottle, increase the shear rate, viscosity falls and suddenly the ketchup comes out. Water behaves much more politely.

Incompressible means that density is treated as constant:

ρ = constant .

Equivalently, a little parcel of water cannot simply compress itself into nothing.

With those assumptions, Navier-Stokes looks intimidating:

ρ (\frac{\partial 𝐮}{\partial t} + (𝐮 \cdot \nabla) 𝐮) = - \nabla p + μ \nabla^{2} 𝐮 + 𝐟

together with

\nabla \cdot 𝐮 = 0 .

This is one of those equations that looks like you need to have spent seven years in a basement at Cambridge to understand it.

You don't. It is mostly two extremely boring statements:

mass is conserved, and $F = m a$ .

That is basically it. The horror comes later.

You cannot delete water

Start with

\nabla \cdot 𝐮 = 0 .

Here $𝐮 (𝐱, t)$ is the velocity field. At every point in space and time, it gives us a vector telling us which direction the fluid is moving and how quickly:

𝐮 = (\begin{matrix} u_{x} \\ u_{y} \\ u_{z} \end{matrix}) .

The divergence of that field is

\nabla \cdot 𝐮 = \frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y} + \frac{\partial u_{z}}{\partial z} .

Very loosely, divergence tells us whether a point behaves like a source or a sink.

Imagine arrows radiating outward from a point. Positive divergence. Arrows collapsing inward. Negative divergence.

But if our fluid is incompressible, water cannot simply disappear from some infinitesimal region or spontaneously materialize inside it. What flows in has to flow out.

Hence

\nabla \cdot 𝐮 = 0 .

Mass conservation. Nothing particularly demonic has happened yet.

$F = m a$ , except everything is moving

Now take Newton's second law:

F = m a .

Apply it to a tiny parcel of fluid.

Instead of mass, it is convenient to work per unit volume, so mass becomes density $ρ$ . Instead of the acceleration of one rigid object, we need the acceleration experienced by a little parcel being carried around by the flow.

You might initially write acceleration as

\frac{d 𝐮}{d t} .

But there is an annoying problem. The velocity field can change with time, while the fluid parcel itself is also physically moving through a velocity field that changes with position.

So the relevant derivative is the material derivative:

\frac{D 𝐮}{D t} = \frac{\partial 𝐮}{\partial t} + (𝐮 \cdot \nabla) 𝐮 .

The first term,

\frac{\partial 𝐮}{\partial t},

is intuitive. Stand at one point and ask how the velocity there changes with time.

The second term,

(𝐮 \cdot \nabla) 𝐮,

is stranger.

The fluid itself is moving through the velocity field, so it can accelerate simply because it has arrived somewhere where the velocity is different.

Written out,

(𝐮 \cdot \nabla) 𝐮 = (u_{x} \frac{\partial}{\partial x} + u_{y} \frac{\partial}{\partial y} + u_{z} \frac{\partial}{\partial z}) 𝐮 .

Notice what has happened.

$𝐮$ is multiplying derivatives of $𝐮$ .

The fluid velocity is acting on itself. This is the nonlinear term. Remember it. It causes problems. Newton now becomes

ρ \frac{D 𝐮}{D t} = forces per unit volume .

What forces?

Pressure, viscosity and whatever external nonsense we have decided to subject the fluid to.

Pressure wants to push things around

Consider drinking through a straw. Sucking creates an area of lower pressure. The pressure difference moves the liquid. Spatial changes in pressure are represented by the pressure gradient:

\nabla p .

Fluid accelerates from high pressure toward low pressure, so the corresponding force per unit volume is

- \nabla p .

Then there is viscosity.

Pour water into a cup and it moves quickly. Pour honey and it moves slowly. Neighboring layers of fluid resist moving at wildly different velocities.

For a Newtonian fluid with constant dynamic viscosity $μ$ , this contributes

μ \nabla^{2} 𝐮,

where

\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}

is the Laplacian.

The Laplacian is doing something very important here. It tends to smooth spatial differences in velocity.

Sharp velocity variation:

fast | slow

gets smeared out by viscosity.

Finally, dump all external body forces into $𝐟$ . If gravity is the only one,

𝐟 = ρ 𝐠 .

Put everything together:

ρ (\frac{\partial 𝐮}{\partial t} + (𝐮 \cdot \nabla) 𝐮) = - \nabla p + μ \nabla^{2} 𝐮 + 𝐟

There. Navier-Stokes. Conservation of momentum wearing a trench coat.

There isn't some special airplane equation and another equation for lakes and another one for honey. We are basically writing down fundamental laws of physics and asking every infinitesimal piece of fluid to obey them simultaneously. Unfortunately, there are rather a lot of infinitesimal pieces.

The million-dollar problem

So why is there a million dollars attached to this? The problem is not writing down Navier-Stokes, but rather proving what the equations are capable of doing.

Start with a smooth initial velocity field

𝐮 (𝐱, 0) = 𝐮_{0} (𝐱),

satisfying

\nabla \cdot 𝐮_{0} = 0 .

Now evolve it according to Navier-Stokes.

Does

𝐮 (𝐱, t)

remain smooth for every

t > 0 ?

Or can some perfectly reasonable initial flow evolve into a singularity in finite time?

Very loosely, could something like

‖ \nabla 𝐮 (\cdot, t) ‖_{\infty} \to \infty

t \to T < \infty ?

Nobody knows. This is the Navier-Stokes existence and smoothness problem.

And mathematics is annoyingly unwilling to accept "water has never done this when I looked at it" as a proof.

Splash around in a bathtub and the water does not explode. It does not spontaneously produce a point moving infinitely fast. You get turbulence, waves, little vortices, and then eventually everything settles down.

This happens 100% of the times you have personally interacted with a bathtub.

Unfortunately the theorem is asking about every admissible smooth initial configuration, including extremely special configurations no sane person would ever deliberately construct.

If something works $99.999999 %$ of the time, most engineers are going home. The mathematician would like to discuss the remaining $0.000001 %$ . For several decades.

Why doesn't viscosity just win?

At first this still feels like it should be easy. Water is viscous. Viscosity dissipates motion. A moving incompressible fluid has kinetic energy

E (t) = \frac{ρ}{2} \int_{ℝ^{3}} | 𝐮 (𝐱, t) |^{2} d 𝐱 .

Ignoring external forcing and assuming appropriate boundary behavior, Navier-Stokes gives an energy relation of the form

\frac{d E}{d t} = - μ \int_{ℝ^{3}} | \nabla 𝐮 |^{2} d 𝐱 .

Therefore

\frac{d E}{d t} \leq 0 .

Excellent.

The fluid cannot spontaneously manufacture kinetic energy.

Integrating in time gives

E (t) + μ \int_{0}^{t} \int_{ℝ^{3}} | \nabla 𝐮 |^{2} d 𝐱 d s = E (0) .

Initial energy either remains as kinetic energy or gets dissipated through viscosity. Splash water around. Wait. Water stops moving. Problem solved right? Except not, because controlling how much energy exists is not the same thing as controlling where that energy goes.

This distinction is basically the entire problem. The energy estimate gives us control over something like

\int | 𝐮 |^{2} d 𝐱 .

But regularity requires much stronger control over the fine spatial structure of the velocity field. In particular,

E (t) < \infty

does not automatically imply

‖ \nabla 𝐮 (t) ‖_{\infty} < \infty .

Imagine velocity varying over some characteristic length scale $ℓ$ . Then very roughly,

| \nabla 𝐮 | ~ \frac{U_{ℓ}}{ℓ} .

Now shrink the length scale:

ℓ \to 0 .

Even if $U_{ℓ}$ remains finite,

\frac{U_{ℓ}}{ℓ}

can become enormous.

This was the part that initially broke my intuition. The nightmare is not necessarily infinite energy. The nightmare is concentration. The equation can obey its global energy budget while potentially doing increasingly pathological things locally. It can be completely honest about the accounting and still hide the money in a room of zero volume.

Big eddies become little eddies

This is where turbulence enters. Imagine a large swirling structure in a fluid with characteristic size

ℓ_{0} .

It becomes unstable and transfers energy into smaller structures:

ℓ_{0} \to ℓ_{1} \to ℓ_{2} \to ℓ_{3} \to \dots

with

ℓ_{n + 1} < ℓ_{n} .

This is the basic picture of an energy cascade.

In real turbulence, a large eddy doesn't normally transfer all of its energy into one perfectly organized smaller eddy. Energy spreads across many structures and many scales.

Eventually the relevant structures become small enough that viscosity dominates and dissipates their kinetic energy into heat.

Big eddy → Smaller eddies → Even smaller eddies → Heat.

Good, but mathematics has to rule out something much more malicious. What if the energy does not disperse? What if the nonlinear dynamics somehow keep concentrating it?

Suppose the characteristic scale shrinks geometrically:

ℓ_{n} = 2^{- n} ℓ_{0} .

And suppose the characteristic time for each transfer also shrinks:

Δ t_{n} = 2^{- n} Δ t_{0} .

Then the total time required for infinitely many transfers is

T = \sum_{n = 0}^{\infty} Δ t_{n} .

But

T = Δ t_{0} \sum_{n = 0}^{\infty} 2^{- n} = 2 Δ t_{0} .

Infinite steps. Finite time.

So merely saying "it would have to pass through infinitely many scales" does not save us. A geometric sequence is perfectly happy to fit infinity inside a finite interval. This opens the conceptual door to finite-time blowup.

Some organized structure at scale $ℓ_{0}$ transfers its energy into a smaller structure at $ℓ_{1}$ .

That one evolves faster.

It transfers into $ℓ_{2}$ .

Faster again.

ℓ_{n} \to 0

while

t_{n} \to T < \infty .

The fluid does not need to manufacture energy. It needs to become increasingly good at concentrating the energy it already has.

Maxwell's demon has access to the bathtub

Terence Tao gives an analogy I really like. Maxwell's demon.

Imagine oxygen and nitrogen mixed together inside a box. Statistically, they stay mixed. There is no reason for every molecular collision to cooperate so that oxygen slowly migrates to one side and nitrogen to the other. It would require an absurd conspiracy. But "absurdly unlikely" and "mathematically impossible" are different statements.

Navier-Stokes has a similar problem.

Ordinary turbulence disperses energy. A large eddy breaks into several smaller eddies, which break into several more, spreading energy around until viscosity can kill it.

Schematically,

E \to E_{1} + E_{2} + E_{3} + E_{4} .

But imagine some demon arranging the nonlinear interactions so that instead, energy remains coherent:

E_{n} \to E_{n + 1},

with $E_{n + 1}$ concentrated at a smaller spatial scale. Again and again, and again. The demon does not violate conservation of energy. That would be too easy. He follows the rules. He is just extremely annoying about how he follows them. And mathematics has to prove that no such conspiracy is possible.

The small scales are the problem

It is useful here to divide the momentum equation by density and define the kinematic viscosity

ν = \frac{μ}{ρ} .

Then, ignoring external forcing,

\frac{\partial 𝐮}{\partial t} + (𝐮 \cdot \nabla) 𝐮 = - \frac{1}{ρ} \nabla p + ν \nabla^{2} 𝐮 .

There is a competition buried inside this equation.

Nonlinear transport:

(𝐮 \cdot \nabla) 𝐮

versus viscous dissipation:

ν \nabla^{2} 𝐮 .

At a characteristic velocity $U$ and length scale $L$ , their magnitudes scale roughly as

(𝐮 \cdot \nabla) 𝐮 ~ \frac{U^{2}}{L}

and

ν \nabla^{2} 𝐮 ~ ν \frac{U}{L^{2}} .

Take their ratio:

\frac{U^{2} / L}{ν U / L^{2}} = \frac{U L}{ν} .

Which is the Reynolds number:

R e = \frac{U L}{ν}

High Reynolds number means inertia and nonlinear transport matter strongly relative to viscosity. Low Reynolds number means viscosity dominates and the fluid behaves itself.

This already gives us an engineering intuition for why a slowly crawling layer of oil and a rocket plume do not have quite the same personality.

But the Millennium problem goes deeper than merely asking whether a flow is turbulent.

The basic energy estimate for three-dimensional Navier-Stokes is supercritical relative to the natural scaling of the equation. Roughly, the control supplied by total energy becomes too weak to automatically control what might happen at increasingly fine scales.

This is why

E (t) \leq E (0)

is enormously useful and still not enough. You know the total budget. You don't know whether the dynamics can arrange that budget into something pathological.

Two dimensions behave themselves

There is another clue that something genuinely structural is happening. In two dimensions, global regularity is known. The 2D incompressible Navier-Stokes equations do not have the same unresolved blowup problem.

One useful way of seeing the difference is through vorticity:

ω = \nabla \times 𝐮 .

In three dimensions, the vorticity equation is

\frac{\partial ω}{\partial t} + (𝐮 \cdot \nabla) ω = (ω \cdot \nabla) 𝐮 + ν \nabla^{2} ω .

Look at this term:

(ω \cdot \nabla) 𝐮 .

Vortex stretching.

A vortex tube can be stretched, which can intensify its vorticity.

Very schematically, if a vortex tube is stretched while its circulation is maintained, its cross-sectional area can shrink and its rotation can intensify.

Three-dimensional fluids therefore have a mechanism for dynamically amplifying vorticity that has no direct analogue in the same form in two-dimensional incompressible flow.

In 2D, the vorticity equation simplifies to

\frac{\partial ω}{\partial t} + (𝐮 \cdot \nabla) ω = ν \nabla^{2} ω .

No vortex-stretching term.

This does not by itself solve the entire conceptual problem, but it gives some intuition for why adding one spatial dimension turns a solved regularity problem into a Millennium Prize problem.

Apparently water gains one axis and immediately starts causing problems for mathematicians.

So Tao changed the laws of physics

This is where the story becomes properly strange. People had repeatedly tried to prove global regularity using conservation of energy, viscosity and other broad structural properties of Navier-Stokes. And repeatedly failed.

Terence Tao wanted to understand whether these methods were failing because nobody had found the clever enough estimate yet, or because those properties were fundamentally insufficient.

He couldn't make actual Navier-Stokes blow up. Nobody can. That is the problem. So he did something mathematicians are allowed to do and aerospace engineers generally are not.

He changed the laws of physics.

Tao constructed an averaged three-dimensional Navier-Stokes equation. It retained important features of the original equation, including an analogue of its energy structure, while modifying the nonlinear interactions.

The idea was roughly this:

Suppose energy at one scale has several possible channels through which it can interact with smaller scales. Instead of allowing all of them, turn some off.

Engineer the interactions so that energy preferentially follows a particular path:

ℓ_{0} \to ℓ_{1} \to ℓ_{2} \to ℓ_{3} \to \dots

Now try to build the demon deliberately. And in the modified equation, he got finite-time blowup. This did not prove that actual Navier-Stokes blows up. It did something subtler. Suppose an argument uses only properties

P_{1}, P_{2}, \dots, P_{n}

shared by both real Navier-Stokes and Tao's averaged equation.

P_{1} + \dots + P_{n}

were sufficient to guarantee regularity, Tao's equation could not blow up.

But it does.

Therefore

P_{1} + \dots + P_{n} ⇏ global regularity .

Any successful proof for real Navier-Stokes must exploit some additional structure that Tao's artificial equation does not possess.

This is a wonderful way to attack a hard problem. If you cannot find the road to the answer, start blowing up roads that cannot possibly get there.

Trying to make water explode

The naive way to engineer blowup sounds obvious. As soon as energy reaches one scale, immediately push it into the next:

E_{0} \to E_{1} \to E_{2} \to E_{3} \to \dots

Push harder.

Push faster.

Get to arbitrarily small scales before viscosity notices.

Except this creates another problem.

Suppose you begin moving $E_{1}$ into $E_{2}$ before all of $E_{0}$ has reached $E_{1}$ . Then you start moving $E_{2}$ into $E_{3}$ while both previous transfers are still occurring.

Now the energy is spread across many scales:

E = E_{0} + E_{1} + E_{2} + E_{3} + \dots .

This dispersion gives viscosity opportunities to damp everything out. Trying to do everything at once kills the blowup. So Tao needed something like a delay. An airlock. Transfer the energy into the next scale. Wait. Close the door behind it. Then open the next one.

E_{0} ⟶ E_{1} ⟶ E_{2} ⟶ E_{3} ⟶ \dots

The energy advances scale by scale while remaining localized. Which means the equation now needs something resembling state. It needs thresholds, gates, and it needs timing.

At which point our fluid mechanics problem has somehow become computer engineering.

The water computer

Tao describes constructing the modified nonlinearity almost like an electronic circuit. A resistor does one thing. A capacitor does another. Connect enough primitive components correctly and suddenly you have a clock, a latch, an AND gate, a computer.

So imagine that certain stable fluid configurations could play the role of information-bearing states.

Maybe

0 \leftrightarrow fluid configuration A

and

1 \leftrightarrow fluid configuration B .

Now imagine two such structures interacting so that the resulting configuration depends predictably on the inputs.

You could begin constructing operations such as

A \land B

A \lor B .

Enough reliable primitive operations and you can start talking about computation. A computer made entirely out of fluid. This sounds completely deranged. There is precedent.

Conway already did this

Conway's Game of Life has almost comically simple local rules.

Each cell is either

0

1 .

Look at its neighbors.

Apply a tiny ruleset.

Repeat.

Symbolically,

X_{n + 1} = F (X_{n}) .

That's basically the whole universe.

And yet inside those simple rules people discovered gliders: small configurations that propagate across the grid.

Then glider guns.

Then logic gates.

Then memory.

Then universal computation.

Then self-replicating structures.

The underlying rules never changed.

The complexity was hiding inside them the entire time.

Or, more precisely, the rules permitted that complexity.

That distinction matters. The Game of Life is discrete. Navier-Stokes is continuous. You cannot take a glider gun and throw it into a bathtub. But mathematically there is a deeper question connecting them:

how much computation can be embedded inside the evolution of a sufficiently rich dynamical system?

A differential equation says

\frac{d X}{d t} = F (X) .

Give it a state and the law tells you how the state changes.

A discrete computer does something structurally similar:

X_{n} \mapsto X_{n + 1} .

If a physical dynamical system can contain persistent states, interactions, memory and sufficiently controllable transitions between states, then computation can emerge from the dynamics themselves.

Which brings us back to water.

A fluid von Neumann machine

Take the idea one step further.

Imagine a fluid machine operating at characteristic length scale

ℓ_{0} .

Its entire purpose is to construct a smaller copy of itself:

ℓ_{1} = λ ℓ_{0}, 0 < λ < 1 .

Once the child is ready, the larger machine transfers its energy into it and shuts itself down.

The child activates.

It constructs another copy:

ℓ_{2} = λ ℓ_{1} = λ^{2} ℓ_{0} .

After $n$ generations,

ℓ_{n} = λ^{n} ℓ_{0} .

Therefore,

{lim}_{n \to \infty} ℓ_{n} = 0 .

Now suppose the characteristic operating time shrinks too:

τ_{n} = α^{n} τ_{0}, 0 < α < 1 .

The total time required for infinitely many generations is

T = \sum_{n = 0}^{\infty} τ_{n} = τ_{0} \sum_{n = 0}^{\infty} α^{n} .

Therefore,

T = \frac{τ_{0}}{1 - α}

which is finite.

The machine reproduces.

Shrinks.

Transfers its energy.

Dies.

The child wakes up.

Reproduces.

Shrinks.

Transfers its energy.

Dies.

Again.

While

n \to \infty,

we get

ℓ_{n} \to 0

but

t_{n} \to T < \infty .

An infinite hierarchy of smaller and faster fluid machines completed in finite time.

A fluid computer whose only program is:

become singular.

This is one of the most ridiculous objects I have encountered in mathematics. I love it.

Unfortunately we don't know how to build the water computer

There are several minor engineering issues.

We don't know the required fluid logic gates. We don't know whether suitable vortex structures can robustly encode states. Analog computation is horrible because errors accumulate continuously. We don't know how to make the larger fluid machine completely disappear without interfering with the smaller machine.

So, minor details.

Tao managed the relevant cascade construction for his modified averaged equation, not for actual Navier-Stokes.

That distinction is extremely important. The water computer is not a proof that real water blows up.

It is closer to a roadmap for how sufficiently structured nonlinear interactions could defeat dissipation, and an obstruction showing why broad energy-based arguments alone cannot settle the actual equation.

Real Navier-Stokes may contain some additional structural feature that makes the entire scheme impossible. But if so, we still need to identify it strongly enough to prove that it wins for every smooth initial condition.

That is the million-dollar problem.

The equation knows more than we do

This is the part I cannot stop thinking about. The equation itself is not mysterious:

\frac{\partial 𝐮}{\partial t} + (𝐮 \cdot \nabla) 𝐮 = - \frac{1}{ρ} \nabla p + ν \nabla^{2} 𝐮 .

Pressure accelerates the fluid.

Viscosity smooths the fluid.

The nonlinear term transports the fluid's velocity through itself.

Locally, everything has an explanation.

The difficulty appears when these completely understandable operations are composed with themselves continuously, everywhere in space, across every accessible length scale.

That is a different kind of ignorance. We do not lack the law but we do lack complete knowledge of what the law implies. For Navier-Stokes, the local evolution is specified:

𝐮 (𝐱, t) ⟶ 𝐮 (𝐱, t + d t) .

The Millennium problem asks whether repeated application of that perfectly explicit local rule can ever produce

smooth initial data ⟶ singularity in finite time .

The obvious reason to say no is dissipation:

\frac{d E}{d t} \leq 0 .

But

E < \infty

does not automatically give

‖ \nabla 𝐮 ‖_{\infty} < \infty .

Global control does not automatically imply local control. A finite amount of energy may still have access to infinitely many spatial scales.

And because

\sum_{n = 0}^{\infty} α^{n} < \infty for 0 < α < 1,

an infinite hierarchy of scales is not automatically protected by an infinite amount of time.

So in very compressed form, the problem becomes a competition:

nonlinear concentration vs. viscous dissipation

If dissipation necessarily wins before arbitrarily fine structure develops, smoothness survives. If nonlinear transport can organize the flow so that concentration repeatedly outruns dissipation, blowup may be possible.

Nobody knows which statement is true for general smooth 3D incompressible Navier-Stokes. There is a broader point here that I think is much more interesting than "fluids are complicated." We tend to implicitly assume that once the fundamental law of a system is known, everything else is bookkeeping.

Find the equation.

Measure the initial conditions.

Add enough compute.

Done.

But

simple local laws ⇏ simple global behavior

The rule can be simple while the space of behaviors generated by the rule is enormous.

Conway's Game of Life makes this obvious in a toy universe. A handful of local rules can contain gliders, logic gates, computers and self-replicating machines without any of those objects appearing explicitly in the rules themselves.

Navier-Stokes asks a much less comfortable version of the same philosophical question.

How much structure can hide inside

F = m a

once every piece of matter is allowed to interact with every neighboring piece, continuously, nonlinearly, across scale?

We started with Newton's second law and conservation of mass.

We got

ρ (\frac{\partial 𝐮}{\partial t} + (𝐮 \cdot \nabla) 𝐮) = - \nabla p + μ \nabla^{2} 𝐮 .

Then we asked whether the solution stays smooth.

That led to energy cascades.

Then concentration.

Then supercriticality.

Then modified laws of physics.