# The Equations of Flow

> What the Navier–Stokes equations say, how Euler, Navier, Stokes, Reynolds and Kolmogorov built our picture of flow and turbulence, and the million-dollar question about blow-up behind an unverified 2026 AI claim. With a live fluid simulation you can stir.

Interactive film: https://www.planetpast.com/stories/navier-stokes-equations/  
Publisher: PlanetPast (https://www.planetpast.com/) · Published 2026-10-12 · Math, Physics

## Prologue

### The equations of flow
*September 2026 · A claim about a 200-year-old equation*

One set of equations describes the wind, the ocean, the blood in your arteries and the air over a wing.

Everything that flows, the wind, the sea, the blood in your veins, the air over a wing, obeys one set of equations. They are called the Navier–Stokes equations, and they are almost two hundred years old. Yet mathematicians still could not prove one basic thing about them: that their solutions never break down. In September 2026 an AI company claimed to have settled that question. The claim is still being checked. This is the story of the equations behind it.

Prologue · Why the news matters

#### The equations of flow

The Navier–Stokes equations describe how fluids move: liquids, and gases such as air. Engineers use them to design aircraft and cars, doctors to study blood flow, forecasters to predict the weather. Yet a basic mathematical question about them has stayed open: in three dimensions, do their solutions always stay smooth, or can they “blow up”? In 2000 the Clay Mathematics Institute made it one of seven Millennium Prize Problems, each worth US$1 million.

**In the news:** on 8 September 2026 OpenAI announced that an internal AI system had found a solution that blows up, and released a Lean formalization. It has not been confirmed by peer review or by the Clay Institute, and its credit is disputed. Chapter VI explains what was claimed and what is still open.

- **First written**: 1822–1845 Navier, then Stokes, building on Euler’s 1757 equations

- **Millennium Prize**: US$1 million offered since 2000 by the Clay Mathematics Institute

The globe shows an illustrative pattern of winds: trade winds, westerlies and spinning storms. It is drawn to show the idea, not real weather data.

How to use: pick a chapter below the scene or press ← → to change step · ▶ plays the step · drag to rotate, scroll or pinch to zoom · in Chapters III and V you can stir a live fluid with your finger or mouse · “Play the film” runs every step full-screen.

Sources: [Wikipedia: Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations); [Clay Mathematics Institute: Navier–Stokes Equation](https://www.claymath.org/millennium/navier-stokes-equation/); [OpenAI: On the Navier–Stokes Millennium Prize Problem (8 Sep 2026, updated 10 Sep)](https://openai.com/index/navier-stokes-solution/); [Smithsonian Magazine (10 Sep 2026)](https://www.smithsonianmag.com/smart-news/ai-may-have-solved-a-longstanding-math-problem-with-a-million-dollar-prize-it-ignited-a-controversy-over-who-gets-credit-180989472/)

## Chapter I · Ideal fluids

### Newton’s law for a drop of fluid
*1757 · Berlin*

Leonhard Euler applied Newton’s laws to every tiny piece of a moving fluid.

In 1757 the Swiss mathematician Leonhard Euler published equations for a moving fluid, in the memoirs of the Berlin Academy. His idea: follow one tiny parcel of fluid and apply Newton’s second law to it. Force equals mass times acceleration. Euler’s parcel feels only pressure from the fluid around it, and outside forces like gravity. Watch it bend around the curve, pushed by higher pressure on the outside. But Euler’s fluid has no friction at all. That turned out to be a serious problem.

Chapter I · Step 1 · 1757

#### Euler’s ideal fluid

Leonhard Euler’s equations first appeared in print in 1757, in “Principes généraux du mouvement des fluides” in the *Mémoires de l’Académie des Sciences de Berlin*. They describe a fluid with no internal friction (an **inviscid** fluid). For each small parcel, its acceleration is set by the push of pressure from its neighbours and by outside forces such as gravity.

- Mass × acceleration of a parcel (yellow and orange arrows)…

- …equals the push from differences in pressure (red arrow, from high to low pressure)…

- …plus outside forces such as gravity (green arrow).

In the scene the parcel follows a curving stream. To turn, it needs a sideways push, and that push comes from the pressure being higher on the outside of the bend than the inside. Arrow lengths are illustrative.

Source: [Wikipedia: Euler equations (fluid dynamics)](https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics))

### Adding friction
*1822 and 1845 · Paris and Cambridge*

Real fluids are sticky. Navier and Stokes put that stickiness, viscosity, into Euler’s equations.

Euler’s frictionless fluid gave strange answers. In 1752 d’Alembert showed that, in this kind of ideal flow, a body moving steadily through a fluid feels no drag at all. Real fluids are sticky. Each layer drags on its neighbours. This internal friction is called viscosity. In 1822 the French engineer Claude-Louis Navier added a friction term to Euler’s equations. In 1845 George Gabriel Stokes, in Cambridge, put it on a firmer footing. Here the top plate drags the fluid along. Friction passes the motion down, layer by layer, to the fixed wall, where the fluid does not move at all.

Chapter I · Step 2 · 1822 and 1845

#### Viscosity: the friction inside a fluid

With no friction, theory said a body moving steadily through a fluid would feel no drag, which contradicts experience (d’Alembert’s paradox, 1752). Claude-Louis Navier presented a memoir with a viscous term to the French Academy of Sciences in March 1822; Siméon Denis Poisson independently reached the same results. George Gabriel Stokes published his treatment of “internal friction of fluids in motion” in 1845.

- **Navier**: 1822 French civil engineer (1785–1836)

- **Stokes**: 1845 Later Lucasian Professor at Cambridge (1819–1903)

- **Water at 20 °C**: ~1 mPa·s dynamic viscosity

- **Honey at 20 °C**: ~2,000–10,000 mPa·s thousands of times more viscous

The scene shows the classic picture of viscosity: fluid between a moving plate and a fixed one. The fluid touching each plate moves with it (the “no-slip condition”), and friction between layers sets up a straight-line velocity profile. Use the buttons on the scene to make the fluid runnier or thicker and watch how quickly the motion spreads down.

Sources: [Wikipedia: d’Alembert’s paradox](https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox); [Wikipedia: Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations); [Wikipedia: Claude-Louis Navier](https://en.wikipedia.org/wiki/Claude-Louis_Navier); [Wikipedia: George Gabriel Stokes](https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet); [Britannica: Navier–Stokes equation](https://www.britannica.com/science/Navier-Stokes-equation); [Wikipedia: Viscosity](https://en.wikipedia.org/wiki/Viscosity); [Wikipedia: No-slip condition](https://en.wikipedia.org/wiki/No-slip_condition)

## Chapter II · The equation

### One line, five ideas
*The incompressible Navier–Stokes equations*

It is Newton’s second law, written for every point in a fluid at once.

Here is the equation in its most common form, for a fluid like water that cannot be squeezed. It is Newton’s second law for a parcel of fluid. On the left, how the parcel’s velocity changes: in time, and because it moves to a place where the flow is different. That second part, the advection term, is what makes the equation so hard. On the right, the forces. Pressure pushes the parcel from high pressure to low. Viscosity smooths out differences in speed between the parcel and its neighbours. And outside forces, like gravity, act on it too. A second line says the fluid cannot be compressed: whatever flows into a region must flow out.

Chapter II · Step 3 · What the equation says

#### Newton’s second law for a fluid

The incompressible Navier–Stokes equations, for a fluid of constant density ρ and kinematic viscosity ν, with velocity **u**, pressure *p* and outside force per unit mass **f**:

∂ u ∂ t +
( u · ∇ ) u
=
− 1 ρ ∇ p +
ν ∇ 2 u +
f
∇ · u = 0

- **∂u/∂t**How fast the velocity changes with time at a fixed spot.

- **(u·∇)u**Advection, or inertia: the parcel carries its own velocity into regions where the flow is different. It multiplies velocity by velocity, so it is *nonlinear*, and that is the source of turbulence and of the mathematical difficulty.

- **−(1/ρ)∇p**Pressure pushes from high to low pressure.

- **ν∇²u**Viscosity spreads momentum out, like heat spreading, smoothing differences between neighbours. Remove it (ν = 0) and you have Euler’s equations.

- **f**Outside forces acting on the fluid, such as gravity.

- **∇·u = 0**Incompressibility: no fluid is created or destroyed, and parcels keep their volume.

Watch the arrows on the parcel in the scene: each colour matches a term in the equation. The incompressible form works well for liquids and for air moving well below the speed of sound (up to about Mach 0.3).

Sources: [Wikipedia: Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations); [Wikipedia: Reynolds number](https://en.wikipedia.org/wiki/Reynolds_number)

## Chapter III · Laminar or turbulent

### A thread of dye
*1883 · Owens College, Manchester*

Osborne Reynolds watched a streak of coloured water in a glass pipe. Then he opened the tap.

In 1883 Osborne Reynolds, an engineering professor in Manchester, ran water through a long glass pipe and fed a thin stream of dye into its centre. When the water flowed slowly, the dye stayed a neat, straight thread. This is laminar flow: smooth layers sliding past each other. When he opened the tap wider, the thread suddenly broke up, and the dye swirled through the whole pipe. Turbulent flow. Try it yourself. Use the slider on the scene to raise the Reynolds number, and stir the water with your finger or mouse.

Chapter III · Step 4 · 1883

#### Reynolds’s experiment

Reynolds introduced a small jet of dyed water into the centre of clear water flowing through a glass pipe, with a valve at the end to control the speed. At low speed the dyed layer stayed distinct along the whole tube. At higher speed it broke up at a certain point and spread across the pipe. His results were published in 1883.

#### The Reynolds number

Whether a flow is smooth or turbulent depends on one combination of quantities, now called the Reynolds number: **Re = uL/ν**, the speed *u* times a size *L*, divided by the kinematic viscosity *ν*. It compares the advection term (inertia) with the viscous term in the equation.

- **Laminar pipe flow**: Re for fully developed flow in a round pipe

- **Turbulent pipe flow**: Re > ~2,900 between the two: intermittent bursts

The scene is a live 2D simulation solving the equations on a grid as you watch, a slice through the pipe. A real pipe is 3D and turns turbulent at higher Re than this coarse model can show directly, so the slider is a guide to the real value; the simulation adds small disturbances at the inlet to stand in for the real ones.

Interactive: drag the slider, or drag across the water to stir it.

Sources: [Wikipedia: Osborne Reynolds](https://en.wikipedia.org/wiki/Osborne_Reynolds); [Wikipedia: Reynolds number](https://en.wikipedia.org/wiki/Reynolds_number)

### The vortex street
*1911–1912 · Göttingen*

Behind a post in a river, behind an island in the wind, whirlpools peel off in a neat zigzag line.

Put an obstacle in a steady flow. At a very low Reynolds number, the fluid closes smoothly behind it. Raise it above about fifty and the wake starts to wobble. Whirlpools peel off one side, then the other, in a regular rhythm. Theodore von Kármán explained this pattern in 1911 and 1912. It is called a Kármán vortex street. You can see it in clouds downwind of islands, and you can hear it: it makes wires sing in the wind. Raise the Reynolds number further and the wake turns turbulent.

Chapter III · Step 5 · 1911–1912

#### Vortex shedding and the Kármán vortex street

Flow past a blunt body separates and, within a range of Reynolds numbers, sheds vortices alternately from each side. For a circular cylinder, shedding starts at about Re ≈ 47. Theodore von Kármán analysed the pattern in papers of 1911 and 1912; he acknowledged earlier observations by Arnulph Mallock and Henri Bénard.

- **Onset (cylinder)**: Re ≈ 47 Re measured with the cylinder’s diameter

- **In clouds**: > 400 km streets behind islands can be this long, seen from satellites

- **Why it matters**: The alternating vortices push the body from side to side If the rhythm matches a structure’s natural frequency it can vibrate strongly: singing power lines, car antennas, tall chimneys

Live 2D simulation. The probe behind the cylinder measures the sideways flow, and the readout counts the vortices shed. Orange dye comes off the top of the cylinder, blue off the bottom. The simulation’s numerical errors act like a little extra viscosity, so its Reynolds numbers are approximate.

Interactive: choose a Reynolds number in the panel on the scene, or drag across the water to stir it.

Sources: [Wikipedia: Kármán vortex street](https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street); [Wikipedia: Reynolds number](https://en.wikipedia.org/wiki/Reynolds_number)

### One number to sort every flow
*From bacteria to hurricanes*

The same equations hold at every scale. Only the Reynolds number changes.

The Reynolds number explains why a bacterium and a blue whale live in different fluid worlds. For a swimming bacterium it is about one ten-thousandth. Viscosity rules. Water feels as thick as honey, and the moment it stops pushing, it stops. For a person swimming it is in the millions, for a large ship in the billions, for a tropical cyclone about a trillion. Inertia rules, and flows are turbulent. George Stokes introduced the idea in 1851. Arnold Sommerfeld named it after Reynolds in 1908.

Chapter III · Step 6 · A scale for every flow

#### The Reynolds number, from bacteria to cyclones

Re is the ratio of inertial forces to viscous forces. When it is small, viscosity smooths everything and flows are laminar; when it is large, inertia wins and flows tend to become turbulent. The concept was introduced by George Stokes in 1851; Arnold Sommerfeld named it after Osborne Reynolds in 1908.

#### Typical values (approximate)

- ~10⁻⁴ Bacterium

- ~1 Smallest fish

- ~10³ Blood flow in the aorta

- ~2×10⁵ A baseball pitch

- ~4×10⁶ A person swimming

- ~4×10⁸ Blue whale

- ~5×10⁹ A large ship

- ~10¹² Tropical cyclone

The axis is logarithmic: each tick is ten times the last. Blue dots are viscous, laminar flows; orange dots are turbulent ones.

Source: [Wikipedia: Reynolds number](https://en.wikipedia.org/wiki/Reynolds_number)

## Chapter IV · Whirls within whirls

### Big whirls have little whirls
*1922 and 1941*

In turbulence, energy pours from large eddies into ever smaller ones, until friction turns it into heat.

Turbulence is a tangle of eddies of every size. In 1922 the British scientist Lewis Fry Richardson summed it up in a rhyme. Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity. Energy enters at the largest scale, is passed down to smaller and smaller eddies, and only at the tiniest scales does viscosity turn it into heat. In 1941 the Soviet mathematician Andrey Kolmogorov turned this picture into a theory, predicting how energy is shared among eddies of different sizes.

Chapter IV · Step 7 · 1922 and 1941

#### The energy cascade

Lewis Fry Richardson pictured turbulence as eddies of many sizes, with large ones breaking up into smaller ones. His verse, from *Weather Prediction by Numerical Process* (1922): “Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity.”

In 1941 Andrey Kolmogorov proposed that, at very high Reynolds numbers, small eddies look statistically the same in every direction and depend only on the rate at which energy is dissipated and on viscosity. This predicts that energy is spread across eddy sizes following a **−5/3 power law** (stated independently by Alexander Obukhov the same year), which experiments broadly confirm.

- **Spectrum**: E(k) ∝ k −5/3 k: wavenumber, 1 ÷ eddy size

- **Smallest eddies**: η = (ν³/ε) ¼ Kolmogorov scale; around a millimetre in wind round a tall building

In the scene, colours mark generations of eddies. Following Kolmogorov’s scaling, smaller eddies turn over faster. The glowing red specks at the end stand for energy turned into heat. Richard Feynman called turbulence the most important unsolved problem in classical physics.

Sources: [Wikipedia: Lewis Fry Richardson](https://en.wikipedia.org/wiki/Lewis_Fry_Richardson); [Wikipedia: Energy cascade](https://en.wikipedia.org/wiki/Energy_cascade); [Wikipedia: Kolmogorov microscales](https://en.wikipedia.org/wiki/Kolmogorov_microscales); [Wikipedia: Turbulence](https://en.wikipedia.org/wiki/Turbulence)

## Chapter V · Solving by computer

### A forecast factory
*1922 · Weather Prediction by Numerical Process*

Richardson imagined a hall of human “computers”, each working out the weather for one patch of the globe.

Richardson also wanted to forecast the weather by solving the equations of fluid motion. He divided the atmosphere into a grid of boxes. Working by hand, he spent at least six weeks calculating a six-hour forecast for two points over central Europe. The result was badly wrong, but the method was right. He dreamed of a vast hall of people computing the weather together, like an orchestra. In 1950 the electronic computer ENIAC made the first computer weather forecasts. Today’s forecasts still solve fluid equations on grids, now with boxes a few kilometres wide.

Chapter V · Step 8 · 1922 to today

#### Forecasting the weather

Weather forecasting treats the atmosphere as a fluid: measure its state now, then use the equations of fluid dynamics and thermodynamics to step it forward in time. Lewis Fry Richardson’s hand calculation of a six-hour forecast over two points in central Europe took him at least six weeks. In *Weather Prediction by Numerical Process* (1922) he imagined a “forecast factory” of human computers.

- **First computer forecasts**: 1950 ENIAC, using a highly simplified model

- **Grid boxes today**: ~5–300 km on a side, depending on the model

- **The equations**: The “primitive equations” a version of the fluid equations adapted to a thin layer of air on a rotating planet, together with the ideal gas law

The gold grid on the globe stands for the boxes a model works with; real models are far finer. The two bright points mark roughly where Richardson computed his forecast. The full-screen still is an imaginative AI illustration of his forecast factory.

Sources: [Wikipedia: Numerical weather prediction](https://en.wikipedia.org/wiki/Numerical_weather_prediction); [Wikipedia: Lewis Fry Richardson](https://en.wikipedia.org/wiki/Lewis_Fry_Richardson)

### Cutting the fluid into boxes
*Computational fluid dynamics*

Computers cannot solve the equations exactly. They chop space into cells and step forward in time.

Nobody can write down an exact formula for most real flows. Instead, computers solve the equations approximately. Space is divided into a grid of cells. In each tiny time step, the computer moves the flow along, adds the effect of viscosity, then corrects the pressure so no fluid is created or lost. This simulation uses a method that Jos Stam published in 1999, called Stable Fluids, popular in films and games because it never explodes. The catch: turbulence has eddies down to tiny sizes. Capturing all of them takes a number of grid points that grows faster than the square of the Reynolds number.

Chapter V · Step 9 · How computers solve it

#### Computational fluid dynamics

Computational fluid dynamics (CFD) solves the Navier–Stokes equations numerically. It is used to design aircraft and cars, to simulate weather, in biomedical engineering and in visual effects. The live simulation in these steps follows Jos Stam’s “Stable Fluids” method (SIGGRAPH 1999), run on a coarse 2D grid in your browser.

- **Advect:** carry velocity and dye along with the flow (the (u·∇)u term).

- **Diffuse:** blend each cell with its neighbours (the ν∇²u term).

- **Project:** solve for the pressure that removes any compression, so ∇·u = 0 (the ∇p term).

- **The cost of capturing every eddy**: N³ ≥ Re 9/4 grid points needed for a full 3D simulation (“DNS”); that is why most engineering CFD uses turbulence models such as RANS or LES instead

The faint lines over the tank are the actual cells this simulation solves on. Drag across the water to stir.

Sources: [Wikipedia: Computational fluid dynamics](https://en.wikipedia.org/wiki/Computational_fluid_dynamics); [Wikipedia: Direct numerical simulation](https://en.wikipedia.org/wiki/Direct_numerical_simulation); [ACM SIGGRAPH History Archive: “Stable fluids”, Stam (1999)](https://history.siggraph.org/?p=116411)

## Chapter VI · The million-dollar question

### A million-dollar question
*24 May 2000 · Collège de France, Paris*

Does every smooth flow stay smooth for ever, or can the equations break?

In May 2000 the Clay Mathematics Institute announced seven Millennium Prize Problems, with a million dollars for each solution. One is about the Navier–Stokes equations. Start with a smooth, gentle flow in three-dimensional space, with finite energy. Will the solution stay smooth for ever? Or can the speed somewhere become infinite in a finite time, a blow-up? In two dimensions it was proved in the 1960s that the answer is: smooth for ever. In three dimensions, nobody knew. The reason is a purely three-dimensional effect: vortex stretching.

Chapter VI · Step 10 · 2000

#### The Millennium Prize problem

The seven Millennium Prize Problems were announced in Paris on 24 May 2000. The Navier–Stokes problem, in the official statement by Charles Fefferman, can be solved by proving any one of four statements: (A) smooth solutions always exist in ordinary 3D space, (B) the same in a periodic box, (C) there is a smooth starting flow and a smooth outside force for which the solution breaks down, or (D) the same breakdown in a periodic box.

- **Two dimensions**: Solved smooth solutions exist for ever (proved in the 1960s)

- **Three dimensions**: Open since 2000 for the prize problem

- **Leray (1934)**: Small flows stay smooth and “weak” solutions always exist

- **Tao (2016)**: An averaged version blows up hinting at a possible route for the real equations

Only one Millennium problem, the Poincaré conjecture, has been officially declared solved. Grigori Perelman declined the prize in 2010.

Sources: [Clay Mathematics Institute: Navier–Stokes Equation](https://www.claymath.org/millennium/navier-stokes-equation/); [Wikipedia: Navier–Stokes existence and smoothness](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness); [Wikipedia: Millennium Prize Problems](https://en.wikipedia.org/wiki/Millennium_Prize_Problems)

### A spinning tube that is pulled
*Why three dimensions are hard*

Stretch a vortex and it spins faster, like a figure skater pulling in their arms.

A vortex is a tube of spinning fluid. In three dimensions the flow can pull its two ends apart. Because the fluid cannot be compressed, the tube must get thinner as it gets longer. And as it narrows, it spins faster, just as a skater spins faster by pulling in their arms. Faster spin can stretch the tube further, which makes it spin faster still. Could this feed on itself until the speed becomes infinite? That is the blow-up question. In two dimensions there is no direction to stretch in, so this cannot happen. That is why the 2D problem was solved long ago.

Chapter VI · Step 11 · An intuition for blow-up

#### Vortex stretching

Vortex stretching is the lengthening of vortices in three-dimensional flow, which increases the spin along the stretching direction because angular momentum is conserved. In incompressible flow a stretched tube must also get thinner. It drives the turbulent energy cascade toward small scales, and it is absent in two-dimensional flow.

Viscosity fights back by smoothing out sharp spins. The open question was which wins in the end: can stretching outrun viscosity and make the velocity infinite in finite time?

**Illustrative only.** This animation shows the idea of stretching, not the specific solution claimed in 2026. Use the slider on the scene to stretch the tube yourself and watch its spin rate in the readout.

Source: [Wikipedia: Vortex stretching](https://en.wikipedia.org/wiki/Vortex_stretching)

### A blow-up, announced
*8 September 2026 · An unverified claim*

An AI company says it has found a flow that breaks the equations. Mathematicians are still checking.

On the eighth of September 2026, OpenAI announced that an internal AI system, running about ten thousand agents for eighty-eight hours, had found a solution that blows up. It describes a vortex that spirals inward and gets more and more elongated, like spaghetti, until its speed becomes infinite in a finite time, while the energy stays finite. The construction uses a smooth outside force. The proof was checked by the Lean proof assistant. Lean confirms the logic, but experts must still check that the formal statement matches the Clay problem. It has not been peer reviewed, and OpenAI says it will not claim the prize. Credit is disputed. Tristan Buckmaster and Levent Alpöge had announced closely related results a day earlier, building on methods by Diego Córdoba and Luis Martínez-Zoroa. OpenAI says its work was independent. Some mathematicians also say the version without an outside force, the one many consider the heart of the problem, remains open.

Chapter VI · Step 12 · September 2026

#### The 2026 claim, and what is still open

**Status as of 12 October 2026:** an unverified claim. Not yet peer reviewed. The Clay Mathematics Institute still lists the problem as “Active” and has not awarded a prize; its rules require publication in a qualifying outlet, at least two years, and general acceptance by mathematicians before it considers a solution.

#### What was claimed

On 8 September 2026 OpenAI said an internal model, run as about 10,000 coordinated agents for 88 hours, produced a proof that a smooth, initially resting fluid with a smooth outside force can develop a singularity in finite time with finite energy: statement (C) of the Clay problem. It describes “a vortex… that spirals inward and gets increasingly elongated, like spaghetti.” A Lean formalization was released. OpenAI says it does not intend to claim the prize.

#### How it is being received

- Clay On 11 Sep the institute said it shares the excitement “as we contemplate the announcement that the Navier–Stokes problem has apparently been settled”, and that its process is “deliberately unhurried”.

- Checking Lean checks that each step follows; whether the formal statement captures the Clay problem is human work still under way.

- Scope A 17 Sep preprint by Constantin, Ignatova and Vicol argues the method cannot work without an outside force. Some experts say the force-free problem, which many regard as the central one, remains open.

#### The dispute over credit

Tristan Buckmaster (NYU) and Levent Alpöge (who works at Anthropic) posted related blow-up results, including for the forced 3D Euler equations, the day before. Both teams built on an approach developed by Diego Córdoba and Luis Martínez-Zoroa. Buckmaster, who used AI models including OpenAI’s Codex, raised questions about whether OpenAI’s system had any access to their work and about how a joint release was discussed; he said he was not accusing anyone. OpenAI acknowledged their priority on forced Euler, said its proofs differ, and after an investigation said his Codex prompts could not have influenced its system. These accounts have not been independently resolved.

The animation is illustrative: a swirl tightening toward a point, not a plot of the claimed solution.

Sources: [OpenAI: On the Navier–Stokes Millennium Prize Problem (8 Sep 2026, updated 10 Sep)](https://openai.com/index/navier-stokes-solution/); [Clay Mathematics Institute: Navier–Stokes Announcement (11 Sep 2026)](https://www.claymath.org/news/navier-stokes-announcement/); [Clay Mathematics Institute: Navier–Stokes Equation](https://www.claymath.org/millennium/navier-stokes-equation/); [Clay Mathematics Institute: Rules for the Millennium Prize Problems](https://www.claymath.org/millennium-problems/rules/); [Smithsonian Magazine (10 Sep 2026)](https://www.smithsonianmag.com/smart-news/ai-may-have-solved-a-longstanding-math-problem-with-a-million-dollar-prize-it-ignited-a-controversy-over-who-gets-credit-180989472/); [NYU Shanghai RITS (9 Sep 2026)](https://rits.shanghai.nyu.edu/ai/openai-navier-stokes-proof-credit-dispute/); [Quanta Magazine (8 Sep 2026)](https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/); [Science News (9 Sep 2026)](https://www.sciencenews.org/article/ai-math-puzzle-navier-stokes-openai); [Scientific American: “Did OpenAI solve the wrong Navier–Stokes problem?” (21 Sep 2026)](https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/); [Constantin, Ignatova & Vicol, arXiv:2609.20803 (17 Sep 2026)](https://arxiv.org/abs/2609.20803)

## Epilogue

### Still flowing
*1757 → 2026*

Whatever the verdict on the proof, the equations keep working every day, in every forecast and every wing.

Whether or not the 2026 proof stands, the Navier–Stokes equations do their work every day: in weather forecasts, aircraft design, medicine and climate science. Turbulence, the tangle of whirls within whirls, is still not fully understood. And even a confirmed blow-up would not mean real water does something infinite. Real fluids are made of molecules, so the smooth equations are only a model at the smallest scales. This is a live fluid. Stir it, and watch a little of the mystery for yourself.

Epilogue · 1757 → 2026

#### Two and a half centuries of flow

- 1757 Euler’s equations for an ideal, frictionless fluid

- 1822 Navier adds viscosity

- 1845 Stokes’s paper on internal friction of fluids

- 1883 Reynolds’s dye experiment: laminar and turbulent flow

- 1911–12 Von Kármán explains the vortex street

- 1922 Richardson: the energy cascade and numerical weather prediction

- 1941 Kolmogorov’s theory of turbulence

- 1950 First computer weather forecast, on ENIAC

- 2000 Navier–Stokes becomes a Millennium Prize Problem

- 2026 OpenAI claims a blow-up solution; verification and credit still debated

Mathematician Gregory Eyink told *Science News* that a blow-up would have little practical effect, since real fluids are made of molecules and the continuum equations are already known to break down at very small scales. Its importance is for mathematics. All full-screen stills are AI-generated to help the imagination and are not photographs. The globe uses NASA Blue Marble imagery; its wind pattern is illustrative.

Interactive: drag across the tank to stir. The buttons on the scene clear it or change the viscosity.

#### Sources

[Wikipedia: Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations)
[Britannica: Navier–Stokes equation](https://www.britannica.com/science/Navier-Stokes-equation)
[Wikipedia: Euler equations (fluid dynamics)](https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics))
[Wikipedia: d’Alembert’s paradox](https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox)
[Wikipedia: Claude-Louis Navier](https://en.wikipedia.org/wiki/Claude-Louis_Navier)
[Wikipedia: George Gabriel Stokes](https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet)
[Wikipedia: Viscosity](https://en.wikipedia.org/wiki/Viscosity)
[Wikipedia: No-slip condition](https://en.wikipedia.org/wiki/No-slip_condition)
[Wikipedia: Reynolds number](https://en.wikipedia.org/wiki/Reynolds_number)
[Wikipedia: Osborne Reynolds](https://en.wikipedia.org/wiki/Osborne_Reynolds)
[Wikipedia: Kármán vortex street](https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street)
[Wikipedia: Turbulence](https://en.wikipedia.org/wiki/Turbulence)
[Wikipedia: Energy cascade](https://en.wikipedia.org/wiki/Energy_cascade)
[Wikipedia: Kolmogorov microscales](https://en.wikipedia.org/wiki/Kolmogorov_microscales)
[Wikipedia: Lewis Fry Richardson](https://en.wikipedia.org/wiki/Lewis_Fry_Richardson)
[Wikipedia: Numerical weather prediction](https://en.wikipedia.org/wiki/Numerical_weather_prediction)
[Wikipedia: Computational fluid dynamics](https://en.wikipedia.org/wiki/Computational_fluid_dynamics)
[Wikipedia: Direct numerical simulation](https://en.wikipedia.org/wiki/Direct_numerical_simulation)
[ACM SIGGRAPH History Archive: “Stable fluids”, Stam (1999)](https://history.siggraph.org/?p=116411)
[Wikipedia: Vortex stretching](https://en.wikipedia.org/wiki/Vortex_stretching)
[Clay Mathematics Institute: Navier–Stokes Equation](https://www.claymath.org/millennium/navier-stokes-equation/)
[Clay Mathematics Institute: Navier–Stokes Announcement (11 Sep 2026)](https://www.claymath.org/news/navier-stokes-announcement/)
[Clay Mathematics Institute: Rules for the Millennium Prize Problems](https://www.claymath.org/millennium-problems/rules/)
[Wikipedia: Millennium Prize Problems](https://en.wikipedia.org/wiki/Millennium_Prize_Problems)
[Wikipedia: Navier–Stokes existence and smoothness](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness)
[OpenAI: On the Navier–Stokes Millennium Prize Problem (8 Sep 2026, updated 10 Sep)](https://openai.com/index/navier-stokes-solution/)
[Smithsonian Magazine (10 Sep 2026)](https://www.smithsonianmag.com/smart-news/ai-may-have-solved-a-longstanding-math-problem-with-a-million-dollar-prize-it-ignited-a-controversy-over-who-gets-credit-180989472/)
[NYU Shanghai RITS (9 Sep 2026)](https://rits.shanghai.nyu.edu/ai/openai-navier-stokes-proof-credit-dispute/)
[Quanta Magazine (8 Sep 2026)](https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/)
[Science News (9 Sep 2026)](https://www.sciencenews.org/article/ai-math-puzzle-navier-stokes-openai)
[Scientific American: “Did OpenAI solve the wrong Navier–Stokes problem?” (21 Sep 2026)](https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/)
[Constantin, Ignatova & Vicol, arXiv:2609.20803 (17 Sep 2026)](https://arxiv.org/abs/2609.20803)