Stir a bowl of water. The spoon pushes some water forward, water elsewhere starts moving, and a swirl forms. Once you lift the spoon out, the water keeps turning for a while.
The Navier-Stokes equations describe motion like this. They connect the forces acting on a fluid with how its motion changes. A fluid includes liquids such as water and gases such as air.
The Clay Mathematics Institute set aside $1 million for a solution to a question about these equations: if a three-dimensional flow starts smoothly, can its mathematical description break down later? On 8 September 2026, OpenAI announced that its AI system had produced a proof of such a breakdown.
An arrow map gives us a way to describe that moving water.
Give every spot in the water an arrow
Imagine looking down into the bowl and choosing one spot. Draw an arrow showing which way the water is moving there. Make the arrow longer when the water moves faster. Do the same at other spots.
You now have a map of the water's motion at one moment. Mathematicians call it a velocity field. Velocity means speed together with direction; the field gives a velocity at each position. The picture shows a few arrows so we can read it. The mathematical model assigns one at every point.
The equations treat water as a continuous material. You can keep looking at smaller regions without having to draw individual molecules. That is an idealisation of real water, which is made of molecules.
In school, solving 2x + 1 = 7 gives the number x = 3. For these fluid equations, a solution describes the whole velocity field and the pressure as time passes. Imagine an entire movie of arrow maps. Knowing a solution would tell you what belongs in every frame.
The familiar rule underneath: F = ma
Newton's rule says total force = mass × acceleration. Acceleration means a change in velocity over time. A force can make something speed up, slow down or turn.
Apply that rule to a small portion of water. Water around it exerts pressure on its sides. If the pushes are unequal, they produce a net force. In the picture below, the Pressure panel shows a stronger push to the right than the opposing push to the left.
Neighbouring layers also drag on each other. A faster layer tends to pull a slower one along, while the slower layer holds the faster one back. This internal friction is called viscosity. The red arrows in the Viscosity panel show those opposing friction forces.
External forces can act too, such as gravity. Navier-Stokes puts the force balance into a form that describes a fluid throughout space. Because the water itself moves, it carries its own motion along with it. A fast patch of water arrives somewhere new, and the arrows there change accordingly. The arrow map you have now shapes the map a moment later.
The difficulty is that each region's behaviour is part of the flow you are trying to work out. Even with simple starting conditions, you are solving for many interacting parts of the same moving fluid.
“Smooth” has a precise meaning here
Look at a graph of speed along a line through the water. In the upper sketch, the curve rounds a bend. Its slope changes gradually. In the lower sketch, two different slopes meet at a sharp corner. There is no single slope at that corner.
Slope is a rate of change. The full mathematical requirement for smoothness goes further than the upper picture alone can show: rates of change must exist continuously, and the graph of the slope itself must have a continuously varying slope, and so on repeatedly. Both the velocity and pressure must meet that requirement.
A whirlpool can satisfy this condition. Its arrows turn as you move around the swirl, yet neighbouring arrows can still fit into a smooth pattern. Smoothness describes the mathematical fields; it does not require a motionless bowl or a flat water surface.
Existence, the other word in the prize's name, asks whether there is a flow satisfying the equations and the given starting conditions. Together, “existence and smoothness” ask whether the required smooth description is available for all future times.
What would a breakdown look like?
One possibility is that the speeds predicted by the equations grow beyond every finite limit as a particular moment approaches. That is a velocity blowup, a kind of mathematical singularity: the smooth description cannot continue through that moment. We can picture the idea with a much simpler formula:
value = 1 ÷ (1 − t)
Here, t is a number representing time. Start at 0 and move towards 1. At t = 0.9, the number you divide by, 1 − t, is 0.1, so the value is 10. Get ten times closer to 1, and the value becomes ten times larger.
| Time, t | 1 ÷ (1 − t) |
|---|---|
| 0 | 1 |
| 0.9 | 10 |
| 0.99 | 100 |
| 0.999 | 1,000 |
Name any finite ceiling for the value. You can exceed it by choosing t close enough to 1 from below. The value grows without limit even though the time approaches the ordinary, finite number 1. At exactly 1, the formula divides by zero and is undefined.
This formula illustrates blowup; it is not a solution to the fluid equations. A Navier-Stokes velocity blowup would mean the smooth description fails at a finite time. Actual water cannot reach infinite speed. The result would tell us about a limit of the mathematical model.
Why doesn't friction prevent this?
Viscosity tends to smooth out differences in motion. The difficult question is whether the flow can concentrate into smaller regions quickly enough to defeat that smoothing.
A limit on the fluid's total energy might seem to settle the issue. To see the gap, compare two different amounts of water. The energy of motion, called kinetic energy, is one half of the mass multiplied by the speed squared. Squaring a speed means multiplying it by itself.
Four kilograms moving at 1 metre per second have 2 joules of kinetic energy. One kilogram moving at 2 metres per second also has 2 joules. A joule is a unit of energy. The second amount has a quarter of the mass and twice the speed, so the factors cancel: one quarter multiplied by two squared equals one.
This comparison shows why a limit on the total energy alone cannot put a ceiling on the speed at any one spot. Increasingly fast motion could occupy increasingly tiny regions while the total stays limited. The hard part is showing whether a flow that obeys Navier-Stokes can actually develop that concentration. These arithmetic comparisons do not construct such a flow.
The prize has rules for the fluid, too
The bowl helps us picture motion, but the prize uses settings with no enclosing walls. One is a fluid filling all of three-dimensional space. Another is a repeating box: leaving one side brings you back through the opposite side.
The fluid is incompressible. Follow a particular portion as it moves: it keeps the same volume, although its shape can change. So the earlier comparison used two separate amounts of water. A portion of this fluid cannot be squeezed into a smaller volume to go faster.
The official problem description gives several ways to solve the prize problem. One route is to prove that every permitted smooth starting flow keeps a smooth solution forever when there is no external force. Another is to produce a permitted example where the required smooth solution breaks down. That route allows an external force, subject to smoothness and other technical conditions. OpenAI says its proof follows the second route.
A simulation and a proof answer different questions
Engineers already use computer approximations of the equations to study airflow. NASA's introduction describes this work, called computational fluid dynamics. A simulation estimates the flow using finitely many quantities and time steps.
Picture checking a road with a ruler at selected spots. The measurements tell you about those spots; a sufficiently narrow bump could sit between them. Likewise, a flow simulation needs more than a dramatic-looking spike to establish that an exact mathematical quantity grows without limit. A proof must justify its conclusion for the whole statement, including what happens between numerical samples. Computers can help build and check that reasoning.
What OpenAI says it found
OpenAI's paper describes a fluid that starts at rest, receives a smooth external force and develops unbounded speed while its total kinetic energy stays within a finite limit. The company says this meets the prize's breakdown conditions. In the announcement, OpenAI says the group that found the result used “on the order of 10,000 concurrent agents”, meaning agents running at once.
OpenAI also released a version of the proof written in Lean, software for checking mathematical proofs. Lean checks that the proof follows from the assumptions written with the statement. Mathematicians still need to examine whether these match the intended problem.
Clay's problem page still lists Navier-Stokes as unsolved. Its prize rules require a proposed solution to appear in a qualifying outlet and gain general acceptance among mathematicians. At least two years must pass after publication before Clay will consider it. OpenAI says it does not intend to claim the prize.