Navier–Stokes existence and smoothness

Q: Has the Navier–Stokes problem been solved?

Claimed, not awarded. On 8 September 2026 OpenAI claimed an AI-found proof that 3D Navier–Stokes with a smooth outside force can blow up in finite time. If it holds, it would technically answer Clay's question as written, because Fefferman's options (C) and (D) allow such a force. The proof has not been peer reviewed, and nobody yet fully understands all 166 pages. Clay's statement of 11 September says the problem "has apparently been settled", but it names no one, has awarded nothing, and describes its review as deliberately unhurried. Reports say Clay's rules require journal publication and a waiting period first. OpenAI has said it will not claim the prize. On 6 October a preprint by Bastounis, Circelli and Hansen showed that OpenAI's Lean code does not faithfully match its written proof. No error in the proof itself has been confirmed. Related AI-assisted, Lean-checked work by Alpöge and Buckmaster claims forced blow-up for model equations, not Navier–Stokes, and credit is disputed. The physical question, whether an unforced fluid can blow up (or always stays smooth), is still open. Many experts regard the force as a loophole.

Does a smooth, calm flow of fluid always stay smooth, or can it tie itself into an infinitely violent knot in a finite time?

$1,000,000 · claimed solution under review 18 min read, with things to play with

Stir your tea. Pour a glass of water. Watch smoke curl off a candle. Every one of those motions follows the same short set of rules, the Navier–Stokes equations, written down about two hundred years ago. Engineers feed them to computers every day to design planes, cars and weather forecasts.

Here is the strange part. Nobody has ever proved that those rules always make sense in three dimensions. Start with a perfectly smooth, gentle flow. Could the fluid twist itself so hard that, after a finite time, the speed at some point becomes infinite? In 2000 the Clay Mathematics Institute offered one million dollars to anyone who could settle it either way.

In September 2026 that question suddenly hit the news. OpenAI claims an AI proved it. Mathematicians are still checking. The claim is real and serious, but it answers a version of the question with an outside push on the fluid, and the version most physicists care about is still open. This page explains what the question is, why it is so hard, and exactly what has and has not been shown.

1 · Play first

Stir a real fluid

Below is a genuine fluid simulation, running the same four ingredients as the Navier–Stokes equations: the fluid carries itself along, it rubs against itself (viscosity), pressure keeps it from being squashed, and your finger pushes it (an outside force). Drag on the square to stir and drop dye.

The slider sets the viscosity, the fluid's internal friction. Honey has a lot, water a little. The two numbers underneath are live: the strongest spin anywhere in the box, and the total energy of motion. This toy is 2D, where mathematicians already know the flow stays smooth forever. The million-dollar question is about 3D, where whirlpools can also be stretched, and that is the part no flat screen can show.

2D toy. In 2D, smoothness is already proven. The $1M question is about 3D.

Strongest spin0max |vorticity|, turns of the flow per second

Kinetic energy0.0×10⁻³, in box units

Energy over the last 20 s. Stir, then watch how fast it drains: honey in a second, water hardly at all.

Viscosity (internal friction): olive oilhoneywateralmost zero

Try it: stir with the slider on honey, then slide to almost zero and stir again. Switch to “Show spin” to see the swirls.

Simplified: an 80×80 grid, with walls around the box. Water is already so thin at this size that it behaves almost like “almost zero”. “Almost zero” sets viscosity to 0, but the grid still blurs the tiniest swirls, so a computer can never show a true blow-up.

What to notice

On honey, swirls smear out almost at once and the energy number drains away fast. Friction wins.
On almost zero, the swirls keep curling into thinner and thinner threads, and the spin number stays high. Small scales are where the blow-up question lives.
When you stop stirring, the energy only ever goes down. With no outside push, a fluid with friction cannot create energy from nothing.
Even with viscosity set to zero, the finest threads blur once they are smaller than a grid square. A computer always has a smallest scale, so it cannot see infinity.

2 · The equation, piece by piece

This is the equation Clay asks about. It is Newton's second law, force equals mass times acceleration, written for every tiny parcel of fluid at once. The bold $\mathbf{u}$ is the velocity field: at every point $x$ and time $t$ it gives the fluid's speed and direction. Tap any piece to see what it does.

\htmlClass{t0}{\frac{\partial \mathbf{u}}{\partial t}} + \htmlClass{t1}{(\mathbf{u}\cdot\nabla)\mathbf{u}} = \htmlClass{t2}{-\nabla p} + \htmlClass{t3}{\nu\,\Delta\mathbf{u}} + \htmlClass{t4}{\mathbf{f}}, \qquad \htmlClass{t5}{\nabla\cdot\mathbf{u} = 0}

\frac{\partial \mathbf{u}}{\partial t}

Change at one spot

How fast the velocity changes if you stand still at one fixed point and watch the fluid go past.

Clay adds three conditions. The starting flow $\mathbf{u}_0$ must be smooth (mathematicians write $C^\infty$ : every derivative exists everywhere, so no corners, jumps or infinities). It must fade away quickly far from the origin, or repeat in a box. And the total energy must stay finite. A blow-up at time $T$ means some quantity, like the top speed or the strongest spin, shoots to infinity as $t$ approaches $T$ .

3 · What's going on

1Four ways to win a million dollars

Clay's official description was written by the mathematician Charles Fefferman in 2000. Take the incompressible Navier–Stokes equations in 3D, with viscosity $\nu > 0$ and a smooth starting flow with finite energy. Then prove one of four statements.

(A) In all of space, with zero force, every such flow stays smooth with finite energy for all time. (B) The same, in a box that repeats in every direction (a periodic box), again with zero force. (C) In all of space, there is some smooth starting flow and some smooth, quickly fading force for which no smooth solution exists for all time. (D) The same breakdown in the periodic box, again with a smooth force allowed.

So you can win by proving that fluids always behave (A or B), or by building one example that misbehaves (C or D). The fine print is in the force: it must be zero for the "always smooth" answers, but it may be nonzero for the "breaks down" answers. Keep that in mind. It is the whole story of 2026.

2Smooth versus blow-up

Think of the flow as a landscape of speeds that changes in time. "Smooth" means the landscape never has a cliff, a spike or a tear, no matter how closely you zoom in. Every slope, and every slope of a slope, is a finite number.

A singularity, or blow-up, is a moment when that fails. Picture a whirlpool that shrinks and spins faster. If it halved in size and doubled its spin again and again, with each halving taking half as long as the one before, the total time would be finite, yet the spin would be infinite at the end. That is the nightmare scenario.

A simple cartoon shows how this can happen. Suppose some amount $\omega$ grows at a rate equal to its own square, $d\omega/dt = \omega^2$ . The solution is below. Starting from $\omega_0 = 2$ , it reaches infinity at $t = 0.5$ : not "eventually", but at a definite time. Navier–Stokes is far more complicated, but its nonlinear term has this same "growth feeds growth" flavour.

\omega(t) = \frac{\omega_0}{1 - \omega_0 t} \;\longrightarrow\; \infty \text{ as } t \to \frac{1}{\omega_0}

A cartoon of blow-up, not Navier–Stokes itself: quadratic growth reaches infinity in finite time.

3The tug-of-war: stretching against friction

To see the fight, look at vorticity, $\boldsymbol\omega = \nabla\times\mathbf{u}$ , which measures how fast the fluid spins at each point. Taking the curl of the main equation (with no force) gives the equation below.

The last term, $\nu\Delta\boldsymbol\omega$ , is friction smoothing the spin out. The term $(\boldsymbol\omega\cdot\nabla)\mathbf{u}$ is vortex stretching. When a whirlpool is pulled longer and thinner, it spins faster, just as a figure skater spins faster when pulling in their arms. Stretching can feed on itself: faster spin means stronger stretching means faster spin.

In 2D, a whirlpool cannot be stretched along its own axis, because there is no third direction to pull it into. That stretching term is simply zero. So in 2D friction always wins, and the flow is smooth forever. That is why the toy above is safe, and why the open question is about 3D.

\frac{\partial \boldsymbol\omega}{\partial t} + (\mathbf{u}\cdot\nabla)\boldsymbol\omega = (\boldsymbol\omega\cdot\nabla)\mathbf{u} + \nu\,\Delta\boldsymbol\omega

The vorticity equation. The stretching term

(\boldsymbol\omega\cdot\nabla)\mathbf{u}

vanishes in 2D.

4What we already know

Quite a lot, but never quite enough. In 1934 Jean Leray proved that "weak" solutions exist for all time. A weak solution is allowed to be rough, so it might hide a singularity, and nobody knows whether it is unique.

Smooth solutions are known to exist for a short time from any smooth start, and forever if the starting flow is small enough. In 2D, Olga Ladyzhenskaya and others showed that everything stays smooth forever. In 1982 Caffarelli, Kohn and Nirenberg proved that if 3D singularities happen at all, the set of bad points must be very small, in a precise mathematical sense.

The one quantity we control for all time is energy. With no force it can only go down, which you can watch in the toy. The trouble is that energy is too weak a measure to stop a singularity, as the next section explains.

\int_{\mathbb{R}^3} |\mathbf{u}(x,t)|^2\,dx \;\le\; \int_{\mathbb{R}^3} |\mathbf{u}_0(x)|^2\,dx \quad (\mathbf{f}=0)

The energy bound: without a force, total kinetic energy never grows.

5Evidence, models and computers

Computers solve Navier–Stokes all the time, but a simulation cannot prove anything about infinity. Every grid has a smallest square, and below that size it sees nothing, exactly as in the toy. So researchers look for evidence and study simpler model equations.

In 2014 Guo Luo and Thomas Hou published strong numerical evidence that the Euler equations (Navier–Stokes with zero viscosity) can blow up in a ring near a wall. That same year Terence Tao proved blow-up for an "averaged" version of Navier–Stokes that keeps the same energy law. Tao's result is a warning: any proof that real fluids stay smooth must use some fine detail of the true equations that energy arguments cannot see.

In September 2025 a team of 22 researchers, including people at Google DeepMind, used machine learning to find unstable self-similar singularities numerically for related equations (an incompressible porous media model and Euler with a boundary). "Unstable" means the slightest nudge knocks the flow off course. That work is a numerical discovery, not a proof, and it is not about Navier–Stokes itself. DeepMind has not claimed to solve the problem.

6The 2026 claim, and its fine print

On 7 September 2026, Tao wrote about proofs by Levent Alpöge and Tristan Buckmaster, heavily assisted by AI and checked in the Lean proof assistant, of finite-time blow-up with a smooth force for three model equations, including 3D Euler. Tao said these "do not quite achieve" Navier–Stokes. They build on 2023 work by Diego Córdoba and Luis Martínez-Zoroa.

On 8 September 2026, OpenAI announced that an internal AI system had produced a proof that 3D Navier–Stokes with a smooth outside force can blow up in finite time. It released a 166-page paper, a separate 57-page paper on unforced Euler, and Lean code. If correct, this would technically answer Clay's options (C) and (D), because those options allow a force. It has not been peer reviewed.

On 6 October 2026, Bastounis, Circelli and Hansen posted a preprint showing that OpenAI's Lean code is not a faithful translation of the written proof. That does not show the proof is wrong. It shows that "a computer checked it" is not yet enough, because the computer checked something slightly different.

Credit is also disputed. Buckmaster has alleged that OpenAI learned of his work; OpenAI denies this. Experts quoted by Scientific American say the two methods differ in their details.

7Why the force feels like a loophole

In a forced blow-up, the force $\mathbf{f}$ is designed specially. It is smooth and fades away, so it obeys Clay's rules, but it keeps feeding energy into exactly the right place at exactly the right time, steering the flow into a singularity. Real fluids are not usually pushed by such a carefully tuned invisible hand.

So most experts separate two questions. The Clay question as written may have been answered, if the proof holds up. The physical question, whether a fluid left alone ( $\mathbf{f} = 0$ ) can blow up, is still completely open. As the Chicago mathematician Luis Silvestre told Scientific American, the Clay problem may be settled "but the main problem… is not."

4 · Why it's so hard

The core difficulty is a mismatch of scales. Energy is the only quantity we control forever, but energy is supercritical: as you zoom into smaller and smaller regions, a fixed amount of energy says less and less about what happens there, while the nonlinear term gets relatively stronger. So the known bounds are too weak to forbid a tiny, intense whirlpool.

Going the other way is hard too. Blow-up candidates tend to be unstable, so any tiny error destroys them. That makes them very hard to find on a computer, and harder still to prove exactly. The forced route sidesteps part of this, because a designer force can keep feeding the singularity.

Energy is too blunt

Total energy stays bounded, but it can concentrate into ever smaller regions. A bound on the total says almost nothing about one tiny spot.

Tao's warning

An averaged version of Navier–Stokes with the same energy law does blow up (Tao, 2014). So a smoothness proof must use structure that energy methods cannot see.

Singularities hide

Likely blow-ups are unstable: the smallest nudge sends the flow elsewhere. Finding one numerically needs extreme precision, and proving it needs more.

Computers stop at the grid

No simulation can show infinity. Below the grid size, everything is blurred, so numbers are evidence, never proof.

5 · What a solution would change

Is our fluid model complete?

The real prize is knowing whether the standard equations of fluid motion always give a sensible answer, or whether they can break down on their own.

Turbulence and trust in simulations

Understanding how energy cascades to small scales shapes how we think about turbulence, and how far we can trust the codes used to design planes and forecast weather.

Planes will still fly

A blow-up would live at idealised, infinitely small scales. Real air stops behaving like a smooth fluid at around 70 nanometres, so engineering does not change overnight.

A test case for AI in mathematics

The 2026 episode is a landmark for machine-made proofs, and it has exposed a new problem: making sure the computer-checked version really says what the written proof says.

6 · Where it stands

As of October 2026

Claimed, not awarded. On 8 September 2026 OpenAI claimed an AI-found proof that 3D Navier–Stokes with a smooth outside force can blow up in finite time. If it holds, it would technically answer Clay's question as written, because Fefferman's options (C) and (D) allow such a force. The proof has not been peer reviewed, and nobody yet fully understands all 166 pages.

Clay's statement of 11 September says the problem "has apparently been settled", but it names no one, has awarded nothing, and describes its review as deliberately unhurried. Reports say Clay's rules require journal publication and a waiting period first. OpenAI has said it will not claim the prize.

On 6 October a preprint by Bastounis, Circelli and Hansen showed that OpenAI's Lean code does not faithfully match its written proof. No error in the proof itself has been confirmed. Related AI-assisted, Lean-checked work by Alpöge and Buckmaster claims forced blow-up for model equations, not Navier–Stokes, and credit is disputed.

The physical question, whether an unforced fluid can blow up (or always stays smooth), is still open. Many experts regard the force as a loophole.

1822–1850
Claude-Louis Navier adds viscosity to Euler's fluid equations, and George Stokes refines them.
1934
Jean Leray proves weak solutions exist for all time. Their smoothness is still unknown.
1982
Caffarelli, Kohn and Nirenberg show any 3D singular set must be extremely small.
2000
The Clay Mathematics Institute names Navier–Stokes a Millennium Prize Problem, with Fefferman's official statement.
2014
Luo and Hou report numerical evidence of blow-up for 3D Euler near a wall. Tao proves blow-up for an averaged Navier–Stokes.
Sep 2023
Córdoba and Martínez-Zoroa post a blow-up result for 3D Euler with a force of limited smoothness.
Sep 2025
Wang, Gómez-Serrano, Buckmaster and others, including Google DeepMind researchers, find unstable singularities numerically with machine learning. Not a proof.
7 Sep 2026
Tao describes Alpöge and Buckmaster's AI-assisted, Lean-checked proofs of forced blow-up for model equations, which "do not quite achieve" Navier–Stokes.
8 Sep 2026
OpenAI claims an AI-found proof of finite-time blow-up for 3D Navier–Stokes with a smooth force (Clay options C and D). Not peer reviewed.
11 Sep 2026
Clay says the problem "has apparently been settled", calls its review "deliberately unhurried", names no one and awards nothing.
21 Sep 2026
Scientific American reports that the result fits Clay's wording but is widely seen as a loophole.
6 Oct 2026
Bastounis, Circelli and Hansen show the Lean code does not faithfully match the written proof.
27 Oct 2026
King's College London publicises that finding, stressing it does not show the proof is wrong.

Common mix-ups

MythOpenAI solved Navier–Stokes.

ActuallyOpenAI claims a proof for the forced case. It is unreviewed, nothing has been awarded, and the unforced question is open.

MythLean checked it, so it must be correct.

ActuallyLean checks exactly what was typed in. A 2026 preprint found the Lean version does not faithfully match the written proof.

MythIf the equations blow up, planes might fall out of the sky.

ActuallyBlow-up concerns idealised maths at vanishingly small scales, where real air is not a smooth fluid anyway.

MythWe can't solve the Navier–Stokes equations.

ActuallyWe solve them on computers every day. The open problem is a proof about every possible smooth starting flow.

MythSimulations (or DeepMind) already showed blow-up.

ActuallySimulations give evidence, not proof, because no grid can show infinity. DeepMind's 2025 work was numerical and on related equations.

7 · References

Existence and Smoothness of the Navier–Stokes Equation (official problem statement), Charles L. Fefferman, Clay Mathematics Institute, 2000
Navier–Stokes Equation (problem page), Clay Mathematics Institute, 2026
Navier-Stokes announcement, Clay Mathematics Institute, 2026
NavierStokesAndEuler (Lean repository, links to the papers), OpenAI, 2026
Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations, Terence Tao, 2026
Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs, Bastounis, Circelli and Hansen (arXiv:2610.08144), 2026
OpenAI's Navier-Stokes proof called into question, King's College London, 2026
Did OpenAI solve the wrong Navier-Stokes problem?, Joseph Howlett, Scientific American, 2026
OpenAI's blockbuster Navier-Stokes proof isn't what we thought it was, Joseph Howlett, Scientific American, 2026
Mathematicians still checking the Navier-Stokes proof that OpenAI claims to have solved, Hande Naz Kavas, The Tufts Daily, 2026
The mathematicians published machine-checkable proofs. OpenAI announced its result on a call with reporters, The Next Web, 2026
Discovery of Unstable Singularities (arXiv:2509.14185), Yongji Wang et al., 2025
Finite time blowup for an averaged three-dimensional Navier-Stokes equation (arXiv:1402.0290), Terence Tao, 2014
Potentially singular solutions of the 3D axisymmetric Euler equations, Guo Luo and Thomas Y. Hou, PNAS, 2014

Next problemP vs NP: If a computer can quickly check an answer, can it always quickly find one too?