A viscous fluid, starting from perfect rest, can be driven by a perfectly smooth force into a point of infinite velocity in finite time. The object that does it is a vortex that collapses into a needle — and the reason it works is an exponent smaller than one hundredth.
The official problem asks “for a proof of one of the following four statements.” Most coverage says “Navier–Stokes is solved” and stops. Which one was proven changes the meaning entirely.
Every smooth divergence-free start, with no force at all, stays smooth forever.
The same claim on the periodic torus, again with zero forcing.
There exist a smooth start and a smooth force for which no global smooth, bounded-energy solution exists.
The same breakdown on the periodic torus, inherited from (C) by compact support.
Proving any one of the four resolves the prize problem as posed — so (C) and (D) settle it. But they settle it in the forced direction. The question people usually mean — does a fluid left entirely to itself blow up? — is alternative (A), and it is still open. A separate companion paper does prove finite-time blowup for the unforced 3D Euler equations, where viscosity is switched off.
Fluid spirals inward toward the axis and escapes up and down along it, away from a dividing layer near z = 0. Everything below is the same flow at the same instant — only the camera changes.
Lab · tracking — the window shrinks with the core radius, so the width on screen stays constant. What you see left over is the anisotropy: the column grows longer relative to its width as τh → 0. This is the needle.
The paper fixes 0 < h < 1/100 — the green sliver. This drives every number and chart on the page. The magenta mark is h = 1/6, where the energy bound of Theorem 1.1 dies.
At the true h the needle is invisible: at τ = 10⁻⁶ the core is only about 11% taller than it is wide. This draws the aspect as τh×N instead — picture only, touching no number anywhere on this page. Set it to 1× for the honest shape. The paper's own Figure 1 does the same: “this difference is exaggerated in the schematic.”
Write τ = 1 − t. Near the singular time the whole construction is fixed by five exponents. On log–log axes each one is just a slope — and the slopes are the entire story.
All five quantities are normalized to their value at τ = 1, so they share one axis and every slope is read directly as an exponent. Drag the clock at the top of the page to move the cursor.
Theorem 1.1 demands both at once: limsup ‖u(t)‖L∞ = ∞ while sup ‖u(t)‖L² < ∞. That isn't a trick of definitions. It's arithmetic on two exponents, and you can watch them race.
Energy is speed squared times volume. The speed exponent is −1−2h; the volume exponent is +3⁄2−h. They add to +1⁄2−3h, which is positive — so the energy carried by the core actually falls to zero even as the speed inside it runs away. The fluid concentrates unboundedly fast into a region that is emptying faster still.
Push h past 1/6 with the slider above and 1⁄2−3h turns negative: the core energy would grow without bound and Theorem 1.1's own hypothesis would fail. The admissible window h < 1/100 is far tighter still. The construction lives on a razor, and that narrowness is precisely why it stayed out of reach.
A cross-section through the axis. The difficulty is not the core and not the exterior — it is the annulus that has to join them.
Self-similar, axisymmetric. Inward spiral plus axial outflow away from a dividing layer near z = 0. Radial inflow carries angular momentum inward — r·uθ is conserved for a torque-free parcel, so shrinking r spins it up. Incompressibility stops the fluid piling up: the axial jets carry it away, so the spin-up can keep going.
momentum balance holdsJoining the core to a well-behaved exterior leaves a leftover in the momentum equation. For the background flow alone that residual becomes unbounded as t ↑ 1. A force like that is not smooth, and a blowup driven by it would be coming from outside the fluid — exactly what the Clay statement forbids.
◆ the hard partBeyond the annulus the flow is purely azimuthal and independent of height, chosen to satisfy the radial heat equation for uθ exactly. Its momentum residual vanishes, so it needs no force at all — and it stays smooth right through t = 1, which is what lets the whole field be cut off to compact support.
zero residual by constructionThe fix is to stop treating the force as something applied from outside, and make the fluid supply the missing momentum itself.
Rings of small, fast, oscillating pulses are placed in the annulus on successively finer scales in space and time. Each is seeded by an exponentially small force and then fed by the background shear — it grows by extracting energy from the very flow it is correcting.
Individually their velocities average to nothing around the ring. Their momentum fluxes do not. Outward motion carrying a surplus of swirl and inward motion carrying a deficit both push angular momentum outward; reverse both and the product is unchanged. That averaged quadratic flux — a Reynolds stress — is engineered to cancel the singular part of the background's residual, term by term, to every order.
What is left over extends smoothly through t = 1. That leftover is f: smooth, compactly supported, and finite — while the velocity it drives is not.
Schematic, not measured. The two divergent terms are real and so is their cancellation, but the paper performs it order by order across several hundred pages — there is no single pair of curves to plot. This shows the shape of the argument: two quantities that each run to infinity, arranged so their sum stays flat and smooth through the singular time.
Diego Córdoba and Luis Martínez-Zoroa built the programme this rests on: an infinite sequence of layers, each individually a perfectly regular solution, designed in advance so that each larger scale amplifies the next smaller one, activating over time intervals that accumulate at a single instant. By 2023 they had blowup for the forced 3D Euler equations — but their layering made the required force too rough to satisfy the Clay criteria.
To keep the force smooth, the divergent terms — acceleration, pressure, viscous friction — have to cancel each other exactly. Finding that cancellation meant exploring an enormous space of profile choices, pulse orientations, and wavelengths. OpenAI reports roughly 10,000 coordinating agents, 2.7 million messages and about 130 billion output tokens over 88 hours, followed by 17 hours of Lean formalization.
Charles Fefferman, who wrote the official problem statement, calls Córdoba and Martínez-Zoroa “the heroes in this story.” Tristan Buckmaster — who was working on closely related problems with Levent Alpöge — has raised concerns about unpublished work stored in OpenAI's Codex. The Clay Institute says the problem has “apparently been settled” and that evaluation and assignment of credit are deliberately unhurried.
Nothing here is a numerical solve of Navier–Stokes. The geometry is drawn from the paper's stated self-similar scalings and its physical description of the core; the streamlines are illustrative integral curves with the correct qualitative structure, not computed solution data.
A Lean formalization exists and has been independently machine-checked, but the Clay Institute's own review is ongoing. This page explains the announced construction; it does not certify it.
Alternative (A) — a fluid left entirely alone — remains open for Navier–Stokes. The blowup shown here is driven by an external force, which is what alternatives (C) and (D) permit and require.
The aspect ratio in the rendering is deliberately exaggerated, by the factor shown on the slider. Every printed number, exponent and chart uses the true value of h.