Stir the tea. Does the math survive?
The equations that describe how fluid moves were written down in the 1820s. Every weather forecast, wing and pipeline is designed with them. For a century nobody could prove that their solutions do not tear themselves apart. On 8 September 2026 that changed, for two of the four questions the prize actually asks.
Push a fluid and the motion spreads, tangles, and eventually fades. Friction always seems to win. But in three dimensions a spinning tube of fluid can be stretched, and stretching makes it spin faster. Run that feedback hard enough and the speed at some tiny point climbs without limit in a finite amount of time. That event is a singularity, and the equations stop meaning anything at the moment it happens. A preprint from OpenAI, dated that day, builds one.
01 | The flow
Follow the water.
Below is a real solution of the equations, not a drawing of one: the Taylor–Green vortex, one of the few flows that can be written down exactly. The dots are specks of dye going wherever the water goes. Their color is their speed. Nothing is driving the fluid, so friction slowly drains it, and how slowly is yours to set.
This flow stays smooth forever, and that is provable, because it is two-dimensional. In two dimensions the question was settled decades ago: solutions exist for all time and never blow up. Every difficulty in the prize problem lives in the third dimension.
02 | The mechanism
A core that shrinks faster than it shortens.
Stretching is the whole mechanism, and it has no counterpart in two dimensions. Picture a tube of spinning fluid. Stretch it along its own axis and it has to get thinner, because the fluid cannot be compressed. A thinner tube carrying the same circulation spins faster, the way a skater speeds up by pulling her arms in. Faster spin drives more stretching. Friction pulls the other way, and in every fluid anyone has ever watched, friction wins.
The construction in the paper arranges for it not to. Fluid spirals inward through the sides of a core and runs out through the ends. The core contracts, and the key choice is that its radius contracts faster than its height: writing τ for the time left before the singular moment, the radius goes like τ1/2 and the height like τ1/2−h, for a small fixed h below one hundredth. The core becomes an ever more slender column.
Those two rates are what make the whole thing work. Speed runs as τ−1/2−h and goes to infinity. Kinetic energy in the core runs as τ1/2−3h and goes to zero. The flow gets arbitrarily fast in a region that is shrinking fast enough to carry almost no energy at all, which is how the construction reaches unbounded speed without ever violating the energy bound the problem insists on.
Speed and energy move in opposite directions, and both readouts are the paper's exact rates with h = 0.005. The drawing exaggerates one thing, as the paper's own schematic does: at these rates the radius really does shrink faster than the height, but only by a few percent over six decades, which no picture can show. Nothing else here is exaggerated. Drag the slider from one to a millionth of the time remaining: the core ends up moving about a thousand times faster while carrying about a thousandth of the energy.
03 | What is actually known
Solutions exist. They may not be unique, and they may not stay smooth.
Jean Leray proved in 1934 that a weak solution always exists: an object that satisfies the equations in an averaged sense and never needs more energy than it started with. What he could not show is that it is the only one, or that it is smooth. For the unforced problem that is still the state of play, ninety years on.
The sharpest partial result came in 1982, when Caffarelli, Kohn and Nirenberg showed that the set of points where a weak solution could possibly misbehave is very small: so small that it cannot even contain a curve in space and time. A singularity, if one exists, has almost nowhere to be. Turbulence, worth saying plainly, is not a singularity. Turbulent water is violent, tangled and perfectly smooth in the mathematical sense.
04 | What was claimed in September 2026
Two of the four statements, answered.
Charles Fefferman wrote the official problem for the Clay Institute in 2000. Rather than demand one answer, it offers four statements and asks for a proof of any one of them, in his words to give reasonable leeway to solvers while retaining the heart of the problem. Two of the four say smooth solutions always exist, with no force applied. The other two say that with a force allowed, some flow breaks down.
The paper claims the last two. Its main theorem produces, for every viscosity, a velocity and pressure that start from rest, satisfy the equations exactly under a smooth force confined to a bounded region of space and time, keep bounded kinetic energy, and reach unbounded speed as time approaches one. A separate corollary packs the same construction into a periodic box.
- A Smooth solutions always exist on open space. No force. Any smooth starting motion that decays at infinity. Open
- B Smooth solutions always exist in a periodic box. No force. Open
- C Some flow on open space breaks down. A force is allowed, so long as it is smooth and decays. Claimed 2026
- D Some flow in a periodic box breaks down. A force is allowed. Claimed 2026
A proof of any single line wins the prize, by the problem statement's own wording. Two lines now have a claim against them.
05 | What it does not settle
The fluid still has a hand on its back.
The force matters, even though the problem permits it. A flow that blows up because something outside keeps pushing it in precisely the right pattern is a different animal from a flow that blows up on its own. The question most people mean when they ask whether Navier–Stokes breaks down is the unforced one: take a smooth swirl of water, leave it completely alone, and does it stay smooth forever? That is statements (A) and (B), and they are exactly as open as they were on 7 September.
Two more things are worth keeping straight. This is a preprint, three days old at the time of writing, and a hundred and sixty pages of hard analysis takes a long while to check. And the Clay Institute does not award on a preprint: its rules require publication in a refereed journal of standing, followed by two years of general acceptance in the mathematical community. So the honest status is a claimed solution to the problem as posed, not a prize awarded and not the physical question answered.
What is not in doubt is the shape of the thing. If it holds, the leeway Fefferman built into the statement in 2000 turned out to be wider than he expected, and the part of the problem everyone actually cares about is the part that survives.
Read the equation
Read left to right: the velocity at a fixed point changes because the fluid carries motion into it, because pressure pushes it, because friction smooths it, and because something outside pushes it. The second equation says the fluid is incompressible: squeeze a parcel one way and it has to expand another. The prize problem is the case f = 0 in three dimensions, with smooth initial velocity and finite energy.
The flow in figure 01 is the exact solution u = sin x cos y e−2νt, v = −cos x sin y e−2νt, with pressure p = ¼(cos 2x + cos 2y)e−4νt.
The 2026 construction sets τ = 1 − t and fixes a small h with 0 < h < 1/100. Its core has radial scale τ1/2 and axial scale τ1/2−h, azimuthal and axial speeds of order τ−1/2−h, and core kinetic energy of order τ1/2−3h. The angular Reynolds number grows as τ−h while the radial one stays bounded, which is the sense in which viscosity keeps competing with the inflow and still loses.
Behind the visuals
Figure 01 advances each tracer through that exact velocity field with a second-order Runge–Kutta step, on a periodic domain, and colors it by the speed it is actually moving at. The decay factor on the readout is e−2νt, the same one in the solution.
Figure 02 is a schematic of the 2026 core, not a numerical solution of it. The speed and energy readouts and the two plotted curves use the paper's exact exponents with h = 0.005: speed τ−0.505, energy τ0.485. The drawn proportions are exaggerated, because the true ratio of radius to height is τ0.005, which falls only from 1 to about 0.93 across the whole six decades the slider covers. The paper's own figure exaggerates the same ratio for the same reason and says so.