Can water’s
equations break?
We have rules for how a fluid moves.
A rule can be useful before we can prove everything it does.
Must it stay smooth for every future time?
Zoom in on a smooth flow: neighbouring arrows fit together without a spike becoming infinite in finite time. Complicated swirls can still be smooth.
Give every point an arrow.
Each arrow says how fast the water there is moving, and in which direction. The whole collection is a velocity field. Navier–Stokes describes how that field changes.
Advance time. In this special flow, viscosity smooths the speed differences between neighbouring layers.
Try the same time with two viscosity settings. Higher viscosity makes this pattern fade faster. Values use model units.
Follow a small parcel of water. Its velocity changes as time passes and as it travels into places with different velocities. The moving water helps transport its own motion.
Now give a swirl room to stretch.
The easy flow above has no vortex stretching. A three-dimensional flow can stretch a swirling region along its axis. To see that geometry, hold the amount of water fixed and pull a short, thick tube into a long, thin one.
The transparent boundary follows a chosen region of water.
Make the tube 4× as long, with the same volume. What happens to its radius?
Stretch it. Length can grow while radius shrinks. The same volume fits inside a very different shape.
*This model isolates stretching: it neglects viscosity and keeps circulation fixed. The spinning rate then rises as the tube narrows. Real Navier–Stokes flow also has viscosity, which spreads vorticity. This finite tube is a mechanism illustration, not a solution of the full equations or an example of blowup.
Here is the difficulty: stretching can intensify local rotation while viscosity spreads it. Describing those effects is easier than proving that every permitted 3D flow stays smooth. In a strictly two-dimensional flow, this stretching mechanism is absent.
A successful run is smaller than a proof.
Finite resolution. Finite time.
Useful evidence about that run. A narrow sample of the possibilities. The bar is a scope metaphor, not a percentage.
Every future time. All spatial scales.
The formal problem fixes the equations, domain and regularity conditions. An alternative prize route is a rigorous breakdown example satisfying the official requirements.
“My simulation of one 3D flow ran for 60 seconds without trouble. The Millennium problem is solved.”
Decide what this evidence actually covers. Then choose a verdict.
The arrows obey a local rule. Three-dimensional stretching creates new difficulties. A proof must control every permitted flow for all future time — or establish an allowed breakdown example.
The question is about the mathematical model. Choppy water, a whirlpool, and a computer crash are not, by themselves, a singularity of its solution.
Status checked 8 September 2026: Clay Mathematics Institute lists this problem as unsolved. It is one of the Millennium Prize Problems, with $1 million allocated to each problem.
For the curious: the equation and the exact example
For an incompressible, constant-density fluid, one normalized form is ∂u/∂t + (u · ∇)u = −∇p + ν∇²u + f, with ∇ · u = 0. Here u is the velocity field, p is pressure divided by density, ν > 0 is kinematic viscosity, and f is external force per unit mass. Incompressibility means a moving parcel keeps its volume; it can still change shape.
The first experiment uses u = (e^(−νt) sin y, 0, 0), p = 0, f = 0, on a repeating domain. This is a specially chosen exact flow, not a general fluid solver. Its advection term is zero, its Laplacian is −u, and its time derivative is −νu. It also works when extended unchanged in the third direction. One easy solution leaves the “every starting flow” question open.
For the tube, volume = πr²L. If length grows by a factor s, fixed volume gives r/r₀ = 1/√s. Uniform vorticity and conserved circulation in the inviscid illustration give spin/spin₀ = s. At s = 4, area is one quarter and radius is one half. The slider has no infinite state.
Teacher notes, model limits and primary sources
This is enrichment for ages 14–17, using arrows, graphs, volume and the difference between examples and proof. It does not teach the analysis needed to solve the problem. “Smooth” here refers to the mathematical velocity and pressure fields, not a visually calm water surface. The first model is a solvable periodic shear; the second is an inviscid stretching illustration. Neither is evidence for a Navier–Stokes singularity.
In the official statement by Charles Fefferman, routes A/B seek global smooth solutions for all admissible smooth divergence-free initial data with zero forcing, on all of 3D space or a periodic domain. Routes C/D allow specified smooth forcing and seek failure of the required global solution. The domain, decay or periodicity, and regularity conditions matter. Two-dimensional global smoothness is known under the analogous conditions; the general three-dimensional question remains open.
Background: NASA: Navier–Stokes equations (a more general compressible system); MIT: vortex structures and stretching (inviscid assumptions). Sources checked 8 September 2026. All drawings are generated locally; no outbound requests are needed.