The problem

The equations, the four alternatives in the Clay statement, and what "blow-up" is allowed to mean.

The Millennium problem asks for a proof of one of four statements. Two say that smooth solutions always exist. Two say that they sometimes do not. The construction on this site proves the third and fourth, and the difference between the pairs is a force.

The equations

A velocity field $u(x,t)\in\mathbb{R}^3$ and a pressure $p(x,t)$ on $\mathbb{R}^3$ satisfy

$$\partial_t u + (u\cdot\nabla)u = \nu\Delta u - \nabla p + f,\qquad \nabla\cdot u = 0,\qquad u(x,0)=u^\circ(x).$$

The left side is the acceleration of a fluid parcel: the local rate of change plus the change a parcel sees because it moves. The second piece, $(u\cdot\nabla)u$, is quadratic in $u$, and it is the whole difficulty. The right side is Newton's law: viscous friction, pressure, and an applied force. Incompressibility says the fluid cannot be squeezed in one direction without stretching in another. Setting $\nu=0$ gives Euler.

Multiply the equation by $u$ and integrate. The transport and pressure terms vanish, which is the energy identity

$$\frac{d}{dt}\,\frac12\int |u|^2\,dx = -\nu\int|\nabla u|^2\,dx + \int f\cdot u\,dx.$$

In English: kinetic energy is lost to viscosity and gained only from the force. Without a force the energy can only fall. This is the one estimate everybody has, and it is not enough: in three dimensions bounded energy does not control the velocity pointwise, and the gap between the two is where a singularity could live.

The four alternatives

Fefferman's 2000 statement restricts attention to initial data $u^\circ$ that decay with all derivatives faster than any power, and to forces that do the same in space and time. A solution counts as physically reasonable if $u$ and $p$ are $C^\infty$ on $\mathbb{R}^3\times[0,\infty)$ and the energy $\int|u|^2\,dx$ is bounded uniformly in time. The periodic version replaces decay by periodicity. The prize goes to a proof of any one of the following.

Alt.	Domain	Force	Claim
(A)	$\mathbb{R}^3$	$f\equiv 0$	For every smooth decaying $u^\circ$, a smooth bounded-energy solution exists for all time.
(B)	$\mathbb{R}^3/\mathbb{Z}^3$	$f\equiv 0$	The same on the torus.
(C)	$\mathbb{R}^3$	smooth, decaying	There exist $u^\circ$ and $f$ for which no smooth bounded-energy solution exists for all time.
(D)	$\mathbb{R}^3/\mathbb{Z}^3$	smooth, periodic	The same on the torus.

Read the table by columns. The existence alternatives fix $f\equiv 0$: they are about the equations alone. The breakdown alternatives allow the prover to choose a force. That asymmetry was written in deliberately, to give leeway to solvers, and it is the asymmetry the 2026 construction uses. It proves (C) with $u^\circ = 0$ and a force that is not merely decaying but compactly supported in space and time, and (D) follows by tiling.

The pairs are not complementary. (A) false does not mean (C) true, because (C) may be witnessed by a force while (A) fails only through data. Likewise (C) true says nothing about (A). After September 2026 the situation is that (C) and (D) are claimed with a machine-checked proof, and (A) and (B) are open exactly as before.

What blow-up means here

For Navier–Stokes a smooth solution that stops existing at a finite time $T$ must have velocity that becomes unbounded as $t\uparrow T$. That is the sense of breakdown in the theorem: $\limsup_{t\uparrow 1}\|u(t)\|_{L^\infty}=\infty$. The energy stays bounded, and the integrated dissipation $\int_0^1\int|\nabla u|^2$ stays finite, so the singularity is not an energy catastrophe. It is a point where speed concentrates faster than energy can follow.

Two known facts bracket what such a point can look like. Caffarelli, Kohn and Nirenberg proved that the singular set of a suitable weak solution has zero one-dimensional parabolic Hausdorff measure, which permits isolated points and nothing larger. Escauriaza, Seregin and Šverák proved that a solution of the unforced equations with bounded $L^3$ norm cannot blow up. The constructed singularity is one isolated point, and its $L^3$ norm is not bounded: the core has volume of order $\tau^{3/2-h}$ and speed of order $\tau^{-1/2-h}$, so $\int|u|^3$ over the core is of order $\tau^{-4h}$. It escapes the criterion, as it must.

Why a force changes everything

With $f\equiv 0$ one has to find a flow that the equations themselves carry to a singularity. With $f$ free one may pick the flow first and let the equations dictate the force. Every incompressible flow solves the forced equations for the right $f$. The question becomes whether the required force can be smooth, and the next page is about that.

This is also why the result is contested as a resolution of the physical problem. Tao's remark that the value of such work lies in insight people can understand, and Silvestre's that the Clay problem is settled but the main problem is not, are both about the force. The status page records who has said what.

Next: the residual trick.