Millennium Prize Problem

The Navier-Stokes Existence and Smoothness Problem

Explore the equations, their exact solutions, the Clay Millennium Problem and a century of research on whether 3D fluid flows stay smooth.

Status update sources reviewed Sep 29, 2026

A claimed proof of forced finite-time blowup

On September 8, 2026, OpenAI announced a proof that 3D incompressible Navier–Stokes solutions can blow up in finite time under a smooth external force—Clay’s alternatives C and D. On September 11, Clay said the problem has “apparently been settled”; evaluation is deliberately unhurried and no prize has been awarded. The result does not settle whether flows without external force always stay smooth.

Start Here

Learn the equations

Learn what the equations describe and what each term does.

Worked solutions and examples

See special flows we can calculate exactly—and why they do not solve the Clay problem.

Understand the Clay problem

What Clay’s official problem asks, why the 3D case is hard, and where the question stands.

See research progress

See what mathematicians have proved, which approaches they have tried, and what remains unknown.

New here? Start with the equations.

A live fluid simulation. Drag to stir.

Water Honey

The equation behind the simulation

\textcolor{#e0e0e0}{\partial_t u} \;+\; \textcolor{#e040fb}{(u \cdot \nabla)u} \;=\; \textcolor{#66bb6a}{-\nabla p} \;+\; \textcolor{#4fc3f7}{\nu \Delta u}, \qquad \textcolor{#90a4ae}{\nabla \cdot u = 0}

What do these terms mean?

Viscosityν∆u

How thick the fluid is — honey resists swirling, water flows freely. The slider above controls this Viscous diffusion — smooths and spreads the flow. Controlled by the ν slider above

Momentum(u·∇)u Movement carries movement — a fast stream drags nearby fluid along, creating swirls Nonlinear advection — velocity transports itself, producing vortex stretching and cascade

Pressure−∇p When fluid bunches up, pressure pushes it apart — the solver handles this automatically Pressure gradient — computed by the projection step to enforce the divergence-free constraint

Change∂ₜu The result — how the fluid's speed changes at each point, computed from all the other terms Net rate of change — the left-hand side, determined by the balance of advection, pressure, and viscosity

Conservation∇·u = 0 The fluid can't compress or expand — it just rearranges, which is what makes water behave like water Divergence-free constraint — satisfied each step by the Helmholtz-Hodge projection

What the 3D Problem Actually Asks

The Navier-Stokes equations are a system of partial differential equations that describe the motion of viscous fluids such as water and air. They balance a fluid’s inertia against pressure, viscosity and outside forces, and they are used for everything from aircraft design and pipe flow to weather prediction and blood flow.

The famous open question is what these equations predict in three dimensions. Start with a smooth flow of finite energy: does the solution stay smooth forever, or can the velocity become infinite after a finite time? Clay’s official statement accepts any one of four answers: a proof that every smooth start stays smooth when there is no external force, on all of space (A) or in a periodic box (B), or an example of breakdown in either setting, where a smooth, rapidly decaying external force may also be chosen (C and D). In two dimensions smooth solutions are known to last forever; without forcing, the 3D question is still open.

The exact conditions are explained in the official Clay problem statement, explained, and the century of results behind them in Progress on the Navier-Stokes problem.

Clay selected Navier–Stokes as one of its seven Millennium Prize Problems in 2000 because it combines fundamental mathematical difficulty with the equations’ central role in fluid dynamics. The September 2026 announcement changes the status, not those reasons for its importance.

Primary sources: Clay (2026-09-11) · OpenAI (2026-09-08) · Lean / GitHub · Official Clay statement.

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

This site is centered on the 3D incompressible Navier-Stokes global regularity problem on $\mathbb{R}^3$ or $\mathbb{T}^3$ : for smooth, divergence-free initial data satisfying the official decay (or periodicity) conditions and $f\equiv0$ , is there a smooth solution for all $t\ge0$ , with bounded energy on $\mathbb{R}^3$ ? Clay’s formulation accepts a proof of global existence (A on $\mathbb{R}^3$ , B on $\mathbb{T}^3$ , both with $f\equiv0$ ) or a breakdown example in which a smooth force satisfying the official decay conditions may also be chosen (C, D). The 2D analogue has long been settled; for $f\equiv0$ the 3D question remains open.

Formal conditions: the official Clay statement, explained. Results and methods: Progress.

Primary sources: Clay (2026-09-11) · OpenAI (2026-09-08) · Lean / GitHub · Official Clay statement.

Frequently asked questions

What are the Navier-Stokes equations?

Have the Navier-Stokes equations been solved?

Why is the Navier-Stokes problem a Millennium Prize Problem?

Research/arXiv feed (Updated September 30, 2026)

A daily-updated carousel of new, revised, and cross-listed arXiv papers matching Navier-Stokes topics.

Sep 29, 2026 New math.AP

Boundary Layers and Initial-Boundary Corner Asymptotics for the 2D Navier-Stokes Equations with Vanishing Vertical Viscosity

Siwei Chen, Yinghui Wang +1

Motivated by anisotropic viscosity in geophysical fluid dynamics, where vertical momentum diffusion is often much weaker than horizontal diffusion and...