What are the Navier-Stokes equations?

The Navier-Stokes equations are a system of partial differential equations that describe the motion of viscous fluids. They model velocity, pressure, and viscosity to predict fluid flow in applications from aircraft design to weather prediction. The equations were developed by Claude-Louis Navier (1823) and George Gabriel Stokes (1845).

What is the difference between the Euler and Navier-Stokes equations?

The Euler equations describe inviscid (frictionless) fluid flow, while the Navier-Stokes equations add a viscous diffusion term. The difference is a single term — the viscous Laplacian — which provides smoothing and energy dissipation. This distinction is central to the regularity question: viscosity should help prevent singularities, but in 3D it has not been proven sufficient.

What is the difference between incompressible and compressible Navier-Stokes?

The incompressible Navier-Stokes equations assume constant density and enforce a divergence-free velocity field. The compressible system adds density as a variable with its own evolution equation, an energy equation, and an equation of state. The Clay Millennium Problem specifically concerns the incompressible 3D system.

Millennium Prize Problem

The Navier-Stokes Existence and Smoothness Problem

Explore the equations, exact solutions, the September 2026 announcement and the remaining unforced regularity questions.

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

September 2026: OpenAI announces smooth-forced 3D blowup (Clay C/D); Clay acknowledges apparent settlement. Review and prize award are separate.

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 unforced 3D Navier–Stokes global-regularity question remains open: the announced forced construction does not settle it. In the technical discussion below, unresolved regularity and uniqueness questions concern the unforced system unless another setting is explicitly stated.

The Navier-Stokes equations describe how fluids move. They govern air, water, blood, weather, and turbulence.

Source status reviewed 2026-09-12 (JST). On September 8, OpenAI announced finite-time blowup for smooth-forced 3D incompressible Navier–Stokes, claiming Clay alternatives C and D, with a manuscript and Lean formalization. On September 11, Clay said the problem has “apparently been settled”; evaluation and assignment of credit are deliberately unhurried. This is not a prize award or our verification of the proof.

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

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 $\mathbb{T}^3$ .

Research/arXiv feed (Updated September 12, 2026)

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

Sep 10, 2026 New math.AP

Limiting Pointwise Decay for the compressible isentropic Navier-Stokes equations

Rui Li, Xu Song +2

We study the long-time pointwise behavior of small localized perturbations of a constant state for the one-dimensional compressible isentropic Navier-...