Millennium Prize Problem

The Navier-Stokes Existence and Smoothness Problem

Learn how Navier-Stokes describes fluid motion, where exact solutions exist, and why the 3D Millennium Problem remains open.

Start Here

01

Learn the equations

Learn what the equations describe and what each term does.

 02

Worked solutions and examples

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

 03

Understand the open problem

See why the 3D problem is still open and what a proof must establish.

 04

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

∂tu  +  (u⋅∇)u  =  −∇p  +  νΔu,∇⋅u=0\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 describe how fluids move. They govern air, water, blood, weather, and turbulence.

But this site isn't about whether the equations work. They do, and they are extraordinarily successful in applications. The real question is the one behind the Clay Millennium Prize: if you start with a perfectly smooth 3D flow, does it stay smooth forever? Or can it blow up?

Nobody knows. That's what makes this problem extraordinary.

Below, we break the subject into distinct paths: the equations themselves, the current solved-or-open status, the formal Clay problem statement, the mathematical obstacles, standard reductions, and the proof strategies people have tried.

This site is centered on the 3D incompressible Navier-Stokes global regularity problem on R3\mathbb{R}^3 or T3\mathbb{T}^3.

The equation is

∂tu+(u⋅∇)u=−∇p+νΔu,∇⋅u=0.\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u, \qquad \nabla \cdot u = 0.

In the Clay setting, we consider smooth divergence-free initial data, either rapidly decaying on R3\mathbb{R}^3 or smooth periodic on T3\mathbb{T}^3. The question: does such data always produce a unique global smooth solution, or can smoothness break down in finite time? Leray's 1934 theory gives global weak solutions. Global smoothness and uniqueness in three dimensions? Still open.

The sections below separate the PDE itself, the formal Clay statement, the scaling obstacles, the standard subproblems, and the approaches that have shaped the field.

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

Every page has two versions. The Simple / Formal toggle in the header switches between plain-English explanations and the full mathematical treatment. You can change modes at any time without losing your place.

Every page is written in parallel. Simple mode gives physical intuition; Formal mode gives the PDE-level statements. Toggle freely — the structure mirrors across both modes.