Millennium Prize Problem

The Navier-Stokes Existence and Smoothness Problem

A guide to the unsolved 3D regularity question, from intuition to rigorous mathematics

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 describe how fluids move. They govern air, water, blood, weather, turbulence. They govern many familiar fluid flows, including air, water, 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 formal 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 $\mathbb{R}^3$ or $\mathbb{T}^3$.

The equation is

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

In the Clay setting, we consider smooth divergence-free initial data, either rapidly decaying on $\mathbb{R}^3$ or smooth periodic on $\mathbb{T}^3$. 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.

Start Here

01

The Equations

What each term means physically and where the equations come from

Incompressible system, pressure projection, and physical interpretation

→ 02

Exact Solutions

Poiseuille, Couette, and Stokes flow — cases where the equations can be solved

Closed-form solutions for pipe flow, shear flow, and impulsively started plates

→ 03

Is It Solved?

No — but here is exactly what is known and what remains open

Weak vs. strong vs. smooth — the gap that remains

→ 04

Millennium Problem

The precise question Clay posed and what a valid solution requires

The Clay global regularity statement and its boundary conditions

→ 05

Why It's Hard

Why the three-dimensional case defeats every technique tried so far

Supercritical scaling, the energy gap, and why known estimates fail

→

Go Deeper

06

Subproblems

The smaller questions researchers can actually attack

Partial regularity, continuation criteria, blowup classification

→ 07

Approaches

The main proof strategies from energy methods to regularity criteria

Leray-Hopf, CKN, critical-space methods, geometric regularity

→ 08

Reynolds & Turbulence

The physical picture that explains why three dimensions are different

Reynolds number, energy cascade, and the small-scales barrier

→

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.