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).

Have the Navier-Stokes equations been solved?

The 3D incompressible Navier-Stokes existence and smoothness problem remains unsolved as of 2026. Leray proved global weak solutions in 1934, and global smooth well-posedness is known in two dimensions, but it is still unknown whether smooth 3D incompressible data can develop a finite-time singularity. This is one of the seven Clay Millennium Prize Problems.

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

The Clay Mathematics Institute named the Navier-Stokes existence and smoothness problem a Millennium Prize Problem in 2000 because it combines deep mathematical significance with practical importance. The equations underpin fluid dynamics across science and engineering, yet mathematicians still cannot prove that every smooth 3D incompressible initial state in the Clay setting produces a global smooth solution. A correct solution carries a $1 million prize.

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

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

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 open problem.

Understand the open problem

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

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 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 $\mathbb{R}^3$ 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.

Research/arXiv feed (Updated July 13, 2026)

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

Jul 10, 2026 New math.AP

On rotated backwards self-similar solutions of the incompressible 3D Navier-Stokes equations

Ben Pineau, Vlad Vicol

We consider backwards globally self-similar solutions of the 3D incompressible Navier-Stokes equations which are invariant under the joint action of...