Navier-Stokes Equation Explained: Meaning, Terms, and Formula

Q: What is the Navier-Stokes equation in simple terms?

The Navier-Stokes equation describes how a fluid's velocity changes because of inertia, pressure, viscosity, and external forces. In the incompressible form, it is paired with the condition that local fluid volume is preserved.

Q: What do the terms in the Navier-Stokes equation mean?

The time derivative describes local acceleration, the nonlinear advection term describes the fluid carrying its own velocity, the pressure gradient pushes the fluid, the viscosity term smooths velocity differences, and the forcing term represents outside forces. The divergence-free condition enforces incompressibility.

Q: Is the Navier-Stokes equation solved?

The equations have exact solutions in special cases and are solved numerically in engineering, but the 3D incompressible existence and smoothness problem remains unsolved as of 2026. No accepted proof or blowup counterexample has resolved the Clay Millennium Prize problem.

Navier-Stokes Equation Explained: Meaning, Terms, and Formula

A clear introduction to the fluid-motion equation: what each term means, why pressure and viscosity matter, and how it connects to the Clay problem

Published: March 22, 2026

What the Navier-Stokes equations are

Short answer: the Navier-Stokes equation describes how fluid velocity changes under inertia, pressure, viscosity, and external forces. In incompressible form, it pairs a momentum equation with the condition that fluid volume is locally preserved.

Common incompressible formula:

$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u + f$

$\nabla \cdot u = 0$

In simple terms: acceleration plus self-transport equals pressure force plus viscous smoothing plus outside force.

The Navier-Stokes equations are a system of partial differential equations that describe the motion of viscous fluids such as water and air.

They are used to describe water in a pipe, air around an airplane wing, blood in an artery, and countless other flows.

At a high level, they say: a fluid changes its motion because of pressure, viscosity, and any forces acting on it. Pressure pushes fluid around, viscosity smooths out sharp differences in motion, and external forces such as gravity can drive the flow.

These equations are not just a physics slogan. They are the working language of much of fluid dynamics, engineering, and computational simulation.

Short answer: the incompressible Navier-Stokes system evolves a velocity field $u(x,t)$ by balancing local acceleration, nonlinear advection, pressure, viscosity, and forcing, while enforcing $\nabla \cdot u = 0$ .

Standard constant-density incompressible form:

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

Here $u$ is velocity, $p$ is pressure divided by constant density, $\nu$ is kinematic viscosity, and $f$ is external forcing.

The Navier-Stokes equations are the momentum-balance equations for a viscous Newtonian fluid. In the incompressible setting, they couple the velocity field $u(x,t)$ and pressure $p(x,t)$ through a nonlinear PDE system.

Strictly speaking, Navier-Stokes is a system of equations rather than a single equation. They model conservation of momentum together with the constitutive law that viscous stress is proportional to the symmetric velocity gradient. Incompressibility adds the constraint that volume is preserved locally.

For the Clay Millennium problem and for most of this site, the relevant setting is the 3D incompressible system.

Formula and term-by-term meaning

The incompressible Navier-Stokes equation is easiest to read one term at a time:

Term	Meaning
$\partial_t u$	How velocity changes at a fixed point in time
$(u \cdot \nabla)u$	The fluid carrying its own velocity along as it moves
$-\nabla p$	Pressure pushing fluid from high pressure toward low pressure
$\nu \Delta u$	Viscosity smoothing out sharp velocity differences
$f$	External force, such as gravity or a driven body force
$\nabla \cdot u = 0$	Incompressibility: local fluid volume is preserved

The left side describes how the velocity changes and how the fluid transports its own motion. The right side contains the forces that push, smooth, or drive the flow.

The incompressible Navier-Stokes system on $\mathbb{R}^3$ is

$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u + f,$

$\nabla \cdot u = 0, \qquad u(x,0)=u_0(x).$

Term	Role in the PDE
$\partial_t u$	Local time derivative of the velocity field
$(u \cdot \nabla)u$	Nonlinear advection; the term that makes the equation self-coupled
$-\nabla p$	Pressure gradient, acting as a Lagrange multiplier for incompressibility
$\nu \Delta u$	Viscous diffusion with kinematic viscosity $\nu$
$f$	External forcing
$\nabla \cdot u = 0$	Divergence-free constraint for incompressible flow

This is the form used throughout the site's pages on the Millennium problem, difficulty, and proof strategies.

Where the Navier-Stokes equations come from

The equations come from a simple idea: apply Newton's second law to a tiny parcel of fluid. The mass of that parcel times its acceleration must equal the total force acting on it.

For a viscous fluid, those forces come mainly from pressure and internal friction. When you write that balance carefully at every point in the fluid, you get the Navier-Stokes equations.

So the equations are not arbitrary. They are a continuum-mechanics version of force equals mass times acceleration. For the full step-by-step derivation, see How the Navier-Stokes Equations Are Derived.

The derivation starts from conservation of momentum for a continuum. One writes balance of linear momentum on a material volume and then localizes the identity to obtain a PDE.

For a Newtonian incompressible fluid, the Cauchy stress tensor has the form

$T = -pI + 2\mu D(u), \qquad D(u)=\frac{1}{2}(\nabla u + \nabla u^T),$

where $\mu$ is the dynamic viscosity. Substituting this constitutive law into the momentum equation and dividing by density yields the familiar incompressible system with kinematic viscosity $\nu = \mu/\rho$ .

In the standard constant-density incompressible setting, the continuity equation reduces to $\nabla \cdot u = 0$ . For the full derivation from momentum balance and the Newtonian constitutive law, see Derivation.

Why the Navier-Stokes equations are hard

The difficult part is the nonlinear term $(u \cdot \nabla)u$ . The fluid does not just respond to outside forces; it also pushes itself around. That feedback is what makes turbulence and chaotic-looking motion possible.

In two spatial dimensions, the equations are much better behaved. In three dimensions, we still do not know whether every smooth starting flow stays smooth forever.

That is why these equations are famous far beyond engineering: they lead directly to the Navier-Stokes Millennium Problem.

The main analytical difficulty is that the natural energy estimate is weaker than the scaling of the 3D equation. Roughly speaking, the standard $L^2$ control is not strong enough to rule out very small-scale concentration.

This is the source of the gap between what is known for global weak solutions and what would be needed to prove global smoothness. The nonlinear advection term is energy-supercritical: the natural energy-level control is supercritical with respect to the equation's scaling, so it is too weak to rule out concentration at small scales.

For a fuller discussion, see Why It's Hard and Approaches.

What they are used for

The Navier-Stokes equations are used every day in science and engineering. Typical applications include:

airflow around wings and vehicles
weather and climate models
ocean circulation
industrial fluid transport
blood flow and other biological transport problems

In practice, people solve approximations of these equations numerically, often with additional modeling assumptions. That practical success is one reason the remaining mathematical questions are so striking.

Applied work typically uses numerical approximations of Navier-Stokes or related models under specific regimes: incompressible flow, compressible flow, turbulence closures, boundary-layer approximations, and reduced models.

Direct numerical simulation, large-eddy simulation, and Reynolds-averaged closures all trace back to the same continuum PDE framework, but they do not eliminate the foundational regularity question in three dimensions.

This split between practical effectiveness and incomplete theory is part of what makes the subject so compelling.

What to read next

If your main question is whether the problem is solved, start with Is the Navier-Stokes Problem Solved?.

If you want the broad mathematical stakes, continue to The Millennium Problem.

If you want to understand how Navier-Stokes differs from the inviscid Euler equations and why viscosity matters, read Euler vs. Navier-Stokes.

If you want the physical intuition behind turbulence and small scales, read Reynolds Number, Turbulence, and Why Small Scales Matter.

If you want the main obstacles, go to Why It's Hard.

And if your question is what it even means to solve these equations (by formula, by computer, or in the Millennium Prize sense), read Solving the Navier-Stokes Equations.

Natural next steps on this site:

Is the Navier-Stokes Problem Solved? for the status question and the distinction between weak existence and global smooth theory
The Millennium Problem for the precise Clay formulation
Incompressible Navier-Stokes Equations for the constant-density system and pressure constraint
Compressible Navier-Stokes Equations for the density, momentum, and energy system
Euler vs. Navier-Stokes for the role of viscosity and the key differences between the inviscid and viscous systems
Incompressible vs. Compressible Navier-Stokes for the difference between constant-density and variable-density formulations
Reynolds Number, Turbulence, and Why Small Scales Matter for the regime-level intuition behind multiscale flow
Why It's Hard for scaling, supercriticality, and blowup scenarios
Solving the Navier-Stokes Equations for the relationship between exact solutions, numerical approximation, and the well-posedness question