What Are the Navier-Stokes Equations?

The Navier-Stokes equations are the standard mathematical model for how fluids move. 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.

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.

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.

In their simplest common form, the equations look like this:

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

$$\nabla \cdot u = 0$$

Here:

• $u$ is the fluid's velocity
• $p$ is the pressure
• $\nu$ is the viscosity
• $f$ is any external force, such as gravity

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

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

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

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

The terms have standard interpretations:

• $\partial_t u$: local time variation
• $(u \cdot \nabla)u$: nonlinear advection, meaning the fluid transports its own velocity
• $-\nabla p$: pressure force
• $\nu \Delta u$: viscous diffusion
• $f$: external forcing
• $\nabla \cdot u = 0$: incompressibility constraint

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

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.

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 $\nu = \mu/\rho$.

Incompressibility corresponds to constant density and gives the divergence-free condition $\nabla \cdot u = 0$.

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-critical relative to the regularity one would like to propagate.

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

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.

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

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
• Why It's Hard for scaling, supercriticality, and blowup scenarios