Incompressible Navier-Stokes Equations: Form, Constraint, and Meaning
The constant-density Navier-Stokes system, the divergence-free condition, and why this is the version behind the Clay Millennium Problem
The system
The incompressible Navier-Stokes equations describe viscous fluids whose density is treated as constant. The standard form is
$$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u + f$$
$$\nabla \cdot u = 0.$$
Here $u$ is velocity, $p$ is pressure, $\nu$ is kinematic viscosity, and $f$ is an external force. The second line is the key incompressible constraint: fluid cannot locally pile up or thin out.
On $\mathbb{R}^3$, the incompressible system is
$$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u + f, \qquad \nabla \cdot u = 0, \qquad u(x,0)=u_0(x).$$
The unknowns are $u : \mathbb{R}^3 \times [0,T) \to \mathbb{R}^3$ and $p : \mathbb{R}^3 \times [0,T) \to \mathbb{R}$. The initial datum must satisfy $\nabla \cdot u_0 = 0$.
What incompressible means
Incompressible does not mean the fluid cannot move. It means small parcels keep their volume as they move. Mathematically, that is the condition $\nabla \cdot u = 0$.
This is a good approximation for water, oils, and many low-speed air flows. If density changes matter, use the compressible Navier-Stokes equations instead.
The condition $\nabla \cdot u = 0$ implies local volume preservation for the flow map. If $\Phi_t$ is the particle flow, then formally $\det D\Phi_t = 1$ when the velocity is divergence-free.
In the constant-density derivation, the continuity equation reduces from $\partial_t \rho + \nabla \cdot(\rho u)=0$ to $\nabla \cdot u=0$.
Pressure as a constraint force
Pressure has a special role in the incompressible equations. It is not set by an equation of state such as the ideal gas law. Instead, pressure adjusts to keep the velocity field divergence-free.
That is one reason incompressible flow is mathematically different from compressible flow: pressure is tied to a global spatial constraint.
Taking divergence of the momentum equation and using $\nabla \cdot u = 0$ gives a pressure Poisson equation, for example
$$-\Delta p = \partial_i\partial_j(u_i u_j) - \nabla \cdot f.$$
Thus pressure acts as a Lagrange multiplier for the incompressibility constraint. This elliptic coupling is one of the defining structural features of the incompressible system.
Why this is the Clay problem setting
The Clay Millennium Problem is about the three-dimensional incompressible Navier-Stokes equations. It asks whether every smooth divergence-free initial velocity produces a smooth solution for all time, or whether a smooth flow can break down.
For the current solved/open status, see Is the Navier-Stokes problem solved?. For the exact Clay statement, see Navier-Stokes existence and smoothness.
Fefferman's Clay statement specifies smooth divergence-free data on $\mathbb{R}^3$ or $\mathbb{T}^3$ and asks for either global smooth existence or an admissible breakdown scenario.
The compressible system has its own deep questions, but the Millennium Prize problem isolates the incompressible regularity question: the gap between global Leray-Hopf weak solutions and global classical smooth solutions in 3D.
Related pages
For a broader comparison, read incompressible vs. compressible Navier-Stokes. For a term-by-term introduction, read what the Navier-Stokes equations are. For where the equations come from, read the derivation.