The Navier-Stokes equations are a family of systems. The difference between incompressible and compressible flow isn't cosmetic. It changes the unknowns, the mathematics, and the open problems.
The physical split: density that changes vs. density that doesn't
"Incompressible vs compressible" boils down to density. Does it stay constant, or does it change?
Try it. Fill a syringe with water and push the plunger. The water moves, but under everyday conditions it doesn't compress noticeably. Water resists compression so strongly that treating it as incompressible is an excellent approximation. Incompressible. Now fill that syringe with air and seal the end. Push the plunger in and you'll feel the air give way, the same mass of air now packed into less volume as it compresses under your thumb. That's compressible flow.
In incompressible flow, the density $\rho$ is constant throughout the fluid, and every tiny parcel keeps its volume as it moves through space. Compressible flow is different. Density becomes a variable, free to change from place to place and moment to moment. Air around a jet engine, gas in an explosion, the atmosphere at large scales: all compressible, all driven by density variations.
Why care? Because this distinction reshapes what the Navier-Stokes equations look like, what they predict, and how difficult they are to analyze and solve.
The incompressible Navier-Stokes equations
The incompressible Navier-Stokes equations describe fluids whose density is constant. They're the version that appears in the Clay Millennium Problem and the version this site focuses on.
The system has two parts. The momentum equation:
$$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u + f$$
and the incompressibility constraint:
$$\nabla \cdot u = 0$$
The constraint $\nabla \cdot u = 0$ says the velocity field is divergence-free: fluid neither piles up nor thins out anywhere. Whatever flows into a tiny region must flow out at the same rate. This single condition replaces the entire density equation. Density doesn't change, so you don't need an equation to track it.
Pressure plays a special role here. It isn't determined by a thermodynamic law (like the ideal gas law). Instead, it adjusts instantaneously everywhere to keep the flow divergence-free. Mathematically, $p$ solves a Poisson equation derived from the constraint. Pressure changes propagate infinitely fast. There's no "speed of sound" in incompressible flow.
The incompressible Navier-Stokes system has two unknown fields: velocity $u$ and pressure $p$. That simplicity is deceptive. The nonlinear term $(u \cdot \nabla)u$ still makes the system extremely difficult in three dimensions.
The compressible Navier-Stokes equations
The compressible Navier-Stokes equations govern flows where density varies. Bigger system. More unknowns. More equations.
You still have a momentum equation, but now density $\rho$ appears explicitly:
$$\partial_t (\rho u) + \nabla \cdot (\rho u \otimes u) = -\nabla p + \nabla \cdot \tau + \rho f$$
The constraint $\nabla \cdot u = 0$ is gone. In its place, you get a continuity equation that tracks how density evolves:
$$\partial_t \rho + \nabla \cdot (\rho u) = 0$$
This says mass is conserved: density changes because the flow compresses or expands fluid parcels.
The system also needs an energy equation and an equation of state, a thermodynamic relation like $p = \rho R T$ (the ideal gas law) that ties pressure to density and temperature. Pressure is no longer a passive enforcer of a constraint. It has its own physics, its own dynamics, and it propagates at a finite speed: the speed of sound.
The compressible system is essential for aerodynamics at high speeds, astrophysical gas dynamics, combustion, and any flow where density changes matter. But it's a genuinely different mathematical object from the incompressible equations. More unknowns, more equations, different PDE structure entirely.
The Mach number: when does compressibility matter?
When does compressibility matter? One number decides: the Mach number.
$$\text{Ma} = \frac{|u|}{c}$$
$|u|$ is flow speed. $c$ is the speed of sound. Their ratio tells you how fast the flow moves compared to the speed at which pressure disturbances can propagate through the medium, and that comparison determines whether you can safely ignore density changes or whether they'll dominate the physics.
When $\text{Ma} < 0.3$, density changes by less than about 5%. Incompressible equations work. Air in a room, water in a pipe, wind around a building: all low-Mach flows where pressure disturbances travel so much faster than the flow itself that density barely budges.
Above $\text{Ma} \approx 0.3$, compressibility starts to bite, and around $\text{Ma} \approx 1$ you hit the transonic regime where local supersonic pockets appear and shock waves form. Fighter jets. Rocket nozzles. Re-entering spacecraft.
Not a binary switch. Most everyday fluid flows, and the Clay Millennium Problem, sit firmly in the low-Mach regime where the incompressible equations apply.
Why the Millennium Problem is about the incompressible case
The Clay Millennium Problem asks a precise question: given a smooth, divergence-free initial velocity on $\mathbb{R}^3$, does the incompressible Navier-Stokes system always produce a smooth solution that exists for all time?
Why incompressible specifically? Three reasons.
First, it's already hard enough. The incompressible 3D equations have resisted proof of global regularity since Leray's foundational work in 1934. Adding variable density, thermodynamics, and shock waves would make the problem vastly harder, not more tractable.
Second, the difficulty is pure fluid mechanics. The incompressible system isolates the core mathematical challenge, the competition between nonlinear advection $(u \cdot \nabla)u$ and viscous dissipation $\nu \Delta u$, without thermodynamic or acoustic complications. It's the cleanest arena to ask the regularity question.
Third, the physics is clean. The incompressible equations model the most common everyday flows. Whether they can produce singularities from smooth data is a fundamental question about the mathematical consistency of classical fluid mechanics.
The compressible system has its own deep open problems (existence of global solutions with large data, formation and interaction of shocks), but those are different problems with different structures. The Clay prize targets the incompressible case because that's the specific regularity question Fefferman formulated for 3D Navier-Stokes.
What to read next
Start here. Want every term in the incompressible system pulled apart, with the physical meaning and mathematical role of each piece explained from scratch? What Are the Navier-Stokes Equations?
Where does this system come from? Derivation of the Navier-Stokes Equations.
Drop viscosity and you get the Euler equations, which are a century older, look simpler on the page, and in some ways are even harder to understand mathematically because you lose the smoothing effect of the diffusion term. Euler vs. Navier-Stokes.
The prize. The Navier-Stokes Existence and Smoothness Problem.
Originally published on navier-stokes.org. The site covers the Navier-Stokes existence and smoothness problem with toggleable simple/rigorous explanations.
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