The Euler equations ignore viscosity. The Navier-Stokes equations include it. That single difference reshapes the physics, the mathematics, and the million-dollar question.
The short answer
The Euler equations describe a fluid with zero internal friction. No viscosity at all. The Navier-Stokes equations describe the same fluid with viscosity included.
Mathematically, the whole difference is one term: $\nu \Delta u$, the viscous diffusion term. Remove it and Navier-Stokes becomes Euler. Keep it and the equation gains a smoothing mechanism that changes both the physics and the analysis in ways you wouldn't expect from a single extra term.
That one term is why smoke dissipates, why boundary layers form along surfaces, and why the Navier-Stokes Millennium Problem has a completely different character from the corresponding Euler question.
The two equations side by side
Both equations in their standard incompressible form, written so the comparison is obvious:
Euler:
$$\partial_t u + (u \cdot \nabla)u = -\nabla p$$
Navier-Stokes:
$$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u$$
Same left side: the rate of change of velocity plus the nonlinear self-transport term $(u \cdot \nabla)u$. Both enforce incompressibility through $\nabla \cdot u = 0$. The only structural difference is the viscous term $\nu \Delta u$ on the right side of Navier-Stokes.
The parameter $\nu$ is the kinematic viscosity, a physical constant of the fluid. Honey has a large $\nu$. Air has a small one. The Euler equations correspond to $\nu = 0$: a perfectly frictionless idealization that can approximate some high-Reynolds-number flows away from boundaries, but doesn't exist in any real fluid.
What viscosity does physically
Viscosity is friction between neighboring layers of fluid. Fast layer next to a slow one? Viscosity transfers momentum between them, smoothing out the velocity difference. Simple concept. The consequences are enormous and they split Navier-Stokes from Euler in three ways.- Dissipation. Kinetic energy converts to heat. Stir coffee, then stop. It eventually comes to rest because viscosity bleeds the motion away as thermal energy. Euler can't predict this at all, since there's no mechanism in the equations to drain kinetic energy into heat.- Boundary layers. Real fluids stick to surfaces (the no-slip condition), creating thin layers of rapid velocity change near walls. These generate drag on aircraft wings, friction losses in pipes, and turbulence onset at high speeds. Euler flows satisfy a slip condition instead, so they miss viscous wall drag entirely.- Small-scale smoothing. Viscosity kills the sharpest velocity gradients. Without it? Nothing stops the flow from developing infinitely fine structure, sharper and sharper forever. This smoothing is exactly what makes the regularity question for Navier-Stokes a different beast from Euler.
Is Euler just Navier-Stokes with zero viscosity?
Formally? Yes. Set $\nu = 0$ and you get Euler. But that's a terrible place to stop thinking about it.
The limit $\nu \to 0$ is singular. Viscosity carries the highest-order derivatives in the equation, so removing it doesn't make a small tweak. It completely changes what kind of PDE you're dealing with. Boundary layers don't thin out gracefully. They can blow up into turbulence. Solutions that were perfectly smooth under Navier-Stokes can develop wildly different behavior under Euler.
Yes, the two equations share their mathematical DNA. But the zero-viscosity limit is one of the deepest open problems in all of fluid dynamics, not a napkin calculation.
When do people use Euler instead of Navier-Stokes?
Whenever viscosity is negligible compared to the other forces in play. This happens more often than you'd think:- High-speed aerodynamics away from surfaces. Far from a wing, airflow is nearly inviscid. Engineers routinely use Euler solvers for the bulk flow and patch in boundary-layer corrections near the wall.- Astrophysical flows. Interstellar gas clouds, stellar interiors, accretion disks around black holes. At those scales, molecular viscosity is completely irrelevant (though turbulent effective viscosity may not be).- Compressible gas dynamics. Shock waves. Detonations. Supersonic flight. The physics that dominates is pressure and inertia, not friction.- Pure theory. Euler is worth studying in its own right, not just as a stepping stone toward Navier-Stokes. It connects to Riemannian geometry, vortex dynamics, and deep questions about the structure of turbulence itself.But for anything where friction, drag, or boundary behavior matters (pipe flow, vehicle aerodynamics near surfaces, blood circulation, weather at human scales), you need Navier-Stokes. Full stop.
What the difference means for regularity
This is where the gap matters most, and it's where things get genuinely interesting.
The Navier-Stokes Millennium Problem asks a question that sounds almost too simple: if you start with a smooth, well-behaved flow in three dimensions, does the solution stay smooth forever, or can it blow up? Nobody on Earth knows the answer.
The same question for Euler is also open in 3D. But the two problems feel completely different:- Navier-Stokes has viscosity on its side. Always smoothing, always dissipating energy, always damping the sharpest gradients. The real question is whether that smoothing is strong enough to overpower the nonlinear term before it creates a singularity.- Euler has nothing. Zero smoothing. Zero dissipation. The nonlinear term can amplify velocity gradients with absolutely no opposing force, and whether this actually produces a finite-time singularity from smooth 3D initial data is one of the biggest open questions in PDE theory.In 2D, both equations are globally well-posed for smooth initial data. Settled. Done. The mystery lives entirely in three dimensions, for both equations, but for fundamentally different reasons.
Viscosity, turbulence, and the cascade
Turbulence. This is where the Euler-vs-Navier-Stokes comparison becomes physically vivid, almost tangible.
In a turbulent flow, energy enters at large scales (the size of the pipe, the wing, the storm) and cascades down to smaller and smaller eddies. This is the energy cascade, and it's one of the most striking phenomena in all of physics. At the very bottom of the cascade, viscosity finally converts kinetic energy into heat. End of the line.
Euler captures the inertial-range dynamics: energy transfer across scales driven by nonlinearity. But it has no viscous cutoff. No bottom to the cascade. No mechanism to convert kinetic energy into heat at any definite scale. Whether energy can still dissipate in the inviscid limit, what's called anomalous dissipation, remains a deep open question.
This is why turbulence modeling almost always uses Navier-Stokes. The Reynolds number $\mathrm{Re} = UL/\nu$ tells you how wide the cascade is: high $\mathrm{Re}$ means many decades of scales separating energy input from viscous burnoff. Real turbulence lives in the tension between the inviscid cascade pouring energy downward and the viscous cutoff destroying it at the smallest scales.
Summary: one term, two different worlds
The difference between Euler and Navier-Stokes is one term: $\nu \Delta u$. That term changes everything.EulerNavier-StokesViscosityNone ($\nu = 0$)Present ($\nu > 0$)EnergyConserved (formally)DissipatedBoundary layersNoYes (no-slip)PDE typeFirst-order nonlinear + nonlocal pressureSecond-order parabolic + nonlocal pressure2D regularitySolvedSolved3D regularityOpenOpen (Millennium Problem)Euler isn't a simplified Navier-Stokes. It's a fundamentally different system that happens to share most of its structure. And the choice matters in practice: picking the wrong model (Euler where viscosity matters, Navier-Stokes where it doesn't) can wreck a simulation entirely. For the full equations, see What Are the Navier-Stokes Equations?. For the obstacles, see Why It's Hard. For the prize, see The Millennium Problem. For incompressible vs. compressible flow, see Incompressible vs. Compressible Navier-Stokes.
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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