All Navier-Stokes Guides
Find every published guide in one place, grouped by the question it helps you answer.
Equations
7- 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
- Deriving the Navier-Stokes Equations Where the equations come from: a step-by-step derivation from Newton's second law to the incompressible system behind the Millennium Problem
- Euler vs. Navier-Stokes: What's the Difference? The Euler equations ignore viscosity. The Navier-Stokes equations include it. That single difference reshapes the physics, the mathematics, and the million-dollar question.
- Incompressible vs. Compressible Navier-Stokes Incompressible flow treats density changes as negligible; compressible flow tracks density, energy, and acoustic effects. Here is how the equations—and the problems they pose—differ.
- Incompressible Navier-Stokes Equations The constant-density Navier-Stokes system, the divergence-free condition, and why this is the version behind the Clay Millennium Problem
- Compressible Navier-Stokes Equations The variable-density Navier-Stokes system for gases and high-speed flows, and how it differs from the incompressible equations
- Reynolds Number, Turbulence, and Why Small Scales Matter A bridge from physical intuition to the regularity problem
Exact Solutions
3- Exact Solutions to the Navier-Stokes Equations From Poiseuille pipe flow to Couette shear and Stokes diffusion: the classical solutions you can write in closed form, and why they don't settle the big open problem
- Solving the Navier-Stokes Equations Engineers solve the Navier-Stokes equations every day. Mathematicians can write down exact solutions. And yet a $1 million prize for "solving" them sits unclaimed. All three statements are true, because "solution" means three different things.
- Poiseuille Flow and the Hagen–Poiseuille Equation A step-by-step derivation of the parabolic pipe-flow profile, pressure–flow law, assumptions, and limits directly from the Navier–Stokes equations
Open Problem
6- The Navier-Stokes Problem A broad guide to the open 3D regularity problem, what is known, why it is hard, and where to go deeper
- Why 2D Navier-Stokes Is Easier Than 3D In two dimensions, vorticity obeys a maximum principle and energy estimates close. In three dimensions, vortex stretching breaks both controls, and the global regularity question remains wide open.
- Is Navier-Stokes Solved? Official 2026 Status: Still Open The current status of the Clay Millennium Prize problem, what is known, and why no proof or blowup has been accepted
- Navier-Stokes Existence and Smoothness: Official Clay Statement Fefferman's official Clay Mathematics Institute formulation, the 3D incompressible regularity question, accepted proof targets, and current status
- Why the Navier-Stokes Problem Is Hard The core mathematical obstacles standing in the way
- Weak, Strong, and Smooth Solutions to the Navier-Stokes Equations The Millennium Prize asks for smooth solutions. All we can prove exist globally for arbitrary data are weak solutions. That gap is the entire problem.