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 Exact solutions establish smooth behavior in special symmetric settings. They do not settle global regularity for arbitrary smooth unforced 3D data. The Clay formulation is broader because its breakdown alternatives also permit admissible forcing.
- Solving the Navier-Stokes Equations Exact solutions, numerical approximations and the unforced global-regularity question are different meanings of solving Navier–Stokes. The September 2026 announcement concerns forced breakdown.
- 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
Clay Problem
6- The Navier-Stokes Problem What the 3D Navier–Stokes problem asks, what has been proved, why it resists proof, and which guide to read next.
- Why 2D Navier-Stokes Is Easier Than 3D Without external forcing: 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? September 2026 Update The short answer, what the September 2026 announcement claims, how it maps onto Clay’s official problem, and what remains open.
- Navier-Stokes Existence and Smoothness: Official Clay Statement Fefferman’s official statement in plain language and in full: the equations, admissible data, the four alternatives, and how the 2026 announcement maps onto them.
- 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 Weak solutions exist globally; smoothness and uniqueness in the unforced 3D Leray–Hopf setting remain central questions.
Progress
3- Progress on the Navier-Stokes Problem A century of Navier–Stokes results, from Leray’s weak solutions to the 2026 forced-blowup claim, and the questions that remain open.
- Navier-Stokes Subproblems Breaking the big question into tractable pieces
- Approaches to the Navier-Stokes Problem Weak solutions, regularity criteria, and the main proof strategies