Understanding the Navier-Stokes equations — from intuition to rigorous mathematics.https://navier-stokes.org/enhttps://navier-stokes.org/poiseuille-flow/https://navier-stokes.org/poiseuille-flow/Derive the Hagen–Poiseuille equation from Navier–Stokes, including the parabolic velocity profile, flow rate, assumptions, example, and limits.Fri, 10 Jul 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-equation-solution/https://navier-stokes.org/navier-stokes-equation-solution/What does it mean to "solve" the Navier-Stokes equations? From exact analytical solutions to CFD to the Millennium Prize, the three meanings of solution explained.Fri, 03 Jul 2026 00:00:00 GMThttps://navier-stokes.org/compressible-navier-stokes-equations/https://navier-stokes.org/compressible-navier-stokes-equations/The compressible Navier-Stokes equations explained: continuity, momentum, energy, equation of state, Mach number, shocks, and how they differ from the incompressible system.Wed, 10 Jun 2026 00:00:00 GMThttps://navier-stokes.org/incompressible-navier-stokes-equations/https://navier-stokes.org/incompressible-navier-stokes-equations/The incompressible Navier-Stokes equations: velocity, pressure, divergence-free constraint, pressure Poisson equation, and why this is the Clay Millennium Problem setting.Wed, 10 Jun 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-weak-solutions/https://navier-stokes.org/navier-stokes-weak-solutions/What weak, strong, and smooth solutions to the Navier-Stokes equations are, why the distinctions matter, and how the gap between weak and smooth is the Millennium Prize problem.Mon, 30 Mar 2026 00:00:00 GMThttps://navier-stokes.org/why-2d-is-easier-than-3d/https://navier-stokes.org/why-2d-is-easier-than-3d/Why 2D Navier-Stokes has global regularity while 3D remains open: vorticity control, scaling, vortex stretching, and the supercriticality gap.Sun, 29 Mar 2026 00:00:00 GMThttps://navier-stokes.org/euler-vs-navier-stokes/https://navier-stokes.org/euler-vs-navier-stokes/The difference between Euler and Navier-Stokes equations explained: viscosity, regularity, modeling, and what it means for the Millennium Prize problem.Wed, 25 Mar 2026 00:00:00 GMThttps://navier-stokes.org/incompressible-vs-compressible-navier-stokes/https://navier-stokes.org/incompressible-vs-compressible-navier-stokes/The difference between incompressible and compressible Navier-Stokes: what changes, what stays the same, and why the Millennium Problem is about the incompressible case.Wed, 25 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-derivation/https://navier-stokes.org/navier-stokes-derivation/Where the Navier-Stokes equations come from: a step-by-step derivation from Newton's second law through viscous stress to the incompressible system.Wed, 25 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-solutions/https://navier-stokes.org/navier-stokes-solutions/Exact solutions to the Navier-Stokes equations: Poiseuille pipe flow, Couette shear flow, Stokes problems, and why they don't resolve the Millennium Problem.Wed, 25 Mar 2026 00:00:00 GMThttps://navier-stokes.org/reynolds-number-turbulence/https://navier-stokes.org/reynolds-number-turbulence/What Reynolds number measures, why higher-Re flow becomes turbulent, and why smaller scales matter for the 3D Navier-Stokes problem.Wed, 25 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-approaches/https://navier-stokes.org/navier-stokes-approaches/Major proof strategies for the 3D Navier-Stokes problem: energy methods, partial regularity, critical spaces, convex integration, and proof barriers.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/why-navier-stokes-is-hard/https://navier-stokes.org/why-navier-stokes-is-hard/Five deep reasons the 3D Navier-Stokes problem resists every known technique, from nonlinearity and scaling to turbulence and pressure.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-equations/https://navier-stokes.org/navier-stokes-equations/A clear introduction to the Navier-Stokes equations: what they model, what the terms mean, where they come from, and why the 3D case is hard.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-existence-and-smoothness/https://navier-stokes.org/navier-stokes-existence-and-smoothness/The official Clay Mathematics Institute Navier-Stokes problem statement, explained: Fefferman's exact criteria, both proof targets, and the {{currentMonthYear}} status.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-problem-solved/https://navier-stokes.org/navier-stokes-problem-solved/No — as of {{currentMonthYear}}, the 3D Navier-Stokes problem is still open. The official Clay status, what is known, and new arXiv research tracked as it appears.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/progress/https://navier-stokes.org/progress/A field guide to the major approaches, milestones, and open questions in the Navier-Stokes regularity problem.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-subproblems/https://navier-stokes.org/navier-stokes-subproblems/The million-dollar problem splits into smaller pieces: weak solutions, singularities, blowup scenarios, and other core subproblems.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/the-problem/https://navier-stokes.org/the-problem/A broad overview of the Navier-Stokes problem: the 3D regularity question, known results, why it is hard, and where the Clay statement fits.Sun, 22 Mar 2026 00:00:00 GMT