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/Weak, strong, and smooth Navier-Stokes solutions: distributional and Leray–Hopf classes, weak–strong uniqueness, and the remaining unforced regularity question.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 unforced 3D regularity 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/Structural obstacles to unforced 3D Navier-Stokes regularity, 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/Clay’s official Navier–Stokes statement explained: admissible data, bounded energy, alternatives A–D, forcing conditions, and how the 2026 claim maps to C and D.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-problem-solved/https://navier-stokes.org/navier-stokes-problem-solved/Short answer: a 2026 proof claims Clay’s forced-breakdown alternatives, and Clay calls the problem “apparently” settled. What it shows and what stays open.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/progress/https://navier-stokes.org/progress/Leray, Caffarelli–Kohn–Nirenberg, Beale–Kato–Majda, Tao and non-uniqueness results, and how the 2026 forced-blowup announcement connects to them.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/navier-stokes-subproblems/https://navier-stokes.org/navier-stokes-subproblems/Remaining Navier-Stokes research questions: weak solutions, unforced regularity, singularities and blowup scenarios, alongside the September 2026 announcement.Sun, 22 Mar 2026 00:00:00 GMThttps://navier-stokes.org/the-problem/https://navier-stokes.org/the-problem/An orientation to the 3D Navier–Stokes regularity problem: the question, known results, why it resists proof, Clay’s four alternatives and a reading map.Sun, 22 Mar 2026 00:00:00 GMT