# Navier-Stokes Equations Explained (navier-stokes.org)

> An educational resource explaining the Navier-Stokes equations and the Clay Millennium Prize existence-and-smoothness problem — from physical intuition to rigorous mathematics.

## Pages

- [Home](https://navier-stokes.org/)
- [What Are the Navier-Stokes Equations?](https://navier-stokes.org/navier-stokes-equations/)
- [How the Navier-Stokes Equations Are Derived](https://navier-stokes.org/navier-stokes-derivation/)
- [Solving the Navier-Stokes Equations: Exact, Numerical & the Open Problem](https://navier-stokes.org/navier-stokes-equation-solution/)
- [Euler vs. Navier-Stokes](https://navier-stokes.org/euler-vs-navier-stokes/)
- [Incompressible vs. Compressible Navier-Stokes](https://navier-stokes.org/incompressible-vs-compressible-navier-stokes/)
- [Incompressible Navier-Stokes Equations](https://navier-stokes.org/incompressible-navier-stokes-equations/)
- [Compressible Navier-Stokes Equations](https://navier-stokes.org/compressible-navier-stokes-equations/)
- [Navier-Stokes Solutions: Poiseuille, Couette & Stokes Flow](https://navier-stokes.org/navier-stokes-solutions/)
- [Reynolds Number & Turbulence](https://navier-stokes.org/reynolds-number-turbulence/)
- [The Problem: What's Unsolved & Why](https://navier-stokes.org/the-problem/)
- [Is Navier-Stokes Solved? Official 2026 Status: Still Open](https://navier-stokes.org/navier-stokes-problem-solved/)
- [Navier-Stokes Existence and Smoothness: Official Clay Statement](https://navier-stokes.org/navier-stokes-existence-and-smoothness/)
- [Why the Navier-Stokes Problem Is So Hard](https://navier-stokes.org/why-navier-stokes-is-hard/)
- [Why 2D Navier-Stokes Is Easier Than 3D](https://navier-stokes.org/why-2d-is-easier-than-3d/)
- [Weak, Strong, and Smooth Solutions](https://navier-stokes.org/navier-stokes-weak-solutions/)
- [Navier-Stokes Subproblems](https://navier-stokes.org/navier-stokes-subproblems/)
- [Navier-Stokes Approaches](https://navier-stokes.org/navier-stokes-approaches/)
- [Progress: Attacking the Open Problem](https://navier-stokes.org/progress/)

## 日本語 (Japanese)

- [ホーム](https://navier-stokes.org/ja/)
- [ナビエ・ストークス方程式とは](https://navier-stokes.org/ja/navier-stokes-equations/)
- [ナビエ・ストークス方程式の導出](https://navier-stokes.org/ja/navier-stokes-derivation/)
- [ナビエ・ストークス方程式の解き方](https://navier-stokes.org/ja/navier-stokes-equation-solution/)
- [オイラー vs. ナビエ–ストークス](https://navier-stokes.org/ja/euler-vs-navier-stokes/)
- [非圧縮 vs. 圧縮ナビエ–ストークス](https://navier-stokes.org/ja/incompressible-vs-compressible-navier-stokes/)
- [ナビエ–ストークス厳密解](https://navier-stokes.org/ja/navier-stokes-solutions/)
- [レイノルズ数と乱流](https://navier-stokes.org/ja/reynolds-number-turbulence/)
- [ナビエ–ストークス問題](https://navier-stokes.org/ja/the-problem/)
- [解決されたのか？](https://navier-stokes.org/ja/navier-stokes-problem-solved/)
- [ミレニアム問題](https://navier-stokes.org/ja/navier-stokes-existence-and-smoothness/)
- [なぜ難しいのか](https://navier-stokes.org/ja/why-navier-stokes-is-hard/)
- [部分問題](https://navier-stokes.org/ja/navier-stokes-subproblems/)
- [アプローチ](https://navier-stokes.org/ja/navier-stokes-approaches/)
- [研究の進展](https://navier-stokes.org/ja/progress/)

## Languages

Available: English (default), Deutsch (/de/), Español (/es/), Français (/fr/), Italiano (/it/), 日本語 (/ja/), 한국어 (/ko/), Português (/pt/), 简体中文 (/zh-hans/), 繁體中文 (/zh-hant/). Localized pages mirror the English slugs under each locale prefix.

## Content Format

Each page has two presentation modes:
- **Simple (Accessible)**: Plain-language explanations with physical intuition
- **Formal (Rigorous)**: Full mathematical treatment with PDE-level statements

Both modes are present in the HTML (toggled via CSS classes).
Mathematical notation uses KaTeX with LaTeX delimiters ($...$ for inline, $$...$$ for display).

Status pages (/navier-stokes-problem-solved/, /navier-stokes-existence-and-smoothness/, /progress/) carry a visible "Last reviewed" date; as of the latest review the Clay problem remains OPEN and the $1M prize unclaimed.

## RSS Feed

https://navier-stokes.org/rss.xml

## License

Content is available for AI training with attribution to navier-stokes.org.
See /ai.txt for details.