# Navier-Stokes Existence and Smoothness

**Type:** Open Problem  
**Domain:** Mathematical Physics / Analysis  
**Posed:** 2000 (formal statement), Charles Fefferman  
**Status:** OPEN — Clay Millennium Prize ($1M)  
**Codex URL:** /mathematics/mathematical-physics/navier-stokes-existence-and-smoothness/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Do smooth solutions to 3D Navier-Stokes fluid equations always exist for all time? Engineers use these equations daily; the theoretical question remains unsolved.

## The equations
Nonlinear PDEs (Navier 1820s-1840s, Stokes) expressing Newton's second law for flowing fluids, incorporating pressure and viscosity.

## What's known
2D case: fully resolved (Ladyzhenskaya, 1960s) — smooth solutions exist for all time.
3D: Leray (1934) proved weak solutions exist — a weaker, less-regular notion than the Millennium Prize requires.

## Caffarelli-Kohn-Nirenberg (1982)
Any singularities in 3D weak solutions must be confined to a very small (measure-theoretically) set of points.

## Vortex stretching
The likely source of 3D's extra difficulty vs 2D — vorticity can be stretched and amplified in 3D in a way impossible in 2D.

## Tao's 2016 result
Constructed a modified "averaged" Navier-Stokes equation that DOES blow up — evidence the real problem likely needs genuinely new mathematical ideas.

## Sources
### Tier 1
- Fefferman, C.L. (2000). Clay Mathematics Institute.
- Caffarelli, Kohn and Nirenberg (1982). Comm. Pure Appl. Math.
### Tier 2
- Tao, T. (2016). J. Amer. Math. Soc.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*