---
tag: 46Z1TVS2K6
kind: conjecture
name: Navier–Stokes existence and smoothness
msc: [35Q30, 76D05, 76D03]
uses: [18KXZHVF4M, 0V33BNKZ9S, 0MWV6ZTK6S]
about: [18KXZHVF4M]
sources:
  - url: https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
same_as:
  wikidata: Q1098081
  wikipedia: Navier–Stokes existence and smoothness
history:
  posed_by: "Clay Mathematics Institute"
  first_stated: "2000"
---

## Statement

Let $\nu > 0$.
Let $u_0 \colon \R^3 \to \R^3$ be a [[smooth|0V33BNKZ9S]] vector field with [[divergence|0MWV6ZTK6S]] $\nabla \cdot u_0 = 0$.
Assume that for every multi-index $\alpha$ and every $K > 0$ there is a constant $C_{\alpha K}$ with $\abs{\partial^\alpha u_0(x)} \le C_{\alpha K} (1 + \abs{x})^{-K}$ for all $x \in \R^3$.
Then there exist smooth functions $u \colon \R^3 \times [0, \infty) \to \R^3$ and $p \colon \R^3 \times [0, \infty) \to \R$ such that:

1. $u$ and $p$ satisfy the [[Navier–Stokes equations|18KXZHVF4M]] $\partial_t u + (u \cdot \nabla) u = \nu \Delta u - \nabla p$ and $\nabla \cdot u = 0$ on $\R^3 \times [0, \infty)$, with no external force.
2. $u(x, 0) = u_0(x)$ for all $x \in \R^3$.
3. There is a constant $E$ with $\int_{\R^3} \abs{u(x, t)}^2 \, dx \le E$ for all $t \ge 0$.

## Remarks

The problem is open and is one of the seven Millennium Prize Problems of the Clay Mathematics Institute.
The official problem asks for a proof of this statement or of its analogue for periodic data on the torus, or for a counterexample showing a breakdown of smooth solutions.
Charles Fefferman wrote the official problem description.
Smooth solutions are known to exist for a short time for such data, and for all time when the data are small in a suitable norm.
In two space dimensions the corresponding statement is true.
Jean Leray proved that global weak solutions with finite energy exist, see [[existence of Leray weak solutions|DHHR5PZZ7E]].
Caffarelli, Kohn and Nirenberg proved that the singular set of a suitable weak solution has one-dimensional parabolic Hausdorff measure zero.

## Explanations

### L1 human

The Navier–Stokes equations describe how water and air flow.
The question is whether a smooth, calm initial flow in all of space can develop an infinitely sharp feature in finite time.
An example of such a feature would be an infinite speed at a point.
Nobody knows, even though the equations are used every day in engineering.
A proof either way earns a prize of one million dollars.