# P17: Navier-Stokes existence and smoothness

- ID: `P17`
- Reference: `navier-stokes-existence-and-smoothness`
- Page: https://theoremdb.org/statements/P17
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(u_0:\mathbb{R}^3\to\mathbb{R}^3\) be smooth, divergence-free, and rapidly decaying. For every viscosity \(\nu>0\), the three-dimensional incompressible Navier-Stokes equations with initial data \(u_0\) have a globally defined smooth solution satisfying the standard energy bounds.

### Context

Smooth solutions are known locally in time under standard hypotheses. The unresolved issue is whether singular behavior can occur in three dimensions.

### Problem setup

- **Definition (The incompressible Navier-Stokes equations are partial differential equations for a velocity field u and pressure p, with div(u) = 0 and positive viscosity).** The incompressible Navier-Stokes equations are partial differential equations for a velocity field u and pressure p, with div(u) = 0 and positive viscosity.
- **Definition (A global smooth solution exists for every nonnegative time and has the regularity and energy bounds required by the standard problem formulation).** A global smooth solution exists for every nonnegative time and has the regularity and energy bounds required by the standard problem formulation.
- **Remark.** Smooth solutions are known locally in time under standard hypotheses. The unresolved issue is whether singular behavior can occur in three dimensions.

### What counts as a solution

- Prove global existence and smoothness under one of the official three-dimensional formulations, or construct initial data and a solution behavior satisfying an official breakdown alternative, with every analytic condition verified.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Global Leray-Hopf weak solutions exist in three dimensions. Smooth solutions are global in two dimensions and local, or global for suitable small data, in three dimensions. Exact unresolved remainder: For every admissible smooth finite-energy three-dimensional datum, prove global smooth existence and uniqueness, or construct admissible data producing finite-time breakdown. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Global Leray-Hopf weak solutions exist in three dimensions. Smooth solutions are global in two dimensions and local, or global for suitable small data, in three dimensions. Exact unresolved remainder: For every admissible smooth finite-energy three-dimensional datum, prove global smooth existence and uniqueness, or construct admissible data producing finite-time breakdown.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Global Leray-Hopf weak solutions exist in three dimensions. Smooth solutions are global in two dimensions and local, or global for suitable small data, in three dimensions.

Exact unresolved remainder: For every admissible smooth finite-energy three-dimensional datum, prove global smooth existence and uniqueness, or construct admissible data producing finite-time breakdown.

### Background and intake notes

- Original intake status: The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- Status and formulation were checked against the Clay Mathematics Institute page on 2026-07-22.
- The official problem allows specified Euclidean and periodic formulations. Consult it for exact decay, forcing, energy, and breakdown conditions.

### Open directions

- **Route 1** (reported): Prove global existence and smoothness under one of the official three-dimensional formulations, or construct initial data and a solution behavior satisfying an official breakdown alternative, with every analytic condition verified. [1](#reference-1)

### Computational notes

- Numerical simulations can test proposed mechanisms and bounds, but finite-resolution calculations cannot alone rule out or establish a singularity.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `navier-stokes-existence-and-smoothness`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Clay Mathematics Institute, Navier-Stokes Equation, official Millennium Prize Problem page, checked 2026-08-01. Official Problem Description by Charles L. Fefferman; listed under Unsolved Millennium Prize Problems https://www.claymath.org/millennium/navier-stokes-equation/
   - Also cited at Unsolved label and official overview
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source use: original_summary
   - The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Provides the authoritative current status and links the official problem description.
   - Source named by the research packet.
2. <a id="reference-2"></a>Charles L. Fefferman, Existence and Smoothness of the Navier-Stokes Equation, official Clay Mathematics Institute problem description. claymath.org checked 2026-08-01. Official problem description https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf
   - website; primary source; checked 2026-08-01
   - Source use: original_summary
   - Defines the exact alternatives and summarizes established theory.