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Analysis and PDE Millennium Prize Problem Status: Open Status last verified

Navier–Stokes Existence and Smoothness

Do smooth, physically reasonable solutions of the 3D incompressible Navier–Stokes equations exist for all time, or can they break down in finite time?

Problem statement

For the incompressible Navier–Stokes equations in R^3 (or on the 3-torus) with smooth, suitably decaying initial data, prove global existence of smooth solutions with bounded energy — or exhibit a breakdown. The four precise alternatives are in Fefferman's official problem description.

∂u/∂t + (u·∇)u = ν Δu − ∇p + f
∇·u = 0
u(x, 0) = u₀(x)

Source: Charles L. Fefferman — Existence and Smoothness of the Navier–Stokes Equation (official problem description)

Introduction

The Navier–Stokes equations describe how a viscous fluid such as water or air moves. Engineers solve them numerically every day, yet mathematicians cannot prove that in three dimensions a smooth starting flow stays smooth forever.

The prize asks for one of two things: a proof that smooth solutions with finite energy exist for all time (in the settings specified), or an example where a smooth solution develops a singularity in finite time.

It is one of the Clay Millennium Prize Problems.

Definitions

Velocity u, pressure p
The unknowns: the fluid velocity field and the pressure.
Viscosity ν
A positive constant measuring the fluid's internal friction.
Weak solution
A solution in an averaged (integral) sense that may fail to be smooth.

What is known

Each statement says how certain it is: an established (proved) result, an experimental observation, an open conjecture, or an unverified claim.

  • Established result Global weak solutions with finite energy exist for any finite-energy initial data (Leray, 1934); their uniqueness and smoothness in 3D are open. [Charles L. Fefferman]
  • Established result In two space dimensions, smooth solutions exist for all time. [Charles L. Fefferman]
  • Established result Partial regularity: for suitable weak solutions, the set of possible singular points is small (one-dimensional parabolic Hausdorff measure zero). [L. Caffarelli, R. Kohn, L. Nirenberg]
  • Established result An averaged version of the 3D equations admits finite-time blowup (Tao, 2016) — a barrier for proofs that use only energy and certain harmonic-analysis estimates. [Terence Tao]
  • Established result Weak solutions with finite kinetic energy are not unique in general (Buckmaster and Vicol, 2019). [Tristan Buckmaster, Vlad Vicol]
  • Experimental observation Numerical simulations of turbulent flows show no confirmed blowup, but simulations cannot resolve arbitrarily small scales.

Important limitations

  • A numerical simulation can suggest but not prove regularity or blowup: discretization and round-off hide the smallest scales.
  • Results for the 2D equations, for averaged models or for weak solutions do not settle the 3D smooth problem.

Status history

  1. Status: Open Named one of the seven Millennium Prize Problems by the Clay Mathematics Institute.
  2. Status: Open Checked against the Clay Mathematics Institute page: still unsolved.

This status is maintained by the Zerethon IDE team from authoritative sources. Contributions, runs and AI answers never change it.