navier_stokes_global_regularity

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Navier-Stokes existence and smoothness (Millennium Prize Problem): For any smooth, divergence-free initial velocity field u₀ in R³, does the Navier-Stokes equation have a globally smooth solution for all time? Clay Institute offers $1M for a resolution.

Why this node was retired

The posted statement is

import Mathlib

theorem navier_stokes_global_regularity :
    ∀ u₀ : (Fin 3 → ℝ) → (Fin 3 → ℝ),
      ContDiff ℝ ⊤ u₀ →
      (∀ x : Fin 3 → ℝ,
        ∑ i : Fin 3, fderiv ℝ (fun y => u₀ y i) x (Pi.basisFun ℝ (Fin 3) i) = 0) →
      ∃ u : ℝ → (Fin 3 → ℝ) → (Fin 3 → ℝ),
        (∀ t : ℝ, ContDiff ℝ ⊤ (u t)) ∧ u 0 = u₀ := by
  sorry

The conclusion is only a smooth extension in time; no Navier–Stokes equation is imposed.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

For smooth divergence-free initial data in a specified finite-energy or periodic class and viscosity ν>0, ask for globally smooth u,p on t≥0 satisfying ∂ₜu+(u·∇)u=νΔu−∇p, div u=0, and u(0)=u₀. State the spatial domain, decay/periodicity and energy conditions explicitly. This is the substantive Navier–Stokes problem, not a proved repair.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.