euler_equations_blowup

⚠️ Retired — specification defect

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

3D Euler equations finite-time blow-up: Does there exist a smooth divergence-free initial velocity field for the incompressible Euler equations in ℝ³ for which the solution develops a singularity in finite time? Recent work by Hou–Luo (2014) and Chen–Hou (2022) provides strong numerical and analytical evidence for blow-up.

Why this node was retired

The posted statement is

import Mathlib

theorem euler_equations_blowup :
    ∃ (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, 0 ≤ t → ContDiff ℝ ⊤ (u t)) ∧ u 0 = u₀ := by
  sorry

The supposed Euler blow-up is nonexistence of any smooth extension; with no Euler PDE constraint the constant-in-time extension always exists.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Require u,p to satisfy the incompressible Euler equations ∂ₜu+(u·∇)u=−∇p and div u=0 with the given initial data, in a specified spatial domain and regularity class. Formulate finite-time loss of regularity of the maximal solution, with energy and decay/periodic conditions as appropriate.

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.