All open problems
Source labels openChecked July 26, 2026
Millennium ProblemsPartial differential equations
Existence And Smoothness Of The Navier–Stokes Equation
(D) Breakdown of Navier–Stokes Solutions on ℝ³/ℤ³.
Mathematical statement
(D) Breakdown of Navier–Stokes Solutions on ℝ³/ℤ³.
Statement source: Millennium Problems statement material
Statement terms: Source-specific
Source-specific terms. Therefore does not assert reuse rights beyond attributed display.
Statement artifacts, not proofs
These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.
Pinned Lean formulation 1
navier_stokes_breakdown_periodic
Complete statement target, proof intentionally absentLean 4
theorem navier_stokes_breakdown_periodic (nu : ℝ) (hnu : nu > 0) : ∃ (u₀ : ℝ³ → ℝ³) (f : ℝ³ → ℝ → ℝ³), InitialVelocityConditionPeriodic u₀ ∧ ForceConditionPeriodic f ∧ ¬ (∃ v p, NavierStokesExistenceAndSmoothnessPeriodic nu u₀ f v p) := by sorry- Statement source
- Formal Conjectures
- Lean version
- v4.27.0
- Placeholder
- Present; no proof artifact
- Source evidence
- Pinned source index
- Fidelity review
- Community formulation
References