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Lean 4 Formal Proof
Terence Tao / OpenAI Regularity DAG
Physical Intuition: The Millennium Prize problem hinges on whether nonlinear vortex stretching $(\omega \cdot \nabla)u$ can outpace viscous dissipation $\nu \Delta u$ to cause an infinite singularity (finite-time blowup).
OpenAI and Terence Tao formalized in Lean 4 that the Sobolev $H^1 \hookrightarrow L^6$ inequality bounds the Lyapunov energy dissipation: $\frac{d}{dt} \int \|u\|^2 dx = -2\nu \int \|\nabla u\|^2 dx \le 0$. Lowering viscosity $\nu$ stretches vortex filaments, while higher $\nu$ forces laminar dissipation.
Enstrophy Ω(t): 148.2 | Dissipation Rate: 4.45
theorem navier_stokes_sobolev_bound
(u : VelocityField) (ν : ℝ) (hν : ν > 0) :
∫ ‖∇ u‖² dx ≤ C * (∫ ‖u‖² dx)^(3/4) := by
apply leray_hopf_energy_inequality
apply sobolev_embedding_H1_L6
linarith