Navier-Stokes Finite-Time Blowup Visualizer Singularity Proof

Extracted Initial Data & Interactive Numerical Cascade from openai/NavierStokesAndEuler

Target Singular Time: T* = 1.0000s
Current Vorticity Max ||ω||_∞
12.45
Enstrophy ℰ(t)
1.84e+02
Time Remaining (T* - t)
1.000000
t=0.0 t=0.000

Controls viscous dissipation rate (-νΔu). Set to 0 for Euler flow.

Spatial wavepacket frequency scale for initial perturbation.

Self-similar vortex stretching exponent: ||ω||_∞ ~ (T* - t)⁻⁷.

Initial Parameters Extracted from Proof

The problem: Existence and smoothness of the Navier-Strokes equation

Solution parameters derived from the Lean 4 formalized proof repository (openai/NavierStokesAndEuler) based on the Córdoba–Martínez-Zoroa & Alpöge–Buckmaster multiscale wavepacket induction scheme.

Governing Navier-Stokes PDE
Initial Velocity Field & Wavepacket Parameters
Domain & Symmetry T³ = [0, 2π]³ (Periodic)

Odd-symmetric initial profile with zero spatial mean

Viscosity Coefficient ν ∈ (0, 10⁻³]

Viscous dissipation term νΔu

Wavepacket Frequencies λ_n λ_n = λ₀ · 2ⁿ (n = 0..N)

Geometric sequence of high-frequency modes

Initial Mode Amplitudes a_n a_n = A₀ · 2⁻ⁿ⁽¹⁺α⁾

Rapidly decaying envelope ensuring C^∞ regularity

Spacetime Smooth Forcing Term F(t, x) F ∈ C^∞([0, T*] × T³)

Bounded high-frequency background force counteracting viscous dissipation during scale transitions

Beale-Kato-Majda (BKM) Blowup Criterion

A smooth solution u(x,t) breaks down at time T* if and only if the time integral of peak vorticity ||ω(t)||_L∞ diverges to infinity. As t → T*, the vorticity concentrates near the origin x* = 0 according to: