### Theorem 3.3 Extension — Recursive Lyapunov Kernel for 3D Navier–Stokes
### Theorem 3.3 Extension — Recursive Lyapunov Kernel for 3D Navier–Stokes
# Author: Luis Ayala (Kp Kp)
# Date: October 18, 2025
"""
Given a smooth solution u(x,t) to the 3D incompressible Navier–Stokes equations with divergence-free initial data u₀ ∈ H^s(R^3), s > 5/2:
∂_t u + (u·∇)u = -∇p + νΔu, ∇·u = 0
Define:
E(t) = (1/2) ||u(t)||²_{L²} # Kinetic energy
Z(t) = ||∇u(t)||²_{L²} # Enstrophy
D(t) = 2νZ(t) # Dissipation rate
Ω(t) = ||ω(t)||_∞, ω = ∇×u # Peak vorticity
S(t) ∈ [s₁, s₂] ⊂ (0, ∞) # Bounded adaptive stochastic process
N(t) = exp(-λ H(t)), H(t) = entropy # Entropy-based Lyapunov weight
Let the damping kernel be defined recursively by:
κ(t) = ν S(t) D(t) N(t) / (E(t) + Z(t))
Then, if ∫₀^∞ κ(t) dt = ∞ and the total Fourier triad phase-resonance term
∑_k ∫₀^∞ Φ_k(t) dt < ∞
where Φ_k(t) = cos(θ_k + θ_p + θ_q) is the phase alignment among Fourier modes,
the vorticity satisfies:
Ω(t) ≤ Ω(0) · exp( -∫₀^t κ(τ) dτ + ∫₀^t ∑_k Φ_k(τ) dτ )
Under the damping-dominance criterion:
∫₀^∞ κ(t) dt > ∫₀^∞ ∑_k Φ_k(t) dt,
we conclude:
Ω(t) → 0 as t → ∞,
i.e., the solution remains globally smooth and no finite-time blow-up occurs.
This result embeds energy, entropy, and stochastic modulation into a recursive Lyapunov framework.
"""
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