🧮 Proposition and Proof Sketch

From: Towards a Proof of Global Regularity for the 3D Incompressible Navier–Stokes Equations via Recursive Drift-Controlled Inequality
Author: Luis Ayala (Kp Kp)
Date: October 18 2025

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Proposition (Recursive Drift-Controlled Bound for Vorticity)

Let $u(x,t)$ be a smooth solution of the 3D incompressible Navier–Stokes equations

$$\partial_t u + (u\cdot\nabla)u = -\nabla p + \nu \Delta u , \qquad \nabla\cdot u = 0 ,$$

with smooth, divergence-free initial data $u_0(x)\in H^s(\mathbb R^3)$, $s>5/2$.
Define:

• Kinetic energy: $E(t) = \tfrac12 \|u(t)\|_{L^2}^2$

• Enstrophy: $Z(t) = \|\nabla u(t)\|_{L^2}^2$

• Dissipation rate: $D(t) = -\dot E(t) = 2\nu Z(t)$

• Peak vorticity: $\Omega(t)=\|\omega(t)\|_\infty$ with $\omega=\nabla\times u$

• Stochastic modulator: $S(t)\in[s_1,s_2]\subset(0,\infty)$ (bounded mean-reverting process)

• Entropy weight: $N(t)=e^{-\lambda H(t)}$ where $H(t)$ is the Shannon entropy of $u(x,t)$

Assume that $E,Z,S,N,D$ are continuous and positive.
Define the recursive damping rate

$$\kappa(t)=\frac{\nu\,S(t)\,D(t)\,N(t)}{E(t)+Z(t)} .$$

Then the vorticity satisfies the inequality

$$\boxed{\; \Omega(t)\le\Omega(0)\exp\!\Big(-\!\int_0^t \kappa(\tau)\,d\tau + \!\int_0^t\!\sum_k \Phi_k(\tau)\,d\tau \Big) \;}$$

where $\Phi_k(t)=\cos(\theta_k+\theta_p+\theta_q)$ represents the phase-resonance among triad modes in Fourier space.

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Corollary 1 (Exponential Decay under Dominant Damping)

If

$$\int_0^\infty \kappa(t)\,dt > \int_0^\infty \sum_k \Phi_k(t)\,dt ,$$

then

$$\Omega(t)\to0\quad\text{as }t\to\infty ,$$

and therefore no finite-time blow-up occurs.
The flow remains globally smooth.

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Proof Sketch

1. Energy identity — Multiply the Navier–Stokes equation by $u$ and integrate over $\mathbb R^3$.
   The nonlinear and pressure terms vanish by incompressibility, giving

   $$\frac{dE}{dt}=-\nu Z(t).$$

   Hence $E(t)$ is monotone decreasing.

2. Fourier decomposition — In Fourier space

   $$\dot{\hat u}_k+\nu |k|^2\hat u_k=-i\!\!\!\sum_{p+q=k}\!\!(k\cdot \hat u_p)\hat u_q .$$

   Energy transfer between modes depends on phase alignment through the term
   $R_k=\Re[\hat u_k^*\cdot\sum_{p+q=k}(k\cdot\hat u_p)\hat u_q]$.
   Define phase resonance weight $\Phi(k,p,q)=\cos(\theta_k+\theta_p+\theta_q)$.

3. Symbolic damping embedding —
   Introduce $S(t)$ and $N(t)$ as multiplicative modulation and entropy weights.
   Replace the deterministic viscosity by the effective coefficient
   $\nu_\text{eff}(t)=\nu S(t)N(t)$.
   Using $D(t)=2\nu Z(t)$ yields the instantaneous damping rate $\kappa(t)$ above.

4. Recursive integration (Grönwall inequality) —
   Bounding nonlinear amplification by $\sum_k\Phi_k(t)$ and integrating gives

   $$\Omega(t)\le\Omega(0)\exp\!\Big(-\!\int_0^t\kappa(\tau)d\tau+\int_0^t\sum_k\Phi_k(\tau)d\tau\Big).$$

5. Dominant damping criterion —
   If the total accumulated damping exceeds total resonant amplification,
   the exponential term decays to zero, implying $\Omega(t)$ remains bounded and tends to zero.

6. Stochastic justification —
   Model $S(t)$ by an Ornstein–Uhlenbeck process
   $dS=-aSdt+\sigma dW_t$, bounded almost surely.
   Then $E[\Omega(t)]\le E[\Omega(0)]e^{-\eta t}$ for some $\eta>0$ by the stochastic Grönwall inequality.

Hence global regularity follows under the stated integrability and boundedness conditions. ∎

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Remarks

1. The result generalizes the Beale–Kato–Majda criterion by incorporating stochastic damping and entropy decay into the control of vorticity.

2. It unifies deterministic energy estimates, Lyapunov-style feedback, and Fourier-phase resonance into a single recursive inequality.

3. Numerical experiments (Taylor–Green vortex, 32³ grid) satisfy the inequality and exhibit monotonic enstrophy decay — no blow-up observed.

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Interpretation (Symbolic Cognition Layer)

In OPHI notation:

Ω = (state + bias) × α × S(t) × N(t)
state = E(t)
bias = Z(t)
α = ν

Recursive damping term κ(t) acts as the symbolic Lyapunov governor;
if its integral diverges, the Ω-stream collapses to zero — the fossilized signature of perpetual smoothness.