Recursive Lyapunov Framework for 3D Navier–Stokes Regularity

Author: Luis Ayala (Kp Kp)
Affiliation: OPHI / OmegaNet / ZPE-1
Date: October 18, 2025

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Abstract

We propose a recursive Lyapunov framework for controlling vorticity growth in the three-dimensional incompressible Navier–Stokes equations. The approach integrates classical energy dissipation, enstrophy evolution, entropy weighting, stochastic modulation, and Fourier-space phase resonance into a single stability inequality governing peak vorticity.

The central bound takes the form

\Omega(t) \le \Omega(0)\exp\left(-\int_0^t \kappa(\tau)d\tau + \int_0^t \sum_k \Phi_k(\tau)d\tau\right)

where $\kappa(t)$ represents a recursive damping kernel and $\Phi_k(t)$ measures nonlinear phase-resonant amplification. If accumulated damping dominates total resonance, vorticity decays and finite-time blow-up cannot occur.

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1. Governing Equations

Consider the 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

$$u_0 \in H^s(\mathbb{R}^3), \quad s>5/2.$$

Define the vorticity

$$\omega = \nabla \times u .$$

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2. Energy and Enstrophy

Kinetic energy

$$E(t)=\frac12 \|u(t)\|_{L^2}^2$$

Enstrophy

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

Dissipation rate

$$D(t)=2\nu Z(t)$$

Energy obeys

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

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3. Peak Vorticity

$$\Omega(t)=\|\omega(t)\|_\infty$$

Regularity is tied to vorticity growth.
By the Beale–Kato–Majda criterion,

$$\int_0^T \|\omega(t)\|_\infty dt < \infty$$

precludes finite-time blow-up.

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4. Recursive Lyapunov Kernel

Introduce two bounded modulation terms.

Stochastic modulation

$$S(t)\in[s_1,s_2]\subset(0,\infty)$$

Entropy weight

$$N(t)=e^{-\lambda H(t)}$$

Define the recursive damping kernel

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

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5. Fourier Phase Resonance

In Fourier space

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

Define phase-alignment weight

$$\Phi_k(t)=\cos(\theta_k+\theta_p+\theta_q).$$

These terms measure constructive triad interactions that amplify vorticity.

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6. Recursive Drift-Controlled Inequality

The vorticity evolution obeys

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

Thus flow stability becomes a competition between

damping via $\kappa(t)$

and nonlinear resonance via $\Phi_k(t)$.

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7. Damping-Dominance Criterion

If

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

then

$$\Omega(t)\to0,$$

implying global smoothness.

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8. Technical Clarifications

8.1 Definition of Entropy $H(t)$

Two possible rigorous definitions exist.

Option A — Velocity Distribution Entropy

Construct a probability density $\rho_u(v,t)$ for the velocity field and define

$$H(t)= -\int_{\mathbb{R}^3}\rho_u(v,t)\log\rho_u(v,t)\,dv .$$

This entropy can be estimated via kernel density reconstruction from the velocity field.

Option B — Ensemble Entropy

In statistical turbulence frameworks,

$$H(t)= -\mathbb{E}[\log P(u(x,t))]$$

where $P$ is the distribution of flow realizations.

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8.2 Boundedness of $S(t)$

Model $S(t)$ as an Ornstein–Uhlenbeck process

$$dS=-a(S-\bar S)dt+\sigma dW_t .$$

For suitable parameters the process remains positive and bounded.

A reflecting boundary at $S=\varepsilon>0$ ensures strict positivity.

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8.3 Divergence of the Damping Integral

From

$$\frac{dE}{dt}=-2\nu Z(t)$$

we obtain

$$D(t)=2\nu Z(t).$$

Thus

$$\kappa(t) = \frac{\nu S(t)D(t)N(t)}{E(t)+Z(t)} = \frac{2\nu^2 Z(t)S(t)N(t)}{E(t)+Z(t)}.$$

If

$$S(t)\ge s_1>0, \quad N(t)\ge n_1>0$$

and $E(t)$ does not decay faster than $Z(t)$, then

$$\kappa(t)\sim O(1).$$

Hence

$$\int_0^\infty \kappa(t)dt=\infty .$$

Formal proof requires precise decay estimates.

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8.4 Relation to Classical Lyapunov Approaches

Traditional Lyapunov analyses rely on fixed energy functionals.

The present framework introduces a recursive Lyapunov kernel

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

which dynamically adapts based on energy, entropy, and stochastic modulation.

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9. Numerical Illustration

Taylor–Green vortex simulation

grid: $32^3$

viscosity: $\nu=0.01$

Results show

monotonic enstrophy decay
stable energy dissipation
no blow-up.

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10. Symbolic Interpretation (OPHI Layer)

Within OPHI symbolic notation

$$\Omega=(\text{state}+\text{bias})\alpha S(t)N(t)$$

state → kinetic energy $E(t)$

bias → enstrophy $Z(t)$

α → viscosity $ν$

The kernel $κ(t)$ functions as a recursive stability governor.

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11. Status

The recursive Lyapunov framework proposes a candidate pathway for controlling vorticity growth.

Remaining tasks include

• deriving the damping kernel directly from the Navier–Stokes equations
• establishing rigorous entropy evolution
• bounding the total phase resonance contribution
• verifying the damping-dominance condition for all smooth initial data.

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Fossil Tag

NS_Recursive_Stochastic_Control_001

Proof hash

a34eab6db6f82fe7ac802bac17651c008c67669520d75751c33ccfa88289be67