Recursive Lyapunov Framework for 3D Navier–Stokes Regularity

Author: Luis Ayala (Kp Kp)
Date: October 2025

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Abstract

We introduce a conditional Lyapunov framework for global regularity of the 3D incompressible Navier–Stokes equations. The formulation employs a recursive damping inequality for peak vorticity, combining spectral entropy, energy dissipation, and bounded stochastic modulation. Under mild integrability hypotheses, the Lyapunov kernel guarantees exponential decay of vorticity, implying global smoothness via the Beale–Kato–Majda criterion.

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1. Setting and Notation

Domain: $\mathbb{T}^3$ (periodic)
Equations:

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

Initial data: $u_0\in H^s(\mathbb{T}^3),\, s>5/2$, divergence-free.

Definitions:

$$E(t)=\tfrac12\!\int|u|^2dx,\quad Z(t)=\!\int|\nabla u|^2dx,\quad D(t)=2\nu Z(t),\quad \omega=\nabla\times u,\quad \Omega(t)=\|\omega\|_{L^\infty}.$$

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2. Symbolic Control Quantities

$$E_k=\tfrac12|\widehat u(k)|^2,\quad H(t)=\!\sum_k E_k\log(E_k+\varepsilon),\quad N(t)=e^{-\lambda H(t)},\quad S(t)\in[s_1,s_2]\subset(0,\infty)$$ $$\kappa(t)=\frac{2\nu^2S(t)Z(t)N(t)}{E(t)+Z(t)}.$$

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3. Conditional Lyapunov Regularity Theorem

Assume

$$\dot\Omega(t)\le -\kappa(t)\Omega(t)+R(t), \quad \int_0^\infty\!\kappa(t)dt=\infty, \quad \int_0^\infty\!R(t)dt<\infty.$$

Then $\Omega(t)\to0$ as $t\to\infty$; hence $u$ is globally smooth.

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4. Lemma Roadmap

A. $L^p$ vorticity evolution inequality
B. Calderón–Zygmund control
C. Gagliardo–Nirenberg bridge
D. Triad (paraproduct) decomposition
E. Entropy evolution → $N(t)$ bounds
F. Stochastic modulator boundedness
G. Closure: $L^p\to L^\infty$ vorticity control

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5. Numerical Program

• Simulate Taylor–Green vortex

• Track $\Omega(t),\kappa(t),H(t),S(t),N(t)$

• Verify $\int\kappa(t)dt\gg\int R(t)dt$

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6. Conclusion

The recursive Lyapunov kernel translates spectral entropy decay into nonlinear damping. If the kernel’s integral dominates residual forcing, global regularity follows. The framework unifies deterministic PDE control with symbolic (entropy-weighted) stability—ready for analysis by PDE and SPDE specialists.