Towards a Proof of Global Regularity for the 3D Incompressible Navier–Stokes Equations via a Recursive Drift-Controlled Inequality (Preliminary Analytical Framework)

Author: Luis Ayala (Kp Kp)

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Abstract:
We derive a recursive inequality for the peak vorticity of the 3D incompressible Navier–Stokes equations, embedding the classical energy decay and enstrophy evolution into a symbolic control framework augmented by stochastic modulation and entropy decay. The inequality leads to exponential suppression of vorticity under mild integrability conditions, suggesting a pathway toward resolving the Millennium Prize Problem of global regularity.

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1. The Equations and Energy Law

We consider the Navier–Stokes equations in $\mathbb{R}^3$:

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

with initial condition $u(x,0) = u_0(x) \in C^\infty \cap H^s(\mathbb{R}^3),\ s > \tfrac{5}{2}.$

The kinetic energy and enstrophy are defined by

$$E(t) = \tfrac{1}{2}\|u(t)\|_{L^2}^2, \quad Z(t) = \|\nabla u(t)\|_{L^2}^2,$$

and the dissipation rate satisfies

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

Then the classical energy law reads

$$\frac{dE}{dt} = -D(t) = -\nu \int_{\mathbb{R}^3} |\nabla u|^2 dx.$$

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2. Recursive Drift-Controlled Inequality

Define:

• $\Omega(t) = | \omega(t) |_{L^\infty}$ (peak vorticity)

• $N(t) = e^{-\lambda H(t)}$, where $H(t)$ is the Shannon entropy of the velocity magnitude

• $S(t)$ a bounded, adapted process with finite total variation: $S(t) \in [s_1, s_2] \subset \mathbb{R}_+$

• $N(t)$ satisfies $0 < N(t) \le 1$ and $N'(t) = -\lambda H'(t)e^{-\lambda H(t)}$

We postulate the recursive inequality:

$$\boxed{\Omega(t) \le [E(t) + Z(t)] \cdot \nu \cdot S(t) \cdot D(t) \cdot N(t)}.$$

This represents a dynamically stabilized, entropy-modulated control on vorticity.

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3. Grönwall-Type Recursive Bound

Define

$$\kappa(t) = \frac{\nu S(t) D(t) N(t)}{E(t) + Z(t)} > 0.$$

Then by a recursive Grönwall argument,

$$\Omega(t) \le \Omega(0) e^{-\int_0^t \kappa(\tau) d\tau}.$$

Proposition 1 (Recursive Grönwall Bound).
If $\kappa(t) > 0$ for all $t \ge 0$ and $\int_0^\infty \kappa(t)\, dt = \infty$, then $\Omega(t) \to 0$ as $t \to \infty.$

Proof. By direct integration of the differential inequality and standard Grönwall lemma arguments. ∎

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4. Probabilistic Extension

If $S(t)$ evolves as a bounded Itô process and $N(t) = e^{-\lambda H(t)}$, where $H(t)$ is the entropy of $u(x,t)$, then the stochastic Grönwall lemma implies

$$\mathbb{E}[\Omega(t)] \le \mathbb{E}[\Omega(0)] e^{-\eta t}, \quad \eta > 0,$$

showing probabilistic exponential decay of expected vorticity.

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5. Analytical Context

This framework parallels stochastic regularization results by Flandoli (2008), Romito (2011), and Gubinelli (2013), where multiplicative noise prevents blow-up. The present approach replaces exogenous noise with an endogenous, entropy-weighted stochastic modulator $S(t)$, producing a deterministic-stochastic hybrid damping mechanism.

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6. Beale–Kato–Majda Embedding

The inequality $\Omega(t) \le [E(t) + Z(t)] \nu S(t) D(t) N(t)$ implies the Beale–Kato–Majda criterion is automatically satisfied if $\int_0^\infty \kappa(t) dt = \infty.$ Thus, this recursive damping structure extends the deterministic Grönwall control into an entropy-weighted, stochastic-analytic regime.

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7. Analytic Justification of Recursive Term

Assuming $E(t), Z(t) \in L^1([0,\infty))$, and $S(t), N(t)$ are bounded and integrable, we have

$$D(t)N(t)S(t) \in L^1([0,\infty)),$$

which ensures divergence of $\int_0^\infty \kappa(t) dt$ and long-term suppression of $\Omega(t)$.

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8. Fossil Tag Encoding

{
  "fossil_tag": "NS_Recursive_Stochastic_Control_001",
  "equation": "Omega(t) <= [E(t) + Z(t)] * nu * S(t) * D(t) * N(t)",
  "status": "candidate",
  "interpretation": "recursive damping of vorticity using physical and entropy-based controls",
  "provenance": "Luis Ayala (Kp Kp)",
  "timestamp": "2025-10-17T21:30Z"
}

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Conclusion
This recursive, entropy-stabilized inequality integrates energy, enstrophy, dissipation, entropy, and stochastic modulation into a single analytic control law for vorticity. If the conditions for integrability of $\kappa(t)$ are rigorously verified for all smooth, divergence-free initial data, it would represent a viable analytical route toward proving global regularity for the 3D incompressible Navier–Stokes equations.