Solving the Millennium Problem with AI is not new. Javier Gómez Serrano (Mathematician, Brown University) teamed up with the Google DeepMind team.

Below, I propose a conceptual whitepaper—created with OpenAI ChatGPT—to address one of the Millennium Prize Problems.

Introduction

The Clay Millennium Problem on the 3D incompressible Navier–Stokes equations asks whether smooth solutions with smooth initial data remain smooth for all time, or if finite-time singularities (blow-up) can occur. Despite major progress (local well-posedness, global regularity in 2D, partial regularity results, small-data theorems in critical spaces), the 3D case with large data remains open.

This note outlines a conceptual research program that integrates overlooked structural elements of the equations:

• Geometric depletion of vortex stretching (inspired by Viktor Schauberger’s natural vortex observations),
• Sparsity of intermittent singular sets (Caffarelli–Kohn–Nirenberg), and
• Pressure as a global stabilizer.

The central idea is to convert vortex geometry into a quantitative damping factor in scale-critical estimates, closing the gap in the current proof strategy.

Additionally, in the future, the condition of the boundary layer that defines a liquid will be explored. This is not yet part of the conceptual whitepaper.

Precise physical conditions under which the Navier-Stokes equations’ underlying assumptions break down.

Executive Summary by Storm Gini Stanford AI Tool: 3D Navier Stokes Millenium Problem

Navier–Stokes Framework

We consider the incompressible Navier–Stokes equations on \mathbb{R}^3:

\begin{cases} \partial_t u + (u\cdot\nabla)u = -\nabla p + \nu \Delta u + f, \ \nabla \cdot u = 0, \quad u(x,0) = u_0(x), \end{cases}

with smooth divergence-free u_0 and smooth forcing f.

The unknowns are the velocity field u(x,t) \in \mathbb{R}^3 and pressure field p(x,t) \in \mathbb{R}.

Overlooked Structural Elements

3.1 Vorticity Alignment

Define the vorticity \omega = \nabla \times u. The vortex stretching term is
(\omega \cdot \nabla)u = S\omega, \quad S = \tfrac12(\nabla u + \nabla u^\top).
If \omega aligns with the eigenvector of S corresponding to its maximal eigenvalue, stretching is maximal. Otherwise, the stretching weakens.

We define the alignment deficit:
\mathcal{A}(x,t) := 1 – (\xi(x,t)\cdot e_{\max}(x,t))^2,
where \xi = \omega/|\omega|. Numerics suggest \mathcal{A} is often nontrivial in real flows, but it has not been fully exploited analytically.

3.2 Sparsity of Singular Sets

Caffarelli–Kohn–Nirenberg (1982) proved that possible singularities occupy a set of parabolic Hausdorff dimension ≤ 1. This indicates that regions of extreme steepness are sparse, yet most analyses ignore this sparseness when estimating nonlinear terms.

3.3 Pressure Stabilization

The pressure satisfies the Poisson equation:
\Delta p = -\nabla \cdot \nabla \cdot (u\otimes u).
Traditionally pressure is projected away (Helmholtz–Leray). But as a nonlocal constraint, pressure redistributes stresses and can dampen coherent growth of steepness, especially on sparse sets.

Proposed Lemmas

Lemma 1 (Geometric ε-Regularity, local)

There exists \varepsilon > 0 such that if for a parabolic cylinder Q_r(x_0,t_0):
[
\Big(\fint_{Q_r} |u|^3 + |p|^{3/2}\Big) \cdot \Big(\fint_{Q_r} \mathcal{A}\Big) < \varepsilon,
]
then u is smooth at (x_0,t_0).

This strengthens classical ε-regularity by factoring in alignment.

Lemma 2 (Dyadic Flux Inequality with Alignment)

For dyadic block u_j at frequency scale 2^j:
\frac{d}{dt}|u_j|_2^2 \;\le\; -c\nu 2^{2j}|u_j|_2^2 + C(1-\cos^2\theta_j)\,\Phi_j(u),
where \theta_j is the average vorticity–strain angle at scale 2^j, and \Phi_j(u) the nonlinear flux.

This shows that alignment deficit directly damps the critical energy cascade.

Lemma 3 (Pressure–Sparsity Bound)

On a parabolic cylinder Q_r where the set {|\nabla u| > \Lambda} is \alpha-sparse:
[
\fint_{Q_r} \lambda_{\max}(\nabla^2 p)\, \chi_{{|\nabla u|>\Lambda}} \le C(\alpha)\,\fint_{Q_r} |u|^2/r^2.
]
This prevents pressure Hessian from reinforcing stretching in sparse regions.

Rigidity Argument

Assume blow-up occurs. Rescaling yields a nontrivial ancient mild solution bounded in a critical norm (e.g. L^\infty_t L^3_x or BMO^{-1}).

• Lemma 1 ensures local smoothness wherever alignment deficit persists.
• Lemma 2 ensures top-scale damping of energy flux.
• Lemma 3 ensures pressure cannot sustain coherent stretching on sparse singular sets.

Together, these imply the ancient solution must vanish — a rigidity contradiction, excluding finite-time blow-up.

Interpretation: Schauberger’s “Implosion vs Explosion”

Schauberger described vortices as stabilizers (implosion) vs destabilizers (explosion). In Navier–Stokes terms:

• Implosion = alignment deficit > 0 ⇒ stretching depleted ⇒ smoothness preserved.
• Explosion = perfect alignment ⇒ dangerous stretching ⇒ potential blow-up.

Thus his intuition aligns with the analytic mechanism we propose.

Conclusion & Outlook

This program integrates geometric depletion, sparsity, and pressure redistribution into a single framework. Proving Lemmas 1–3 would yield the missing scale-critical estimate and close the Navier–Stokes global regularity problem.

Next steps:

1. Prove Lemma 1 rigorously by modifying CKN ε-regularity.
2. Establish Lemma 2 via dyadic paraproduct estimates.
3. Develop Lemma 3 with Calderón–Zygmund theory and sparsity.
4. Attempt rigidity proof for ancient solutions.

References

• Caffarelli, Kohn, Nirenberg (1982) — partial regularity.
• Escauriaza, Seregin, Šverák (2003) — L^3-regularity criterion.
• Constantin, Fefferman, Majda (1996) — geometric depletion.
• Koch, Tataru (2001) — critical BMO^{-1} well-posedness.
• Viktor Schauberger (1940s–50s) — vortex observations, implosion vs explosion.

Literature Review with Gemini Advance based on this Research Note

Using State of the Art Problem Solving for the Navier-Stokes Equations