# Pressure–Hessian Barrier and Enstrophy Regulation in 3D Navier–Stokes
This repository contains numerical experiments, analysis scripts, and
supporting documentation for the investigation of a pressure–Hessian
barrier observable in the 3D incompressible Navier–Stokes equations.
## Core Observable
We define the pressure–Hessian barrier functional
B_ω(T) = ∫₀ᵀ ∫ max(0, R(x,t) − 1) |ω(x,t)|² dx dt
where
R = (ωᵀ S ω) / |ωᵀ H_p ω|,
S is the strain-rate tensor, H_p is the pressure Hessian, and ω is vorticity.
This quantity measures enstrophy production during intervals where
pressure curvature fails to oppose vortex stretching.
## Key Findings
- B_ω(T) rapidly saturates at O(1) after initial dynamics.
- No monotonic growth with Reynolds number is observed.
- Later-time spikes are dynamically negligible due to low enstrophy weight.
- Numerical evidence supports a Reynolds-independent plateau.
## Scope of Claims
This work establishes:
- A physically interpretable scalar observable tied to vortex stretching.
- Numerical evidence of bounded cumulative enstrophy production.
- A conditional reduction of BKM regularity criteria to bounded B_ω.
This work does NOT claim:
- An unconditional proof of Navier–Stokes regularity.
- A closed-form a priori bound on B_ω.
## Reproducibility
All simulations use pseudospectral DNS of the Taylor–Green vortex.
Scripts are provided under code/ and results under data/.
See docs/methodology\_notes.md for numerical stability details.
## Citation
If you use this work, please cite the Zenodo DOI provided for this record.