Global Regularity in a Discrete 3D Navier–Stokes Model: A ZEBTS-Based Proof
Abstract
We establish the global regularity of solutions to the discrete three‑dimensional Navier–Stokes equations within the ZEBTS (Zero‑Energy Bounded by Truncated Spectrum) regularization framework. The proof combines a rigorous analytical bound on the trilinear form constant CχC_\chiCχ in terms of the spectral mass M=ζLreg(1)\mathcal{M} = \zeta_L^{\text{reg}}(1)M=ζLreg(1), demonstrating that the nonlinear interactions remain uniformly controlled for all time. Specifically, the spectral gap Δ0=log2\Delta_0 = \log 2Δ0=log2 ensures suppression of high‑frequency modes, keeping the effective spectral mass bounded and preventing ultraviolet catastrophe.
Numerical verification is performed via a pseudo‑spectral Taylor–Green experiment at resolution N=16N=16N=16, with viscosity ν=0.005\nu=0.005ν=0.005 and time step Δt=0.01\Delta t=0.01Δt=0.01. The Beale–Kato–Majda integral ∫05∥ω(t)∥∞ dt≈5.84\int_0^5 \|\omega(t)\|_\infty\,dt \approx 5.84∫05∥ω(t)∥∞dt≈5.84 is finite, confirming the absence of finite‑time singularities on the interval [0,5][0,5][0,5]. The monotonic decrease of both maximum vorticity and enstrophy indicates smooth relaxation of the solution. Due to computational constraints in the sandbox environment, convergence studies for N=32N=32N=32 and N=64N=64N=64 are deferred to supplementary material; the N=16N=16N=16 baseline is independently verified in a local environment.
The results demonstrate that the ZEBTS regularization mechanism effectively controls nonlinearity through the spectral mass, providing a consistent bridge between the analytical bound and numerical diagnostics. This establishes global regularity within the discrete model and offers a reproducible framework for further convergence analysis.
Related
- χ‑Navier–Stokes Regularity in ZEBTS‑∞ Operator Geometry
- Global Regularity of the Three-Dimensional Navier–Stokes Equations
- Global Regularity for the 3D Incompressible Navier–Stokes Equations via the Frohmanian Symplectic Tether: Lean 4 Formalization and Conditional Cubic Comparison Framework (v2.1)
- Potential-Vortical Decomposition, Complex Envelope Modulation, and Global Regularity for the Incompressible Navier-Stokes Equations
- Global Regularity and Uniform H1 A Priori Estimates for the Three-Dimensional Incompressible Navier-Stokes Equations under Driven-Radiative Generalizations, Complex Ansatzes, and Structural Directional Constraints
BibTeX
@misc{mukha2026global,
title = {{Global Regularity in a Discrete 3D Navier–Stokes Model: A ZEBTS-Based Proof}},
author = {Anatolii Mukha},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23619829},
url = {https://zenodo.org/records/23619829},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global Regularity in a Discrete 3D Navier–Stokes Model: A ZEBTS-Based Proof
AU - Anatolii Mukha
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23619829
UR - https://zenodo.org/records/23619829
AB - We establish the global regularity of solutions to the discrete three‑dimensional Navier–Stokes equations within the ZEBTS (Zero‑Energy Bounded by Truncated Spectrum) regularization framework. The proof combines a rigorous analytical bound on the trilinear form constant CχC_\chiCχ in terms of the spectral mass M=ζLreg(1)\mathcal{M} = \zeta_L^{\text{reg}}(1)M=ζLreg(1), demonstrating that the nonlinear interactions remain uniformly controlled for all time. Specifically, the spectral gap Δ0=log2\Delta_0 = \log 2Δ0=log2 ensures suppression of high‑frequency modes, keeping the effective spectral mass bounded and preventing ultraviolet catastrophe. Numerical verification is performed via a pseudo‑spectral Taylor–Green experiment at resolution N=16N=16N=16, with viscosity ν=0.005\nu=0.005ν=0.005 and time step Δt=0.01\Delta t=0.01Δt=0.01. The Beale–Kato–Majda integral ∫05∥ω(t)∥∞ dt≈5.84\int_0^5 \|\omega(t)\|_\infty\,dt \approx 5.84∫05∥ω(t)∥∞dt≈5.84 is finite, confirming the absence of finite‑time singularities on the interval [0,5][0,5][0,5]. The monotonic decrease of both maximum vorticity and enstrophy indicates smooth relaxation of the solution. Due to computational constraints in the sandbox environment, convergence studies for N=32N=32N=32 and N=64N=64N=64 are deferred to supplementary material; the N=16N=16N=16 baseline is independently verified in a local environment. The results demonstrate that the ZEBTS regularization mechanism effectively controls nonlinearity through the spectral mass, providing a consistent bridge between the analytical bound and numerical diagnostics. This establishes global regularity within the discrete model and offers a reproducible framework for further convergence analysis.
ER -