Solving the Three-Dimensional Incompressible Navier–Stokes Global Regularity Problem by Structural Decomposition and Normalized Growth Analysis
Abstract
This work studies the three-dimensional incompressible Navier-Stokes equations through structural decomposition and normalized enstrophy-growth intervals. Starting from a hypothetical finite maximal smooth time T_* < infinity, first-hitting times are constructed at which the enstrophy doubles. Navier-Stokes scaling normalizes every such interval from enstrophy 1 to 2. For the normalized palinstrophy cost q_n and energy cost J_n, we derive the block estimate
J_n >= c_nu * (q_n + 1/2)^4 / q_n^3,
together with weighted summability constraints and exact successive-scale relations. These imply that every hypothetical finite-time doubling cascade must satisfy
integral from t_n to T_* of ||grad omega(t)||_2^2 dt = infinity
for every n. Moreover, for the truncated terminal tails
T_n^(N) = sum from k=0 to N of 2^k * q_(n+k),
the successive tails satisfy the exact relation
T_n^(N) - q_n = 2 * T_(n+1)^(N-1),
and hence
(T_n^(N) - q_n) / T_(n+1)^(N-1) = 2
for every N >= 1. Thus, although both aligned terminal tails diverge as N -> infinity, their relative scale factor remains exactly preserved. The resulting scalar constraints remain mutually compatible and therefore do not exclude T_* < infinity. The argument yields necessary structural conditions for a hypothetical finite-time singularity, but does not establish global regularity.
Related
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- Global Regularity for the 3D Incompressible Navier–Stokes Equations via the Frohmanian Symplectic Tether: Lean 4 Formalization and Conditional Cubic Comparison Framework (v2.1)
- Global Analytic Gevrey Regularity, Solenoidal Feshbach–Schur Dissipation, and Non-Commutative Spectral Turbulence in 3D Incompressible Rotating Navier–Stokes Flows
- Asymptotically Hölder gradient regularity for degenerate or singular parabolic normalized $ p$-Laplace equations
BibTeX
@misc{tran2026solving,
title = {{Solving the Three-Dimensional Incompressible Navier–Stokes Global Regularity Problem by Structural Decomposition and Normalized Growth Analysis}},
author = {Son Tuyet Tran},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23493417},
url = {https://zenodo.org/records/23493417},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Solving the Three-Dimensional Incompressible Navier–Stokes Global Regularity Problem by Structural Decomposition and Normalized Growth Analysis
AU - Son Tuyet Tran
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23493417
UR - https://zenodo.org/records/23493417
AB - This work studies the three-dimensional incompressible Navier-Stokes equations through structural decomposition and normalized enstrophy-growth intervals. Starting from a hypothetical finite maximal smooth time T_* < infinity, first-hitting times are constructed at which the enstrophy doubles. Navier-Stokes scaling normalizes every such interval from enstrophy 1 to 2. For the normalized palinstrophy cost q_n and energy cost J_n, we derive the block estimate J_n >= c_nu * (q_n + 1/2)^4 / q_n^3, together with weighted summability constraints and exact successive-scale relations. These imply that every hypothetical finite-time doubling cascade must satisfy integral from t_n to T_* of ||grad omega(t)||_2^2 dt = infinity for every n. Moreover, for the truncated terminal tails T_n^(N) = sum from k=0 to N of 2^k * q_(n+k), the successive tails satisfy the exact relation T_n^(N) - q_n = 2 * T_(n+1)^(N-1), and hence (T_n^(N) - q_n) / T_(n+1)^(N-1) = 2 for every N >= 1. Thus, although both aligned terminal tails diverge as N -> infinity, their relative scale factor remains exactly preserved. The resulting scalar constraints remain mutually compatible and therefore do not exclude T_* < infinity. The argument yields necessary structural conditions for a hypothetical finite-time singularity, but does not establish global regularity.
ER -