Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier-Stokes Equations

We prove a finite-scale estimate for vortex stretching in spatially filtered three-dimensional Navier--Stokes flow.

The positive near-field part of the filtered stretching is bounded by a pairwise defect of filtered vorticity directions.

A magnitude-weighted direction inequality converts this angular defect into a first-order difference quotient of filtered vorticity, and the resulting term is absorbed by filtered diffusion up to a lower-order enstrophy reservoir.

In the localized filtered enstrophy balance, the remaining positive surplus is assigned to far-field strain, commutator forcing, and localization residuals.

The far-field term is reduced to weighted packing and conditional annular Carleson embedding.

The differentiated commutator stress is controlled by a scale-invariant increment defect adapted to the filter and its derivative.

At the critical exponent, bounded increment defects generate cylindrical generalized Young-measure profiles.