Logarithmic Depletion of Vortex Stretching and Singularity Evasion in the 3D Navier-Stokes Equations

Abstract

We present a geometric-analytic mechanism for the suppression of finite-time singularities in the 3D incompressible (unforced) Navier-Stokes equations for critical point singularities exhibiting $L^{3/2, \infty}$ spatial concentration of vorticity. We demonstrate that if the vorticity direction resides locally in a logarithmically weighted space of bounded mean oscillations, $\mathrm{bmo}_{1/|\log r|}$ -- a space failing the Dini condition and thus permitting wild oscillatory defects -- the non-linear vortex stretching is fundamentally depleted. By isolating a unidirectional geometric cancellation, we recast the stretching eigenvalue as a singular integral commutator. Utilizing a localized Coifman-Rochberg-Weiss estimate coupled with dyadic BMO tail bounds, we prove the stretching potential vanishes as a logarithmic envelope on shrinking super-level sets. This depletion forces the vorticity magnitude into a sub-critical Lorentz-Zygmund space via interpolated De Giorgi energy method. The logarithmic gain is subsequently transferred to the velocity field, forcing the geometric scale of local 1D sparseness below the uniform radius of spatial analyticity, ultimately averting the finite-time blow-up via the harmonic measure maximum principle.