On Decay of the Local Mean Oscillations of the Vorticity Direction in Critical Navier-Stokes Flows

Abstract

We isolate and analyze the geometric PDE governing the evolution of the vorticity direction in the 3D incompressible (unforced) Navier-Stokes equations (NSE), restricted to the case of a critical spatial point singularity where the vorticity magnitude concentrates as $O(|x|^{-2})$, inhabiting the critical Lorentz space $L^{3/2, \infty}$. The PDE consists of the Harmonic Map Heat Flow (HMHF) into the sphere supplemented with the fluid transport, cross-diffusion and tangential strain. The question is whether the NSE mechanics can propagate logarithmic decay of the local mean oscillations of the direction -- the condition $\xi \in \mathrm{bmo}_{1/|\log r|}$ -- which (in this setting) was shown in the companion paper to prevent finite time blow-up. A key observation is that the $O(|x|^{-2})$ concentration, factored out of the viscous cross-diffusion, generates an outward radial drift $4\nu\, x/|x|^2$ at the core. Under a suitable condition we prove a barrier lemma for the total drift-diffusion operator and show that the HMHF nonlinearity is harmless for the quantity $\frac12|\xi - e|^2$ which is a subsolution on any hemisphere. This yields a transfer theorem: the $\mathrm{bmo}_{1/|\log r|}$ regularity of the direction at the core, uniformly up to the singular time, is controlled by a logarithmic modulus in time of the direction at the inner scale, together with the physical strain. Crucially, the strain enters the direction equation only through its tangential component which vanishes precisely when the direction is an eigenvector of the strain tensor. Since this includes the eigenvector carrying the maximal stretching, the result is in contrast to the classical geometric regularity theory which is built on depleting the full vortex-stretching term and thus confined to configurations of weak stretching.