Paper 150F: Coherent Aligned Patch Control in High-Vorticity Navier–Stokes Flow

Paper 150F addresses a specific loophole in the high-vorticity pinching program for the three-dimensional incompressible Navier-Stokes equations. Earlier papers developed the idea that strong vorticity can create directional pinching, and that this pinching may reduce the ability of the strain field to keep stretching the vortex. Paper 150E identified a remaining escape route: even if most of a high-vorticity region is strongly pinched, dangerous positive stretching might still be carried by small coherent aligned patches.

This paper studies whether such aligned patches can remain dominant while surrounded by pinched flow. It defines aligned patches as high-vorticity regions where the vorticity direction remains strongly aligned with the strain tensor. It then introduces positive aligned-patch dominance, which measures how much of the positive high-vorticity stretching reservoir is carried by those patches. This positive-part diagnostic is emphasized because signed stretching can hide dangerous positive stretching through cancellation.

The proposed control mechanism is boundary leakage. If an aligned patch is surrounded by a pinched region, then its boundary must connect a coherent aligned interior to a directionally disordered exterior. That boundary may carry directional mismatch, strain-tracking demand, loss of alignment, or fragmentation cost. The paper formulates a conditional patch-control estimate: aligned patches cannot remain fully dominant if boundary leakage is sustained, leakage reaches the stretching-active core on the relevant time scale, and fragmentation does not preserve stretching at negligible dissipative cost.

The paper does not prove global Navier-Stokes regularity. Instead, it isolates a necessary bridge in the 150-series proof pathway: coherent aligned patches must fail to carry nearly all dangerous positive high-vorticity stretching under sustained boundary leakage. If this patch-control mechanism can be proved or supported in well-resolved simulations, it strengthens the strain-tracking obstruction developed in Paper 150E and prepares the next step: controlling the remaining enstrophy remainder in Paper 150G.