Pressure Anchoring and Weighted Enstrophy in the 3D Navier–Stokes Equations

We introduce a pressure–weighted enstrophy functional for the three–dimensional incompressible Navier–Stokes equations that explicitly couples vorticity amplification to the pressure response while remaining compatible with the Leray–Hopf energy framework. For exponents we derive a closed evolution inequality whose growth coefficient depends only on energy–level quantities and is integrable on finite time intervals. We prove that the weighted enstrophy is locally finite for almost every positive time and upgrade this property to global finiteness almost everywhere via weighted dissipation, Sobolev embedding, and a tail–closure argument. As a consequence, the Beale–Kato–Majda criterion becomes integrable forward in time, yielding smoothness and strong–weak uniqueness beyond a positive time. A geometric interpretation in cylindrical coordinates clarifies how rotational contraction necessarily activates pressure–weighted dissipation, providing a stabilizing anchoring mechanism against singular concentration.