Endpoint Energy Atoms Force Local Pressure Concentration in Three-Dimensional Navier-Stokes Flow

Abstract

We prove that a point atom in an endpoint kinetic-energy measure of a three-dimensional incompressible Navier-Stokes flow on the flat torus forces quantitative concentration of the actual pressure. At each restart time tau, a same-state constraint-response comparison evolves u(tau) by the pressure-free componentwise advection-diffusion equation driven by u, and subtracts the resulting passive field z_tau. Writing r_tau = u - z_tau and g_tau = Q z_tau, we obtain an exact relative pressure-work identity. If the atom has mass m, Nash smoothing makes z_tau terminally non-atomic while r_tau retains terminal atomic mass at least m; hence, after every fixed restart, the terminal limsup of the accumulated pressure work is at least m/2. Mesoscopic localization yields scale-explicit lower bounds for gauge-invariant L2 pressure oscillation and for the local L2 mass of the pressure gradient in shrinking terminal cylinders. Consequently, on every terminal neighborhood of the atomic point, the pressure modulo functions of time and its gradient fail to be square integrable. The relative pressure-work principle extends to constrained solenoidal Oseen evolutions: as tau approaches T*, the zero-initial responses vanish in weak-tail topologies, although every fixed restart retains the same lower bound. This identifies a necessary pressure-work mechanism, generated by the incompressibility constraint, that links endpoint atoms to local pressure concentration, without requiring a full regularity hypothesis.