Finite-Scale One-Component Regularity via Harmonic Pressure for the 3D Navier-Stokes Equations

Abstract

We study a finite-scale one-component regularity mechanism for suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations. The results are organized in three layers. The first layer is unconditional. Under a fixed scale-invariant local bound Phi(1)=A(1)+E(1)+C(1)+D(1) <= M, smallness of the critical vertical-component quantity C_3(1)=int_{Q_1} |u_3|^3 dx dt yields a positive lower bound, depending only on M, for the local regularity radius at the origin. The proof converts one-component smallness into approximation by the two-and-a-half-dimensional limiting class and then into Caffarelli--Kohn--Nirenberg smallness at a smaller scale. The pressure approximation is measured in a quotient by spatially harmonic functions. This pressure topology reflects a genuine obstruction: time-dependent harmonic pressures may have bounded scale-invariant L^{3/2}-oscillation while their pointwise gradients lie beyond the control provided by the available scale-invariant quantities. The second layer is a conditional logarithmic refinement. A prepared two-shadow comparison package replaces the abstract compactness modulus by a logarithmic modulus and gives a logarithmic finite-scale decay. The third layer is a conditional relaxed-shadowing refinement. The comparison class is enlarged to smooth no-stretching horizontal flows V=(v_h,0), with the comparison pressure allowed to have partial_3 pi not equal to 0. The resulting vertical residual pairs with the small component u_3 in the relative-energy identity. Under the buffered strong-flow and localized relaxed stability inputs stated below, this gives a power-type relaxed harmonic approximation and a power-type finite-scale decay. The unconditional theorem is separated from the logarithmic and power-type assumptions; the latter two layers identify the quantitative stability mechanisms needed to upgrade the compactness modulus.