Vanishing boundary geometry and critical pressure compactness for the three-dimensional Navier--Stokes equations

RESUMO

We develop a boundary blow-up scheme for the three-dimensional nonstationary Navier--Stokes equations in domains whose local boundary graphs belong to $W^{2-1/P_b,P_b}(\mathbb R^2)$ with $P_b>3$. The argument is designed to remove the $P_b>15/4$ restriction in the boundary partial-regularity theorem of Breit. After a rigid rotation to the tangent plane and parabolic rescaling, such a graph becomes small simultaneously in the multiplier classes $\mathcal M^{4/3,3/2}$ and $\mathcal M^{16/15,15/14}$ at the rate $r^{1-3/P_b}$. A two-parameter contradiction argument then sends both the fluid excess and the geometric multiplier norm to zero, producing the standard flat Stokes system in the limit. At the critical pressure pair $(5/3,15/14)$, a localized flat-Stokes decomposition separates a strongly vanishing error pressure from forced and homogeneous pressures. Their decay exponents are respectively $6-15/P_f$ and $12/5-9/Q$, where $P_f>5/2$ and $Q>15/4$. Rough-coefficient errors are absorbed only at the critical multiplier level; higher spatial integrability is invoked only for flat Stokes problems. The resulting Campanato iteration gives boundary Hölder regularity outside a relatively closed set of zero parabolic $5/3$-dimensional Hausdorff measure.