Vanishing boundary geometry and critical pressure compactness for the three-dimensional Navier--Stokes equations
Abstract
We prove boundary partial regularity for the three-dimensional nonstationary Navier--Stokes equations in bounded domains with fractional Sobolev boundary graphs whose integrability exponent is greater than three. Within the fractional Sobolev graph setting, this lowers the boundary threshold from fifteen quarters to three. We work in Breit's boundary suitable weak-solution framework, with the mixed maximal-regularity pair $(5/3,15/14)$ explicitly included in the solution assumptions. The main difficulty is to recover pressure compactness without imposing higher regularity on the rough boundary coefficients. After a rigid rotation to the tangent plane and parabolic rescaling, we show that each boundary graph becomes small in both Sobolev multiplier classes needed for the local energy inequality and the critical Stokes estimate. Geometry and fluid excess enter a contradiction argument as independent small parameters, so the limiting problem is the standard flat Stokes system. A localized pressure decomposition separates the vanishing nonlinear and coefficient errors from the forced and homogeneous flat pressures. The error pressure converges strongly at the critical mixed integrability pair, whereas higher spatial integrability is used only for the flat problems. These estimates yield a one-step excess decay bound with a constant independent of the contraction radius. A physical-scale iteration, followed by interior--boundary Campanato patching, gives local parabolic Hölder continuity of the velocity at every regular boundary point when the force has integrability exponent greater than five halves. The boundary singular set is relatively closed and has zero parabolic Hausdorff measure in dimension five thirds. The geometric decay determines a uniform starting scale and does not restrict the final regularity exponent.
Versions
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BibTeX
@misc{wang2026vanishing,
title = {{Vanishing boundary geometry and critical pressure compactness for the three-dimensional Navier--Stokes equations}},
author = {Leyang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.13223},
eprint = {2609.13223},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13223},
}
RIS
TY - UNPB
TI - Vanishing boundary geometry and critical pressure compactness for the three-dimensional Navier--Stokes equations
AU - Leyang Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.13223
UR - https://arxiv.org/abs/2609.13223
AB - We prove boundary partial regularity for the three-dimensional nonstationary Navier--Stokes equations in bounded domains with fractional Sobolev boundary graphs whose integrability exponent is greater than three. Within the fractional Sobolev graph setting, this lowers the boundary threshold from fifteen quarters to three. We work in Breit's boundary suitable weak-solution framework, with the mixed maximal-regularity pair $(5/3,15/14)$ explicitly included in the solution assumptions. The main difficulty is to recover pressure compactness without imposing higher regularity on the rough boundary coefficients. After a rigid rotation to the tangent plane and parabolic rescaling, we show that each boundary graph becomes small in both Sobolev multiplier classes needed for the local energy inequality and the critical Stokes estimate. Geometry and fluid excess enter a contradiction argument as independent small parameters, so the limiting problem is the standard flat Stokes system. A localized pressure decomposition separates the vanishing nonlinear and coefficient errors from the forced and homogeneous flat pressures. The error pressure converges strongly at the critical mixed integrability pair, whereas higher spatial integrability is used only for the flat problems. These estimates yield a one-step excess decay bound with a constant independent of the contraction radius. A physical-scale iteration, followed by interior--boundary Campanato patching, gives local parabolic Hölder continuity of the velocity at every regular boundary point when the force has integrability exponent greater than five halves. The boundary singular set is relatively closed and has zero parabolic Hausdorff measure in dimension five thirds. The geometric decay determines a uniform starting scale and does not restrict the final regularity exponent.
ER -