三维Navier--Stokes方程的边界几何消失与临界压力紧致性
Vanishing boundary geometry and critical pressure compactness for the three-dimensional Navier--Stokes equations
- Mathematics and Science College, Shanghai Normal University（上海师范大学数理学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出一种边界爆破方案，消除Breit定理中$P_b>15/4$的限制，通过双参数矛盾论证和临界压力分解，在粗糙边界上建立Navier--Stokes方程的边界Hölder正则性。
中文摘要 AI 辅助
我们针对三维非定常Navier--Stokes方程，在局部边界图属于$W^{2-1/P_b,P_b}(\mathbb R^2)$（其中$P_b>3$）的区域中，发展了一种边界爆破方案。该论证旨在消除Breit边界部分正则性定理中$P_b>15/4$的限制。在刚性旋转至切平面并进行抛物型重标度后，这样的边界图在乘子类$\mathcal M^{4/3,3/2}$和$\mathcal M^{16/15,15/14}$中同时以$r^{1-3/P_b}$的速率变得小。随后，一个双参数矛盾论证将流体过剩量和几何乘子范数同时趋于零，在极限情形下产生标准的平坦Stokes系统。在临界压力对$(5/3,15/14)$处，一个局部化的平坦Stokes分解将强消失的误差压力与强迫压力和齐次压力分离开来。它们的衰减指数分别为$6-15/P_f$和$12/5-9/Q$，其中$P_f>5/2$且$Q>15/4$。粗糙系数误差仅在临界乘子水平被吸收；更高的空间可积性仅用于平坦Stokes问题。由此得到的Campanato迭代给出了在零抛物型$5/3$维Hausdorff测度的相对闭集之外的边界Hölder正则性。
英文摘要
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.
