We investigate regularity estimates for the stationary Navier–Stokes equations above a highly oscillating Lipschitz boundary with the no-slip boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the boundary layer thickness. We also obtain an improved \(C^{1,\mu }\) estimate and identify the building blocks of the regularity theory, dubbed ‘Navier polynomials’. In the case when some structure is assumed on the oscillations of the boundary, for instance periodicity, these estimates can be seen as local error estimates. Although we handle the regularity of the nonlinear stationary Navier–Stokes equations, our results do not require any smallness assumption on the solutions.

我们研究了在具有无滑移边界条件的高度振荡的Lipschitz边界上方的平稳Navier-Stokes方程的正则性估计。我们的主要结果是在大于边界层厚度的尺度上改进了Lipschitz正则性估计。我们还获得了改进的\（C ^ {1，\ mu} \）估计，并确定了被称为“ Navier多项式”的正则性理论的基本组成部分。在边界的振荡假设某种结构（例如周期性）的情况下，这些估计可以看作是局部误差估计。尽管我们处理了非线性平稳的Navier–Stokes方程的正则性，但我们的结果并不需要对解做任何细微的假设。