Regularity for the stationary Navier–Stokes equations over bumpy boundaries and a local wall law

Abstract

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.