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

TL;DR

This paper establishes improved regularity estimates for the stationary Navier-Stokes equations over highly oscillating boundaries, introducing Navier polynomials and applicable to structured boundary oscillations without smallness assumptions.

Contribution

It provides new Lipschitz and $C^{1,rac{1}{ ext{mu}}}$ regularity estimates for Navier-Stokes solutions over oscillating boundaries, including the concept of Navier polynomials.

Findings

Enhanced Lipschitz regularity at scales larger than boundary layers

Introduction of Navier polynomials as regularity building blocks

Regularity results applicable without smallness conditions on solutions

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.