The Navier Wall Law at a Boundary with Random Roughness

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size epsilon << 1. In the parent paper [8], we derived a homogenized boundary condition of Navier type as epsilon -> 0. We show here that for a large class of boundaries, this Navier condition provides a O(epsilon(3/2)vertical bar ln epsilon vertical bar(1/2)) approximation in L(2), instead of O(epsilon(3/2)) for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.