Convergence rates for the stationary and non-stationary Navier–Stokes equations over non-Lipschitz boundaries

Abstract

In this paper, we consider the convergence rates for the 2D stationary and non-stationary Navier–Stokes Equations over highly oscillating periodic bumpy John domains with C2 regularity in some neighborhood of the boundary point (0,0). For the stationary case, using the variational equation satisfied by the solution and the correctors for the bumpy John domains obtained by Higaki and Zhuge [Arch. Ration. Mech. Anal. 247(4), 66 (2023)] after correcting the values on the inflow/outflow boundaries ({0} ∪ {1}) × (0, 1), we can obtain an O(ɛ3/2) approximation in L2 for the velocity and an O(ɛ3/2) convergence rates in L2 approximated by the so called Navier’s wall laws, which generalized the results obtained by Jäger and Mikelić [J. Differ. Equations 170(1), 96–122 (2001)]. Moreover, for the non-stationary case, using the energy method, we can obtain an O(ɛ3/2 + exp(−Ct)) convergence rate for the velocity in Lx2.