Boundary layer analysis of the Navier-Stokes equations with Generalized Navier boundary conditions

Abstract: We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $\epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1, 1) tensor on the boundary. When the tensor is a multiple of the identity we obtain Navier boundary conditions, and when the tensor is the shape operator we obtain conditions in which the vorticity vanishes on the boundary. By constructing an explicit corrector, we prove the convergence of the Navier-Stokes solutions to the Euler solution as the viscosity vanishes. We do this both in the natural energy norm with a rate of order $\epsilon^{3/4}$ as well as uniformly in time and space with a rate of order $\epsilon^{3/8 - \delta}$ near the boundary and $\epsilon^{3/4 - \delta'}$ in the interior, where $\delta, \delta'$ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.