  We study the inviscid limit problem for the incompressible Navier-Stokes
equation on a half-plane with a Navier boundary condition depending on the
viscosity. On one hand, we prove the $L^2$ convergence of Leray solutions to
the solution of the Euler equation. On the other hand, we show the nonlinear
instability of WKB expansions in the stronger $L^{\infty}$ and $\dot{H}^s$
(s>1) norms.
