Zero-viscosity limit of the Navier-Stokes equations with the Navier friction boundary condition

In this paper, we consider the zero-viscosity limit of the Navier-Stokes equations in a half space with the Navier friction boundary condition $$ (\beta u^{\varepsilon}-\varepsilon^{\gamma}\partial_y u^{\varepsilon})|_{y=0}=0, $$ where $\beta$ is a constant and $\gamma\in (0,1]$. In the case of $\gamma=1$, the convergence to the Euler equations and the Prandtl equation with the Robin boundary condition is justified for the analytic data. In the case of $\gamma\in (0,1)$, the convergence to the Euler equations and the linearized Prandtl equation is justified for the data in the Gevrey class $\frac 1 {\gamma}$.