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

Abstract

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