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 $$(\beta u^{\epsilon }-{\epsilon }^{\gamma }{\partial }_y{u}^{\epsilon }){|}_{y=0}=0,$$ 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 $\frac{1}{\gamma }$.

59 pages

References in corpus (1)

• On the ill-posedness of the Prandtl equation