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

TL;DR

This paper investigates the zero-viscosity limit of the Navier-Stokes equations with Navier friction boundary conditions, establishing convergence to Euler and Prandtl equations under specific conditions on the boundary parameter.

Contribution

It provides a rigorous justification of the zero-viscosity limit for Navier-Stokes with Navier boundary conditions, covering different regimes of the boundary parameter γ.

Findings

Convergence to Euler and Prandtl equations for γ=1 with analytic data.

Convergence to Euler and linearized Prandtl equations for γ in (0,1) with Gevrey class data.

Clarifies the impact of the boundary friction parameter on the limit behavior.

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