The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions

Abstract

We study boundary layer solutions of the isentropic, compressible Navier-Stokes equations with Navier-friction boundary conditions when the viscosity constants appearing in the momentum equation are proportional to a small parameter ε. These boundary conditions are characteristic for the underlying inviscid problem, the compressible Euler equations. The boundary condition implies that the velocity on the boundary is proportional to the tangential component of the stress. The normal component of velocity is zero on the boundary. We first construct a high-order approximate solution that exhibits a boundary layer. The main contribution to the layer appears in the tangential velocity and is of width √ ε and amplitude O(√ ε). Next we prove that the approximate solution stays close to the exact Navier-Stokes solution on a fixed time interval independent of ε. As an immediate corollary we show that the Navier-Stokes solution converges in L∞ in the small viscosity limit to the solution of the compressible Euler equations with normal velocity equal to zero on the boundary.