Uniform Regularity and Vanishing Viscosity Limit for the Compressible Navier--Stokes with General Navier-Slip Boundary Conditions in Three-Dimensional Domains

Abstract

In this paper, we investigate the uniform regularity for the isentropic compressible Navier--Stokes system with general Navier-slip boundary conditions (1.6) and the inviscid limit to the compressible Euler system. It is shown that there exists a unique strong solution of the compressible Navier--Stokes equations with general Navier-slip boundary conditions in an interval of time which is uniform in the vanishing viscosity limit. The solution is uniformly bounded in a conormal Sobolev space and is uniformly bounded in $W^{1,\infty}$. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. In particular, it is proved that the velocity will be uniformly bounded in $L^\infty(0,T;H^2)$ when the boundary is flat and the Navier--Stokes system is supplemented with the vorticity free boundary condition (1.25). Based on such uniform estimates, we prove the convergence of the viscous solutions to the inviscid ones in $L^\infty(0,T;L^2)$, $L^\infty(0,T;H^1)$ and $L^\infty([0,T]\times\Omega)$ with a rate of convergence.