  In this paper we investigate the issue of the inviscid limit for the
compressible Navier-Stokes system in an impermeable fixed bounded domain. We
consider two kinds of boundary conditions. The first one is the no-slip
condition. In this case we extend the famous conditional result obtained by
Kato in the homogeneous incompressible case. Kato proved that if the energy
dissipation rate of the viscous flow in a boundary layer of width proportional
to the viscosity vanishes then the solutions of the incompressible
Navier-Stokes equations converge to some solutions of the incompressible Euler
equations in the energy space. We provide here a natural extension of this
result to the compressible case. The other case is the Navier condition which
encodes that the fluid slips with some friction on the boundary. In this case
we show that the convergence to the Euler equations holds true in the energy
space, as least when the friction is not too large. In both cases we use in a
crucial way some relative energy estimates proved recently by Feireisl, Ja Jin
and Novotn{\'y}.
