  This paper concerns the global existence for classical solutions problem to
the 3D Density-Dependent Viscosity barotropic compressible Navier-Stokes in
$\Omega$ with slip boundary condition, where $\Omega$ is a simply connected
bounded $C^{\infty}$ domain in $\mathbb{R}^3$ and its boundary only has a
finite number of 2-dimensional connected components. By a series of a priori
estimates, we show that the classical solution to the system exists globally in
time under the assumption that the initial energy is suitably small. The
initial density of such a classical solution is allowed to have large
oscillations and contain vacuum states. We also adopt some new techniques and
methods to obtain necessary a priori estimates, especially the boundary
integral terms estimates. This is the first result concerning the global
existence of classical solutions to the compressible Navier-Stokes equations
with density containing vacuum initially and viscosity coefficients depending
on density for general 3D bounded smooth domains.
