Global Classical Solutions to the 3D Density-Dependent Viscosity Compressible Navier-Stokes Equations with Navier-Slip Boundary Condition in a Simply Connected Bounded Domain

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.