Global well-Posedness of classical solutions to the isentropic compressible Navier-Stokes equations with anisotropic viscous-stress tensor in 3D bounded domains

In this paper, we consider the initial boundary value problem of isentropic compressible Navier-Stokes equations with anisotropic viscous stress tensors in a three-dimensional smooth bounded domain under slip boundary conditions. Through a rigorous analysis comprising careful energy estimates, subtle examinations of boundary effects, and commutator estimates for mollifier, we prove the existence and uniqueness of the global classical solution under some small assumptions of the initial value and small perturbation of the viscous stress tensor. Furthermore, we establish the exponential decay rates for the time of both density and velocity. To the best of our knowledge, this result gives the first global classical solution of the compressible Navier-Stokes equations with anisotropic viscous stress tensors in 3D bounded domains.