Global Existence and Incompressible Limit for Compressible Navier-Stokes Equations in Bounded Domains with Large Bulk Viscosity Coefficient and Large Initial Data

We investigate the barotropic compressible Navier-Stokes equations with the Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain. For initial density that is allowed to vanish, we establish the global existence and exponential decay of weak, strong, and classical solutions when the bulk viscosity coefficient is suitably large, without any restrictions on the size of the initial data. Furthermore, we prove that when the bulk viscosity coefficient tends to infinity, the solutions of the compressible Navier-Stokes equations converge to those of the inhomogeneous incompressible Navier-Stokes equations. The key idea is to utilize the logarithmic interpolation inequality on general bounded domains and apply the compensated compactness lemma.