Global Existence and Incompressible Limit for the Two-Dimensional Compressible Navier-Stokes Equations in a Half-Space with Large Initial Data and Vacuum

This paper concerns the barotropic compressible Navier-Stokes equations in a two-dimensional half-space subject to Navier-slip boundary conditions with vacuum or non-vacuum far-field density. The global existence and large-time behavior of weak and strong solutions are established under the assumption that the bulk viscosity coefficient is sufficiently large. It should be remarked that this result is obtained without any restrictions on the size of the initial data. For strong solutions, we derive some a priori decay estimates for the spatial gradient of the velocity field that are uniform with respect to the bulk viscosity coefficient, which play a crucial role in establishing the time-uniform upper bound for the density. Furthermore, we prove that, as 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. In particular, the incompressible limit for weak solutions holds without requiring the initial velocity to be divergence-free.