Global weak solutions and incompressible limit to the isentropic compressible Navier-Stokes equations in the half-plane with ripped density and large initial data

We prove the global existence of weak solutions to the isentropic compressible Navier-Stokes equations with ripped density in the half-plane under a slip boundary condition provided the bulk viscosity coefficient is properly large. Moreover, we show that such solutions converge globally in time to a weak solution of the inhomogeneous incompressible Navier-Stokes equations as the bulk viscosity coefficient tends to infinity. In particular, the large initial data and an initial patch of density as well as a vacuum are allowed. Our method relies on a Desjardins-type logarithmic interpolation inequality and some new techniques based on the effective viscous flux.