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

Abstract

We investigate the barotropic compressible Navier-Stokes equations with Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain. For initial data allowing vacuum, we establish the global existence and exponential decay of weak, strong, and classical solutions when the bulk viscosity coefficient is sufficiently large, without any restrictions on the size of the initial data. Furthermore, we prove that, as the bulk viscosity coefficient tends to infinity, solutions of the compressible Navier-Stokes equations converge to those of the inhomogeneous incompressible Navier-Stokes equations. In particular, we establish the incompressible limit of weak solutions without assuming that the initial velocity field is divergence-free.