Global Existence and Incompressible Limit of the Cauchy Problem for 2D Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data

Abstract

This paper investigates the Cauchy problem for the barotropic compressible Navier-Stokes equations in $\mathbb{R}^2$ with the constant state as far field, which may be vacuum or non-vacuum. Under the assumption of a sufficiently large bulk viscosity coefficient, we establish the global existence and large time behavior of weak, strong, and classical solutions. It should be mentioned that this result is obtained without any restrictions on the size of the initial data. Moreover, we demonstrate 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. The incompressible limit of the weak solutions holds even without requiring the initial velocity to be divergence-free.