Global Strong Solutions to the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Initial Data and Vacuum

Abstract

This paper investigates the three-dimensional axisymmetric compressible Navier-Stokes equations under slip boundary conditions in a cylindrical domain excluding the axis. For initial density allowed to vanish, we establish the global existence and large time asymptotic behavior of strong and weak solutions, provided the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds. It should be noted that these results are obtained without any restrictions on the size of initial data. The key idea is to derive a pointwise estimate of the effective viscous flux by exploiting the axisymmetry of the solutions, along with the conformal mapping and the pull back Green's function, and then to cancel out the singularity using the slip boundary conditions.