Global Solutions to a 3D Axisymmetric Compressible Navier-Stokes System with Density-Dependent Viscosity

In this paper, we consider the 3D compressible isentropic Navier-Stokes equations when the shear viscosity μ is a positive constant and the bulk viscosity is λ ( ρ ) = ρ β with β > 2, which is a situation that was first introduced by Vaigant and Kazhikhov in [1]. The global axisymmetric classical solution with arbitrarily large initial data in a periodic domain Ω = { (r,z)| r = √(x^2 + y^2). , ( x, y, z ) ∈ ℝ 3 , r ∈ I ⊂ (0,+∞), −∞ < z < +∞ is obtained. Here the initial density keeps a non-vacuum state ρ̅> 0 when z → ±∞. Our results also show that the solution will not develop the vacuum state in any finite time, provided that the initial density is uniformly away from the vacuum.