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) = ρβ 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 \(\Omega = \{ (r,z)\left| {r = \sqrt {{x^2} + {y^2}} } \right.\), (x, y, z) ∈ ℝ3, r ∈ I ⊂ (0,+∞), −∞ < z < +∞} is obtained. Here the initial density keeps a non-vacuum state \(\bar \rho > 0\) 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.