Global classical solutions for 3D compressible Navier–Stokes equations with vacuum and a density-dependent viscosity coefficient

In this paper, we prove the global existence of classical solutions to the three-dimensional (3D) compressible Navier–Stokes equations with a density-dependent viscosity coefficient ( λ = λ ( ρ ) ) provided the initial data is of small energy. This in particular implies that the solutions may have large oscillations and contain vacuum states. As a result of the uniform estimates, the large-time behavior of the solution is also studied. The result obtained generalizes those results in Zhang (2011) [39] and Huang et al. (2012) [17] where the non-vacuum initial data and the constant viscosity coefficients are considered, respectively.