Global smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with variable viscosity

Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0 − 1 ∈ Hs+1(ℝ2), u0 ∈ Hs(ℝ2) ∩ Ḣ−ε(ℝ2) for s > 2 and 0 < ε < 1, the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid. Furthermore, the L2 decay rate of the velocity field is obtained.