Global Smooth Solutions to the 2-D Inhomogeneous Navier-Stokes Equations with Variable Viscosity

Under the assumptions that the initial density p0 is close enough to 1 and ρ0 - 1 ∈ H~(s+1)(R~2),u0 ∈ H~s(R~2) ∩ H~(-ε)(R~2) 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 L~2 decay rate of the velocity field is obtained.