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

Abstract Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0 − 1 ∈ H s+1(R2), u0 ∈ H s(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