On the Decay and Stability of Global Solutions to the 3D Inhomogeneous Navier-Stokes Equations

In this paper, we investigate the large-time decay and stability to any given global smooth solutions of the 3D incompressible inhomogeneous Navier-Stokes equations. In particular, we prove that given any global smooth solution (a, u) of (1.2), the velocity field u decays to 0 with an explicit rate, which coincides with the L-2 norm decay for the weak solutions of the 3D classical Navier-Stokes system [a, u] as t goes to infinity. Moreover, a small perturbation to the initial data of (a, u) still generates a unique global smooth solution to (1.2), and this solution keeps close to the reference solution (a, u) for t > 0. We should point out that the main results in this paper work for large solutions of (1.2). (C) 2010 Wiley Periodicals, Inc.