On the decay and stability of global solutions to the 3‐D inhomogeneous Navier‐Stokes equations

Abstract

Abstract In this paper, we investigate the large‐time decay and stability to any given global smooth solutions of the 3‐D 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 3‐D classical Navier‐Stokes system [26,29] as t goes to ∞. 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). © 2010 Wiley Periodicals, Inc.