The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping

This paper concerns the asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping |u|(beta-1)u. We first study the L(2) decay of weak solutions with beta >= 10/3 by developing the classic Fourier splitting method. Second, for 7/2 <= beta < 5, we prove the optimal upper bounds of the higher-order derivative of the strong solution by employing a new analysis technique. Finally, we investigate the asymptotic stability of the large solution to the system with beta >= 7/2 under large initial perturbation. (C) 2010 Elsevier Ltd. All rights reserved.