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|β−1u|u|β−1u. We first study the L2L2 decay of weak solutions with β≥10/3β≥10/3 by developing the classic Fourier splitting method. Second, for 7/2≤β<57/2≤β<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 β≥7/2β≥7/2 under large initial perturbation.