Long time decay of solutions to the 3D incompressible Navier–Stokes equations with nonlinear damping

Abstract

This paper is interested in the decay of solutions to 3D Navier–Stokes equations with a nonlinear damping term αfalse|ufalse|β−1u$$ \alpha {\left|u\right|}^{\beta -1}u $$. We discuss the bounds of the global strong solutions in the negative Sobolev space trueH˙−s$$ {\dot{H}}^{-s} $$ and establish the optimal decay rates in L2$$ {L}^2 $$ of the global strong solutions. Moreover, the upper bound of the derivative is also obtained. Finally, we investigate the asymptotic stability of the solutions to the system.