Asymptotic behavior of the 3D incompressible Navier–Stokes equations with damping

In this paper, we consider the 3D incompressible Navier–Stokes equations with damping term ${\left|u\right|}^{\beta -1}u(\beta \ge 1).$ First, by using a different and simple method from Cai and Lei (2010), Jia et al. (2011), Jiang (2012) and Yu and Zheng (2019), for any $\beta \ge 1,$ we prove that the weak solutions decay to zero in $L^2$ as time tends to infinity; for any $\beta \ge 3,$ we derive optimal decay rates of the $L^2$-norm of the solutions. Second, we obtain the decay rate with some appropriate space weighted estimates, which is the first result on the 3D damped Navier–Stokes equations to our knowledge.