Weak and strong solutions for the incompressible Navier–Stokes equations with damping

In this paper, we show that the Cauchy problem of the Navier–Stokes equations with damping α|u|β−1u (α>0) has global weak solutions for any β⩾1, global strong solution for any β⩾7/2 and that the strong solution is unique for any 7/2⩽β⩽5.