Strong solution of 3D generalized Navier–Stokes equations with damping

This paper studies the Cauchy problem of the 3D generalized Navier-Stokes equations with damping term $\alpha |u|^{\sigma -2}u$ ( $\sigma \geq 2$ ). There are two choices for the stress tensor: $S(\nabla u)=|\nabla u|^{q-2}\nabla u+\nabla u$ and $S(\nabla u)=|\nabla u|^{q-2}\nabla u$ . For both cases, we consider the global existence of strong solutions by using pure energy method.