Decay results of solutions to the incompressible Navier–Stokes flows in a half space

In a half space, we consider the asymptotic behavior of the strong solution for the non-stationary Navier–Stokes equations. In particular, the decay rates of the second order derivatives of the Navier–Stokes flows in (n⩾2) with 1⩽r⩽∞ are derived by using Lq−Lr estimates and a clever analysis on the fractional powers of the Stokes operator. In addition, we prove that the strong solution and its first and second derivatives decay in time more rapidly than observed in general if the initial datum lies in a suitable weighted space.