Asymptotic behavior for the Stokes flow and Navier–Stokes equations in half spaces

Using the solution formula in Ukai (1987) [27] for the Stokes equations, we find asymptotic profiles of solutions in (n⩾2) for the Stokes flow and non-stationary Navier–Stokes equations. Since the projection operator is unbounded, we use a decomposition for P(u⋅∇u) to overcome the difficulty, and prove that the decay rate for the first derivatives of the strong solution u of the Navier–Stokes system in is controlled by for any t>0.