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 L-1 (R-+(n)) (n >= 2) for the Stokes flow and non-stationary Navier-Stokes equations. Since the projection operator P: L-1 (R-+(n)) --> L-sigma(1)(R-+(n)) is unbounded, we use a decomposition for P(u . del 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 L-1 (R-+(n)) is controlled by t(-1/2)(1 + t(-n+2/n)) for any t > 0. (C) 2010 Elsevier Inc. All rights reserved.