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

Abstract

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 σ 1 ( R + n ) 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 L 1 ( R + n ) is controlled by t − 1 2 ( 1 + t − n + 2 2 ) for any t > 0 .