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

摘要

Using the solution formula in Ukai (1987) [27] for the Stokes equations, we find asymptotic profiles of solutions in L 1(R{double-struck} + ~n) (n≥2) for the Stokes flow and non-stationary Navier-Stokes equations. Since the projection operator P:L 1(R{double-struck} + ~n)←Lσ1(R{double-struck} + ~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{double-struck} + ~n) is controlled by t-1/2(1+t-n+2/2) for any t>0.