Digital Object Identifier (DOI) 10.1007/s002090100276 Math. Z. 238, 799–816 (2001) Decay rate for the incompressible flows in half spaces Hyeong-Ohk Bae 1 , Hi Jun Choe 2 1 Department of Mathematics, Ajou University, Suwon, Republic of Korea (hobae@madang.ajou.ac.kr) 2 Department of Mathematics, Yonsei University Seoul, Republic of Korea Received: 3 November 1999; in final form: 10 May 2000 / Published online: 17 May 2001 – c  Springer-Verlag 2001 Abstract. Weshowthatthetimedecayrateof L 2 normofweaksolutionfor the Stokes equations and for the Navier–Stokes equations on the half spaces are t - n 2 ( 1 r - 1 2 )- 1 2 iftheinitialdata u 0 ∈ L 2 ∩ L r and  R n + |y n u 0 (y)| r dy < ∞ for 1 <r< 2. We also show that the decay rate is determined by the linear part of the weak solution. We use the heat kernel and Ukai’s solution formulafortheStokesequations.Ithasbeenknownuptonowthatthedecay rate on the half space was t - n 2 ( 1 r - 1 2 ) , which was obtained by Borchers and Miyakawa [1] and Ukai [9]. 1. Introduction We study the asymptotic behavior in L 2 of weak solutions of the Navier– Stokes equations in half spaces: u t - ∆u +(u ·∇)u + ∇p =0, in R n + × (0, ∞), ∇· u =0, in R n + × (0, ∞), u(x, 0) = u 0 , for x ∈ R n + , u(x,t)=0, for x n =0,t ∈ (0, ∞), (1.1) where n ≥ 2, and R n + = {x ∈ R n : x = (¯ x,x n ),x n > 0} is the upper half space of R n . We denote by ¯ x def = (x 1 , ··· ,x n-1 ) ∈ R n-1 . Here, u def = (u 1 ,...,u n ) and p denote the velocity and pressure, respectively, while u 0 is a given initial velocity. This research was supported by KOSEF, KRF and BK21-prg