Pointwise decay estimate of Navier-Stokes flows in the half space with slowly decreasing initial value

In this paper we study the spatial and temporal decay estimate of the Navier-Stokes flow in the half space corresponding to auniformly but slowly decreasing initial velocity. We show the local in time solvability of the Navier-Stokes equations with |u(x,t)|≤C0(1+|x|+t)−α and |∇u(x,t)|≤C0t−12(1+|x|+t)−αwhen (1+|x|+t)αe−tAh∈L∞(R+n×(0,∞)) and t12(1+|x|+t)αe−tAh∈L∞(R+n×(0,∞)), 0<α≤n. Asymptotically, it holds that u(x,t)=e−tAh(x)+ot(|x|−α),0<α