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

Abstract

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<n, as |x|→+∞. The solution exists globally in time for α≥1 when the data is small enough, and it holds that u(x,t)=e−tAh(x)+o((|x|+t)−α), 1<α<n, as |x|+t→+∞.