Decay rates for the higher-order derivatives of the Navier-Stokes flow in exterior domains

Abstract

This paper makes investigations on the time-decay rates for the 3-dimensional Navier-Stokes flow in exterior domains. Boundedness of the convolution operator with the Gaussian kernel from the intersection L01(Ω)∩L1(Ω,|x|αdx) for some 0<α≤1 to the space Lr(Ω) for all 1≤r<∞ is firstly established, from which, an innovated estimate ‖u(t)‖r≤Ct−α2−32(1−1r) (t≥T0, 1<r≤∞) is derived. Decay rates of the higher-order spatial and temporal derivatives of the flow are mainly studied then. In order to derive the estimates ‖∇∂tku(t)‖r≤Ct−k+12−α2−32(1−1r) adequately, translation transformation v˜kv(t)=∂tku(t+Tk) has been made.