Decay Results of Higher-Order Norms for the Navier–Stokes Flows in 3D Exterior Domains

Abstract

Using the nonstationary Stokes solution formula and the regularity theory of the steady Stokes equations, we establish the decay results (including the weighted cases) of higher-order spatial derivatives for the Navier–Stokes flows in 3D exterior domains, which is a long standing problem. The higher-order norm (weighted) decay results in this article are inspired by the work in Bae and Jin (Proc R Soc Edinburgh Sect A 135:461–477, 2005), Schonbek and Wiegner (Proc R Soc Edinburgh Sect A 126:677–685, 1996), respectively, which are considered in the whole space.