Long-time behavior for Navier–Stokes flows in a two-dimensional exterior domain

Abstract

There are several long standing problems on the incompressible Navier–Stokes flows in 2D exterior domains, which claim how to characterize L 1 -summability of the 2D N-S flows; whether the total net force exerted on the boundary is finite; and how to establish decay results of higher-order spatial derivatives, including the weighted cases. In order to solve these questions, we firstly find some types of new technical inequalities, which are used to overcome the difficulties caused by the domain boundary; using L q − L r properties for nonstationary Stokes flows, together with elliptic estimates for the steady Stokes system, we can avoid the strong singularity and answer these mentioned problems completely. It should be pointed out that main results in this article are motivated by the works in [5] , [37] , respectively.