METHODS AND APPLICATIONS OF ANALYSIS. c  2016 International Press Vol. 23, No. 4, pp. 293–316, December 2016 001 ON THE LARGE TIME APPROXIMATION OF THE NAVIER-STOKES EQUATIONS IN R n BY STOKES FLOWS ∗ PABLO BRAZ E SILVA † , JENS LORENZ ‡ , WILBERCLAY G. MELO § , AND PAULO R. ZINGANO ¶ Abstract. We show, under quite general assumptions, the time asymptotic property t κ n, q ‖ u(·,t) - v(·,t) ‖ L q (R n ) → 0 as t →∞, for each 2 ≤ q ≤∞ and all Leray-Hopf global L 2 solutions u(·,t) of the incompressible Navier-Stokes equations and their associated Stokes flows v(·,t) in R n (n =2, 3), where κ n, q =(n/2) (1 - 1/q) - 1/2. We use the approximation results to de- rive several new related results on Stokes flows. Our method is based on classic tools in real analysis and PDE theory like standard Fourier and energy methods. Key words. Incompressible Navier-Stokes equations, Leray-Hopf (weak) solutions, large time behavior, Leray’s problem, Stokes approximation, supnorm estimates. AMS subject classifications. Primary 35Q30, 76D05; Secondary 76D07. 1. Introduction. We are interested in deriving various fundamental asymptotic properties concerning the so-called Leray-Hopf solutions to the initial value problem for the incompressible Navier-Stokes equations u t + u ·∇ u + ∇p =Δu + f (·,t), ∇· u(·,t)=0, (1.1a) u(·, 0) = u 0 ∈ L 2 (R n ), ∇· u 0 =0, (1.1b) where f (·,t)=( f 1 (·,t), ..., f n (·,t)) ∈ L 1 loc ([0, ∞ [ ,L 2 (R n )) is a given external force field satisfying the general conditions (1.5), (1.8) below, and p(·,t) denotes the kine- matic pressure. We begin with an overview of known results and those obtained here. Considering the Helmholtz decomposition of f (·,t) in L 2 (R n ), i.e., f (·,t)= -∇Φ(·,t)+ g(·,t), ∇· g(·,t)=0, (1.2a) with ∇Φ(·,t), g(·,t) ∈ L 1 loc ([0, ∞ [ ,L 2 (R n )) such that 1 ‖ g(·,t) ‖ L 2 (R n ) ≤‖ f (·,t) ‖ L 2 (R n ) , ‖ D g(·,t) ‖ L 2 (R n ) ≤‖ Df (·,t) ‖ L 2 (R n ) , (1.2b) one can write the equations (1.1) above as ∗ Received March 24, 2015; accepted for publication April 15, 2016. † Departmento de Matem´atica, Universidade Federal de Pernambuco, Recife, PE 50740, Brazil. ‡ Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA. Research partially supported by NSF Grant DMS-1148801. § Departmento de Matem´atica, Universidade Federal de Sergipe, S˜ao Crist´ ov˜ ao, SE 49100, Brazil. Research partially supported by CAPES (Ciˆ encia sem Fronteiras) Grant 2778-13-0. ¶ Departamento de Matem´atica Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91509, Brazil. 1 For the definition of the vector norms involved here, see (1.14), (1.15). If f (·,t)  ∈ H 1 (R n ), then the second condition in (1.2b) is merely the trivial assertion that ‖ Dg(·,t) ‖ L 2 (R n ) ≤∞. 293