Methods and Applications of Analysis

Volume 23 (2016)

Number 4

On the large time approximation of the Navier–Stokes equations in $\mathbb{R}^n$ by Stokes flows

Pages: 293 – 316

DOI: http://dx.doi.org/10.4310/MAA.2016.v23.n4.a1

Authors

Pablo Braz e Silva (Departmento de Matemática, Universidade Federal de Pernambuco, Recife, Brazil)

Jens Lorenz (Department of Mathematics and Statistics, University of New Mexico, Albuquerque, N.M., U.S.A.)

Wilberclay G. Melo (Departmento de Matemática, Universidade Federal de Sergipe, São Cristóvão, Brazil)

Paulo R. Zingano (Departamento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil)

Abstract

We show, under quite general assumptions, the time asymptotic property $t^{\kappa_{n,q}} \Vert u (\cdot , t) - v (\cdot , t) \Vert {}_{L^q (\mathbb{R}^n)} \to 0$ as $t \to \infty$, for each $2 \leq q \leq \infty$ and all Leray–Hopf global $L^2$ solutions $u(\cdot, t)$ of the incompressible Navier–Stokes equations and their associated Stokes flows $v(\cdot, t)$ in $\mathbb{R}^n (n = 2, 3)$, where $\kappa_{n,q} = (n/2)(1 - 1/q) - 1/2$. We use the approximation results to derive 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.

Keywords

incompressible Navier–Stokes equations, Leray–Hopf (weak) solutions, large time behavior, Leray’s problem, Stokes approximation, supnorm estimates

2010 Mathematics Subject Classification

Primary 35Q30, 76D05. Secondary 76D07.

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