WANG Yuanyuan and YANG Yongfu

College of Science,Hohai University,Nanjing 211100,China.

Abstract. The aim of this paper is to investigate homogenization of stationary Navier-Stokes equations with a Dirichlet boundary condition in domains with 3 kinds of typical holes. For space dimension N=2 and 3,we utilize a unified approach for 3 kinds of tiny holes to accomplish the homogenization of stationary Navier-Stokes equations. The unified approach due to Lu[1]is mainly based on the uniform estimates with respect to ε for the generalized cell problem inspired by Tartar.

Key Words: Homogenization;Navier-Stokes equations;perforated domain;cell problem.

1 Introduction

It is well known homogenization describes the asymptotic behavior of fluid flows in perforated domains, as the number of holes increases and goes to infinity, the size of holes will go to zero, the flow will tend to the solution of certain effective or “homogenized”equations which are homogeneous in form.

Let Ω ?RNbe a bounded domain withC1boundary, here and hereafterN=2 or 3.The domain Ω is covered with a regular mesh of sizeε,each cell being a cubeQk. At the center of each cubeQkincluded in Ω there is a holeTε,ksuch that(i=1,2,3)are positive constants.The mutual distance of holes is denoted byε,the size of holes is denoted byaε,andεxk=εx0+εklocates the holes. Without loss of generality,we assume thatx0=0 and 0

The open set Ωεis obtained by removing all the holesTε,kfrom Ω, so Ωεis also a boundedC1,Precisely,Ωεis defined as following:

Consider the Dirichlet problem of the stationary Navier-Stokes equations with external force f∈[L2(Ω)]Nin Ωε:

In recent decades, due to its physical importance, complexity, rich phenomena, and mathematical challenges,there have been a lot of literatures on homogenization problem.Allaire [3,4] did a systematic study on the Stokes equations with a Dirichlet boundary condition in a domain containing many tiny holes, which are periodically distributed in each direction of the axes. For holes of critical size, Allaire established an abstract framework and showed that the limit problem is described by a law of Brinkman type(see[5]). He also proved that for smaller holes, the limit problem reduces to the Stokes equations, and for larger holes, to Darcy’s law. Similar as in [3], we define the ratioσεbetween the size and the mutual distance of the holes:

On the other hand,different from Allaire’s framework,Tartar[6]utilized the so-called cell problem to study the asymptotic behavior of the solution family{uε}ε>0to Dirichlet problem of Stokes equations asε→0, under the assumption that the size of the holes is proportional to the mutual distance of the holes,i.e.

Later,homogenization problems of fluid flows are generalized to more complex models.When the size of holes is proportional to the mutual distance of holes,for the incompressible Navier-Stokes equations,for the compressible Navier-Stokes equations,and for the complete Navier-Stokes-Fourier equations,the limit systems all obey the Darcy’s law(see [7], [8], [9]). Besides, authors in [10,11] considered the case of small holes for the compressible Navier-Stokes equations. Recently,Lu in [12,13] obtained the uniform estimates for Laplace equation and Stokes equation in a domain with small holes inLpframework. And Dieninget al[14] established the homogenization limit of small holes for compressible Navier-Stokes equation inLpframework. Feireislet al[15]furthermore studied the case with critical size of holes for the incompressible Navier-Stokes equations. All of these results are consistent with those for Stokes’equation by Allaire(see[3,4]).More recently,inspired by Tartar,Lu[1]utilized the generalized cell problem to achieve a unified proof of the homogenization for Stokes equation in a perforated domain with different sizes of holes.

The purpose of this paper is to study the homogenization of the stationary Navier-Stokes equations, with a Dirichlet boundary condition, in the domain perforated with tiny holes of different sizes. To be precise, with the uniform estimates with respect toεfor the weak formulation of(3.9),we letε→0 to get the limit system.The key point in this process is the choice of suitable test functions. The test functions inC∞c(Ω)do not match the test functions inC∞c(Ωε),so we have to make a small“surgery”on the test functions,Tartar[6]and Allaire [3,4]achieved this problem in different ways. Compared with homogenization for Stokes equation(see[1]), Navier-Stokes equation is more complicated and the appearance of convective term gives rise to strong nonlinearity and more difficulty for mathematical analysis,so we have to use more delicate estimates to obtain the asymptotic limits.

The paper is organized as follows. In the next section, we state the main result of this paper. Section 3 is devoted to the proof of Theorem 2.1, we recall the generalized cell problem,the permeability tensor,and the scaled cell solutions,establish the uniform estimates for(?uε,Pε(pε)),and finally verify strictly the limits case by case.

2 Main result

For the convenience of presentation,we recall some basic facts established by Allaire[3,4](see also[1,6]). To begin with,one can define velocity’s zero extension as

For Ωεdefined by(1.1),it was shown in[3,4]that there exists a linear operator,named restriction operatorRε,satisfying

In addition,an extension operator for the pressurePε,defined by

In(2.6)and(2.7),the permeability tensor A is a constant positive definite matrix. It is determined by the size of holes and the mutual distance of the holes. If aε=ε, A is defined by

where wi is the solution to the following cell problem and is extended by zero on Q0

Here and hereafter{e1,···,eN}stands for the standard Euclidean coordinate ofRN. On the other hand,if aε?ε, A is determined later by(3.7).

Remark 2.1. Theorem 2.1 covers both the casesaε=εandaε?ε. We can find limε→0σε=0 includes two cases:aε=εandaε?ε,their limit models are both the Darcy’s law,however the related permeability tensorsAare different.

3 Proof of Theorem 2.1

In this section,we are going to use a unified approach to prove Theorem 2.1.

3.1 The generalized cell problem

In particular, whenη→0 asε→0 such thatcη→0, by (3.4), there holds ?wi=0, so that the limitwiis a constant vector.

3.2 The permeability tensor

3.3 The scaled cell solutions

3.4 Uniform estimates for(?uε,Pε(pε))

Taking the inner product of(3.9)with uεand integration by parts,we obtain the classical energy equality

Together with Bogovskii’s operator,we deduce from(3.19)that

with

It follows from Rellich-Kondrachov compactness theorem that

Hence,up to a subsequence,asε→0,we have

3.5 Homogenization process

which implies

3.5.2 The case with large holes: limε→0σε→0

It follows from(3.10)and(3.18)that

and then

Together with the fact that

(3.28) leads to Darcy’s law, which has a unique solution. Because of the uniqueness,all the subsequences of (?uε,Pε(pε)) converge to the same limit,thus the entire sequence converges.

Since the solution (u,p) to (3.31)-(3.32) is unique, the limit process holds for the whole sequence.This completes the proof of Theorem 2.1.□

Acknowledgement

Part of this work was supported by the foundation of School of Mathematics, Fudan University in 2020. The authors would like to thank Associate Prof. Libin Wang for her kind invitation and warm hospitality.The research is also partially supported by the NSF of Jiangsu Province(N0. BK20191296).