Mathematical Problems in Elasticity and Homogenization

By O.A. Oleinik, A.S. Shamaev and G.A. Yosifian

This monograph is based on research undertaken by the authors during the last ten years. The main part of the work deals with homogenization problems in elasticity as well as some mathematical problems related to composite and perforated elastic materials. This study of processes in strongly non-homogeneous media brings forth a large number of purely mathematical problems which are very important for applications. Although the methods suggested deal with stationary problems, some of them can be extended to non-stationary equations. With the exception of some well-known facts from functional analysis and the theory of partial differential equations, all results in this book are given detailed mathematical proof.

It is expected that the results and methods presented in this book will promote further investigation of mathematical models for processes in composite and perforated media, heat-transfer, energy transfer by radiation, processes of diffusion and filtration in porous media, and that they will stimulate research in other problems of mathematical physics and the theory of partial differential equations.

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 15, 2009
• ISBN: 9780080875231

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Studies in Mathematics and Its Applications

Navier–Stokes Equations

Theory and Numerical Analysis

Roger Temam

Université de Paris-Sud, Orsay, France

Ecole Polytechnique, Palaiseau, France

J.L. Lions, G. Papanicolsou, R.T. Rockafellar, Editors

ISSN  0168-2024

Volume 2 • Suppl. (C) • 1977

Table of Contents

Cover image

Title page

Studies in Mathematics and its Applications

Copyright page

Foreword

The Steady-State Stokes Equations

Introduction

§1 Some function spaces

§2 Existence and uniqueness for the Stokes equations

§3 Discretization of the Stokes equations (I)

§4 Discretization of Stokes equations (II)

§5 Numerical Algorithms

§6 Slightly Compressible Fluids

Steady-State Navier–Stokes Equations

Introduction

§1 Existence and Uniqueness Theorems

§2 Discrete inequalities and compactness theorems

§3 Approximation of the Stationary Navier–Stokes Equations

§4 Bifurcation Theory and Non-Uniqueness Results

The Evolution Navier–Stokes Equation

Introduction

§1 The Linear Case

§2 Compactness Theorems

§3 Existence and Uniqueness Theorems (n 4)

§4 Alternate Proof of Existence by Semi-Discretization

§5 Discretization of the Navier—Stokes Equations: General Stability and Convergence Theorems

§6 Discretization of the Navier—Stokes Equations: Application of the General Results

6.4 Numerical algorithms. Approximation of the pressure

§7 Approximation of the Navier—Stokes Equations by the Fractional Step Method

7.2 A scheme with n + 1 intermediate steps

7.3 Convergence of the Scheme

§8 Approximation of the Navier—Stokes equations by the Artificial Compressibility Method

Comments and Bibliography

Chapter I

Chapter II

Chapter III

References

Appendix

0 Test Problems

1 The triangulation

Example (k = 2)

Example (k = 2)

Examples

2 The nodes

3 Computation of the basis function on a given triangle

4 Solution of Stokes’ Problem

Examples

Studies in Mathematics and its Applications

VOLUME 2

Editors:

J. L. LIONS, Paris

G. PAPANICOLAOU, New York

R. T. ROCKAFELLAR, Seattle

NORTH-HOLLAND PUBLISHING COMPANY–AMSTERDAM • NEW YORK • OXFORD

Copyright page

© North-Holland Publishing Company 1977

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 07204 2840 8

Published by:

North-Holland Publishing Company

Amsterdam • New York • Oxford

Sole distributors for the U.S.A. and Canada:

Elsevier North-Holland, Inc.

52 Vanderbilt Avenue

New York, NY 10017

Library of Congress Cataloging in Publication Data

Temam, Roger.

Navier-Stokes equations.

Bibliography: 15 pp.

1. Navier-Stokes equations. I. Title.

QA372.T38 515′.352 76-51536

ISBN 0-7204-2840-8

Printed in England

Foreword

In the present work we derive a number of results concerned with the theory and numerical analysis of the Navier-Stokes equations for viscous incompressible fluids. We shall deal with the following problems: on the one hand, a description of the known results on the existence, the uniqueness and in a few cases the regularity of solutions in the linear and non-linear cases, the steady and time-dependent cases; on the other hand, the approximation of these problems by discretisation: finite difference and finite element methods for the space variables, finite differences and fractional steps for the time variable. The questions of stability and convergence of the numerical procedures are treated as fully as possible. We shall not restrict ourselves to these theoretical aspects: in particular, in the Appendix we give details of how to program one of the methods. All the methods we study have in fact been applied, but it has not been possible to present details of the effective implementation of all the methods. The theoretical results that we present (existence, uniqueness,…) are only very basic results and none of them is new; however we have tried as far as possible to give a simple and self-contained treatment. Energy and compactness methods lie at the very heart of the two types of problems we have gone into, and they form the natural link between them.

Let us give a more detailed description of the contents of this work: we consider first the linearized stationary case (Chapter 1), then the non-linear stationary case (Chapter 2), and finally the full non-linear time-dependent case (Chapter 3). At each stage we introduce new mathematical tools, useful both in themselves and in readiness for subsequent steps.

In Chapter 1, after a brief presentation of results on existence and uniqueness, we describe the approximation of the Stokes problem by various finite-difference and finite-element methods. This gives us an opportunity to introduce various methods of approximation of the divergence-free vector functions which are also vital for the numerical aspects of the problems studied in Chapters 2 and 3.

In Chapter 2 we introduce results on compactness in both the continuous and the discrete cases. We then extend the results obtained for the linear case in the preceding chapter to the non-linear case. The chapter ends with a proof of the non-uniqueness of solutions of the stationary Navier-Stokes equations, obtained by bifurcation and topological methods. The presentation is essentially self-contained.

Chapter 3 deals with the full non-linear time-dependent case. We first present a few results typical of the present state of the mathematical theory of the Navier-Stokes equations (existence and uniqueness theorems). We then present a brief introduction to the numerical aspects of the problem, combining the discretization of the space variables discussed in Chapter 1 with the usual methods of discretization for the time variable. The stability and convergence problems are treated by energy methods. We also consider the fractional step method and the method of artificial compressibility.

This brief description of the contents will suffice to show that this book is in no sense a systematic study of the subject. Many aspects of the Navier-Stokes equations are not touched on here. Several interesting approaches to the existence and uniqueness problems, such as semi-groups, singular integral operators and Riemannian manifolds methods, are omitted. As for the numerical aspects of the problem, we have not considered the particle approach nor the related methods developed by the Los Alamos Laboratory.

We have, moreover, restricted ourselves severely to the Navier-Stokes equations; a whole range of problems which can be treated by the same methods are not covered here. Nor are the difficult problems of turbulence and high Reynolds number flows.

The material covered by this book was taught at the University of Maryland in the first semester of 1972–3 as part of a special year on the Navier-Stokes equations and non-linear partial differential equations. The corresponding lecture notes published by the University of Maryland constitute the first version of this book.

I am extremely grateful to my colleagues in the Department of Mathematics and in the Institute of Fluid Dynamics and Applied Mathematics at the University of Maryland for the interest they showed in the elaboration of the notes. Direct contributions to the preparation of the manuscript were made by Arlett Williamson, and by Professors J. Osborne, J. Sather and P. Wolfe. I should like to thank them for correcting some of my mistakes in English and for their interesting comments and suggestions, all of which helped to improve the manuscript. Useful points were also made by Mrs Pelissier and by Messrs Fortin and Thomasset. Finally, I should like to express my thanks to the secretaries of the Mathematics Departments at Maryland and Orsay for all their assistance in the preparation of the manuscript.

Chapter I

The Steady-State Stokes Equations

Introduction

In this chapter we study the stationary Stokes equations; that is, the stationary linearized form of the Navier-Stokes equations. The study of the Stokes equations is useful in itself? it also gives us an opportunity to introduce several tools necessary for a treatment of the full Navier-Stokes equations.

In Section 1 we consider some function spaces (spaces of divergence-free vector functions with L²-components). In Section 2 we give the variational formulation of the Stokes equations and prove existence and uniqueness of solutions by the projection theorem. In Sections 3 and 4 we recall a few definitions and results on the approximation of a normed space and of a variational linear equations (Section 3). We then propose several types of approximation of a certain fundamental space V of divergence-free vector functions; this includes an approximation by the finite-difference method (Section 3), and by conforming and non-conforming finite-element methods (Section 4). In Section 5 we discuss certain approximation algorithms for the Stokes equations and the corresponding discretized equations. The purpose of these algorithms is to overcome the difficulty caused by the condition div u = 0. As it will be shown, this difficulty, sometimes, is not merely solved by discretization.

Finally in Section 6 we study the linearized equations of slightly compressible fluids and their asymptotic convergence to the linear equations of incompressible fluids (i.e., Stokes’ equations).

§1 Some function spaces

In this section we introduce and study certain fundamental function spaces. The results are important for what follows, but the methods used in this section will not reappear so that the reader can skim through the proofs and retain only the general notation described in Section 1.1 and the results summarized in Remark 1.6.

1.1 Notation

In Euclidean space Rn we write e1 = (1, 0, …, 0), e2 = (0, 1, 0, …, 0), …, en = (0, …, 0, 1), the canonical basis, and x = (x1, …, xn),y = (y1, …, yn), z = (z1, …, zn), …, will denote points of the space. The differential operator

will be written Di and if j = (j1, … jn) is a multi-index, Dj will be the differentiation operator

(1.1)

where

(1.2)

If ji = 0 for some iis the identity operator; in particular if [j] = 0, Dj is the identity.

The set Ω

Let Ω be an open set of Rn with boundary Γ. In general we shall need some kind of smoothness property for Ω. Sometimes we shall assume that Ω is smooth in the following sense:

(1.3)

We will say that a set Ω satisfying (1.3) is of class Cr. However this hypothesis is too strong for practical situations (such as a flow in a square) and all the main results will be proved under a weaker condition:

(1.4)

This means that in a neighbourhood of any point x ∈ Γ, Γ admits a representation as a hypersurface yn = θ(y1, …, yn-1) where θ is a Lipschitz function, and (y1, …, yn) are rectangular coordinates in Rn in a basis that may be different from the canonical basis e1, …, en.

Of course if Ω is of class C¹, then Ω is locally Lipschitz.

Fig. 1

It is useful for the sequel of this section to note that a set Ω satisfying (1.4) is locally star-shaped. This means that each point xj ∈ Γ, has an open neighbourhood Ojis star-shaped with respect to one of its points. According to (1.4) we may, moreover, suppose that the boundary of Oj is Lipschitz.

If Γ is bounded, it can be covered by a finite family of such sets Oj, j ∈ J; if Γ is not bounded, the family (Oj)j ∈ J can be chosen to be locally finite.

It will be assumed that Ω will always satisfy (1.4), unless we explicitly state that Ω is any open set in Rn or that some other smoothness property is required.

LP and Sobolev Spaces

Let Ω be any open set in Rn. We denote by Lp (Ω), 1 < p < + ∞ (or L∞ (Ω)) the space of real functions defined on Ω with the p-th power absolutely integrable (or essentially bounded real functions) for the Lebesgue measure dx = dx1 … dxn. This is a Banach space with the norm

(1.5)

or

For p = 2, L² (Ω) is a Hilbert space with the scalar product

(1.6)

The Sobolev space Wm, p(Ω) is the space of functions in Lp(Ω) with derivatives of order less than or equal to m in Lp(Ω) (m p + ∞). This is a Banach space with the norm

(1.7)

When p = 2, Wm,2(Ω) = Hm(Ω) is a Hilbert space with the scalar product

(1.8)

be the space of C. The closure of D(Ω) in Wm, p

We recall, when needed, the classical properties of these spaces such as the density or trace theorems (assuming regularity properties for Ω).

We shall often be concerned with n-dimensional vector functions with components in one of these spaces. We shall use the notation

and we suppose that these product spaces are equipped with the usual product norm or an equivalent norm (except D, which are not normed spaces).

The following spaces will appear very frequently

The scalar product and the norm are denoted by (·, ·) and |·| on L²(Ω) or L)).

We recall that if Ω is bounded in some direction(1) then the Poincaré inequality holds:

(1.9)

where D is the derivative in that direction and c(Ω) is a constant depending only on Ω which is bounded by 2l, l ) is equivalent to the norm:

(1.10)

is also a Hilbert space with the associated scalar product

(1.11)

(Ω bounded in some direction).

Let V be the space (without topology)

(1.12)

The closures of V in Lare two basic spaces in the study of the Navier-Stokes equations; we denote them by H and V. The results of this section will allow us to give a characterization of H and V.

1.2 A density theorem

Let E(Ω) be the following auxiliary space:

This is a Hilbert space when equipped with the scalar product

(1.13)

It is clear that (1.13) is a scalar product on E(Ω); it is easy to see that E(Ω) is complete for the associated norm (1)

Our goal is to prove a trace theorem: for u ∈ E(Ω) one can define the value on Γ of the normal component u·v, v = the unit vector normal to the boundary. The method we use is the classical Lions-Magenes [1] one. We begin by proving

Theorem 1.1

Let Ω be a Lipschitz open set in Rn. Then the set of vector functions belonging to is dense in E(Ω).

Proof. Let u be some element of E(Ω). We have to prove that u is a limit in E.

(i) We first approximate u by functions of E(Ω) with compact support in Ω.

Let ϕ ∈ D(Rn1, ϕ = 1 for |x1, and ϕ = 0 for |x2. For a > 0 let ϕa be the restriction to Ω of the function x ϕ(x/a). It is easy to check that ϕau ∈ E(Ω) and that ϕau converges to u in this space as a → ∞.

The functions with compact support are a dense subspace of E(Ω) and we may assume that u has compact support.

(ii) Let us consider first the case Ω = Rn; hence u ∈ E(Rn) and u has a compact support.

The result is then proved by regularization. Let ρ ∈ D(Rn) be a smooth Cdenote the function x n) ρ(xconverges in the distribution sense to the Dirac distribution and it is a classical result that(1)

(1.14)

* u belongs to D(Rn) + support u) and components which are C∞. According to *u converges to u in L² (Rn→ 0, and

div (ρ * u) = ρ * div u converges to div u in L² (Rn),

→ 0. Hence u is the limit in E(Rn) of functions of D(Rn).

(iii) For the general case, Ω ≠ Rn, we use the remark after (1.14): Ω is locally star-shaped. The sets Ω2, (Oj-)j∈J. Let us consider a partition of unity subordinated to this covering.

(1.15)

We may write

the sum Σj is actually finite since the support of u is compact.

Since the function ϕu has compact support in Ω it can be shown as in (ii) that ϕu is the limit in E(Ω) of functions belonging to D(Ω) (the function ϕu extended by 0 outside Ω belongs to E(Rn*(ϕu) has compact support in Ω).

Let us consider now one of the functions uj = ϕju = Oj ∩ Ω is star-shaped with respect to one of its points; after a translation in Rn we can suppose this point is 0. Let σλ, λ ≠ 0, be the linear (homothetic) transformation x → λxis Lipschitz, and star-shaped with respect to 0, that:

v denote the function x v(σλ(x)); because of ui, λ > 1, converges to uj in E) (or E1. But if ψj ∈ D)) and ψj = 1 on Oj the function ψju) clearly belongs to E(Rn). Hence we must only approximate in place of the function uj, a function vj ∈ E(Ω) which is the restriction to Ω of a function wj ∈ E(Rn) with compact support (take wj = ψju)). The result follows then from point (ii).

---

(1)* is the convolution operator

If f ∈ L¹(Rn), g ∈ Lp (Rnp < ∞, then f*g makes sense and belongs to Lp(Rn).

It remains only to prove Lemma 1.1 giving some results that we used in the above proof and other results which will be needed later.

Lemma 1.1

Let O be an open set which is star-shaped with respect to 0.

(i) If p ∈ D′(O) is a distribution in O, then a distribution p can be defined in D′ (σλ O) by

(1.16)

The derivatives of p are related to the derivatives of p by the formula

(1.17)

If λ > 1, λ → 1, the restriction to O of p converges in the distribution sense to p.

(ii) If p ∈ Lα(Oα < + ∞, then p ∈ Lα(σoλ O). For λ > 1, λ → 1, the restriction to O of p converges to p in Lα(O).

Proofis linear and continuous on D(σλ Op.

The formula (1.17) is easy,

ϕ have compact support in O for λ – 1 small enough, and converge to ϕ in D(O) as λ → 1.

(ii) It is clear that

p restricted to O converges to p, for the p’s belonging to some dense subspace of Lα(O). But D(O) is dense in Lα(O), and the result is obvious if p ∈ D(O).

1.3 A trace theorem

We suppose here that ft is an open bounded set of class C². It is known that there exists a linear continuous operator γ0 ∈ L(H¹(Ω), L²(γ)) (the trace operator), such that γ0u = the restriction of u to Γ for every function u ∈ His equal to the kernel of γ0. The image space γ0(H¹(Ω)) is a dense subspace of L²(Γ) denoted H¹/²(Γ); the space H¹/²(Γ) can be equipped with the norm carried from HΩ ∈ L(H¹/²(Γ), HΩ = the identity operator in H¹/²(Γ). All these results are given in Lions [1], Lions & Magenes [1].

We want to prove a similar result for the vector functions in E(Ω).

Let H-1/2(Γ) be the dual space of H¹/²(Γ); since H¹/²(Γ) ⊂ L²(Γ) with a stronger topology, L²(Γ) is contained in H-1/2(Γ) with a stronger topology. We have the following trace theorem (which means that we can define u · v|Γ when u ∈ E):

Theorem 1.2

Let Ω be an open bounded set of class C². Then there exists a linear continuous operator γv ∈ L(E(Ω), H-1/2(Γ)) such that

(1.18)

The following generalized Stokes formula is true for all u ∈ E(Ω) and w ∈ H¹(Ω)

(1.19)

Proof. Let ϕ ∈ H¹/²(Γ) and let w ∈ H¹(Ω) with γ0 w = ϕ. For u ∈ E(Ω), let us set

Lemma 1.2

Xu (ϕ) is independent of the choice of w, as long as w ∈ H¹(Ω) and γ0 w = ϕ.

Proof. Let w1 and w2 belong to H¹ (Ω), with

γ0 w1 = γ0 w2 = ϕ

and let w = w1 – w2.

We must prove that

(div u, w1) + (u, grad w1) = (div u, w2) + (u, grad w2),

that is to say

(1.20)

But since w ∈ H¹ (Ω) and γ0 w = 0, w (Ω) and is the limit in H¹(Ω) of smooth functions with compact support: w = lim wm, wm ∈ D(Ω). It is obvious that

(div u, wm) + (u, grad wm) = 0, ∀ wm ∈ D (Ω)

and (1.20) follows as m → ∞.

Let us take now w Ωϕ (see above). Then by the Schwarz inequality

Ω ∈ L(H¹/²(Γ), H¹(Ω))

(1.21)

where cΩ.

Xu (ϕ) is a linear continuous mapping from H¹/²(Γ) into R. Thus there exists g = g(u) ∈ H-1/2(Γ) such that

(1.22)

It is clear that the mapping u g (u) is linear and, by (1.21),

(1.23)

this proves that the mapping u g(u) = γ0u is continuous from E(Ω) into H-1/2(Γ).

The last point is to prove (1.18) since (1.19) follows directly from the definition of γvu.

Lemma 1.3

If , then

γvu = the restriction of u · v on Γ.

Proof. For such a smooth u (or if u and w are twice continuously differentiable in Ω),

Since for these functions w, the traces γ0w form a dense subset of H¹/²(Γ), the formula

is also true by continuity for every ϕ ∈ H¹/²(Γ). By comparison with (1.22), we get γvu = u·v|Γ.

Remark 1.1

Theorem 1.1 is not explicitly used in the proof of Theorem 1.2, but the density theorem combined with Lemma 1.3 shows that the operator γv is unique since its value on a dense subset is known.

Remark 1.2

The operator γv actually maps E(Ω) onto H-1/2(Γ).

Let ϕ be given in H-1/2(Γ), such that <ϕ, 1> = 0. Then the Neumann problem

(1.24)

has a weak solution p = p(ϕ) ∈ H¹(Ω) which is unique up to an additive constant (See lions & Magenes [1]). For one of these solutions p let

u = grad p

It is clear that u ∈ E(Ω) and γvu = ϕ. In addition it is clear that there exists a vector function usuch that γvu0 = 1. Then for any ψ in H¹/²(Γ), writing

(1.25)

one can define a u = u(ψ) such that γvu = ψ by setting

(1.26)

u(ψ) is a linear continuous mapping from H-1/2(Γ) into E(Ω) (i.e., a lifting operator Ω).

Let E0(Ω) be the closure of D(Ω). We have

Theorem 1.3

The kernel of γv is equal to E0(Ω).

Proof. If u ∈ E0(Ω), then by the definition of this space, there exists a sequence of functions um ∈ D(Ω) which converges to u in E(Ω) as m → ∞. Theorem 1.2 implies that γvum = 0 and hence γv · u = limm→∞ γv · um = 0.

Conversely let us prove that if u ∈ E(Ω) and γv · u = 0, then u is the limit in E(Ω) of vector functions in D(Ω)n.

Let Φ be any function in D(Rn), and ϕ the restriction of Φ to Ω. Since γv·u = 0, we have <γv · u, γ0 ϕ> = 0 which means

Hence

and so

(1.27)

denotes the function equal to v Ω. This implies that ũ ∈ E(Rn).

Following exactly the same steps as in proving Theorem 1.1 (in particular points (i) and (ii)) we may reduce the general case to the case where the function u . For such a function u we remark that ũ ∈ E(Rn(or Ω) converges to u in E) (or E(Ω)) as λ → 1. The problem is then reduced to approximating a function u with compact support in Ω in terms of elements of D(Rn); this is obvious by regularization (as in point (ii) in the proof of Theorem 1.1).

Remark 1.3

If the set Ω is unbounded or if its boundary is not smooth, some partial results remain true: for example, if u ∈ E(Ω), we can define γvu on each bounded part Γ0 of Γ of class C², and γvu ∈ H¹(Γ0). If Ω is smooth but unbounded or if its boundary is the union of a finite number of bounded (n − 1)-dimensional manifolds of class C², then γvu is defined, in this way, on all Γ. Nevertheless the generalized Stokes formula