Reconstruction of a stationary flow from boundary data

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Link¨

oping Studies in Science and Technology. Theses

No. 853

Reconstuction of a Stationary Flow

From

Boundary Data

Tomas Johansson

Department of Science and Technology,

Campus Norrk¨

oping, Link¨

opings University, S-601 74 Norrk¨

oping, Sweden

Norrk¨

oping 2000

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Link¨

oping Studies in Science and Technology. Theses

No. 853

Reconstruction of a Stationary Flow

from

Boundary Data

Tomas Johansson

Department of Science and Technology,

Campus Norrk¨

oping, Link¨

oping University, S-601 74 Norrk¨

oping, Sweden

Norrk¨

oping 2000

LiU-TEK-LIC-2000:29

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Abstract

We study a Cauchy problem arising in fluid mechanics, involving the so-called stationary generalized Stokes system, where one should recover the flow from boundary measurements. The problem is ill-posed in the sense that the solution does not depend continuously on data.

Two iterative procedures for solving this problem are proposed and investi-gated. These methods are regularizing and in each iteration one solves a series of well-posed problems obtained by changing the boundary conditions. The advantage with this approach, is that these methods place few restrictions on the domain and on the coefficients of the problem. Also the structure of the equation is preserved.

Well-posedness of the problems used in these procedures is demonstrated, i.e., that the problems have a unique solution that depends continuously on data. Since we have numerical applications in mind, we demonstrate well-posedness for the case when boundary data is square integrable. We give convergence proofs for both of these methods.

Acknowledgement :

I would like to thank both my supervisors G. Bastay and S. Miklavcic for valuable discussions and for helpful suggestions on this thesis.

I express my gratitude to V. A. Kozlov for sharing his great knowledge in mathe-matics and for his invaluable advice during this work.

I also thank the following people: Bengt-Ove Turesson for helping me with LA_TEX,

Fredrik Berntsson for valuable discussions on ill-posed problem, the Department of Mathematics at Link¨opings University for giving interesting courses in mathematics, the Department of Science and Technology at Campus Norrk¨oping for providing an excellent research environment.

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1 Introduction

In this chapter, we give an introduction to ill-posed problems. We provide some background to this field and discuss some examples. We also give an outline of this thesis.

1.1 Ill-posed problems

First, we must understand what an ill-posed problem is. Following the mathe-matician J. Hadamard, by a well-posed problem we mean a problem satisfying the following conditions

1. A solution exists. 2. The solution is unique.

3. The solution depends continuously on the data of the problem.

If one or more of these above conditions are not fulfilled, then the problem is denoted as ill-posed . That there exist ill-posed problems can be evidenced by the examples below. Hadamard, in 1902, gave an example of an ill-posed problem for the Laplace equation, see [29]. Some areas where ill-posed problems arise are in X-Ray tomography, signal processing, image reconstruction, heat conduction and radar technology. For interesting examples in the last two fields, see L. Eld´en et al. [21] and L. E. Andersson et al. [3] respectively.

Ill-posed problems can arise as variations of direct problems. An example of a direct problem could be a problem where one wants to predict the future state of a physical situation. To predict the past state of the same physical problem given current information can lead to an ill-posed problem, denoted as an inverse problem, see J. B. Keller [33].

There is usually no great difficulty in satisfying the first of the above condi-tions. One can often rely on physical reasoning to argue that a solution exists. Furthermore, if a solution is not unique, one can place restrictions on the prob-lem to single out one solution. Thus, the second condition can also be satisfied. It is an unsatisfied third condition that makes an ill-posed problem difficult to handle. For example, it is known that computers have finite precision, i.e., one cannot in general avoid introducing some error in the process of solving a problem on a computer. For ill-posed problems, this means that one can get an unrestricted amount of error in a computer generated solution.

We now give three examples of ill-posed problems. First, we start with a simple one.

Example 1.1. Let f ∈ C1[0, 1] and let δ ∈ (0, 1). For n = 2, 3, . . ., define fn,δ(x) = f (x) + δ sin nx δ , x ∈ [0, 1]. Then fn,δ0 (x) = f0(x) + n cos nx δ , x ∈ [0, 1].

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We then have sup x∈[0,1] |f (x) − fn,δ(x)| = δ, while sup x∈[0,1] |f0(x) − f_n,δ0 (x)| = n.

If we consider f as given exact data and fn,δ as a bounded perturbation of f ,

then the error in the result, the derivate, can be arbitrarily large. Hence, the derivate does not depend continuously on the data (in the supremum norm) and we have an ill-posed problem.

We now recall Hadamard’s classic example. Example 1.2. Consider the following problem

   ∆un= 0 (x, y) ∈ R × (0, ∞), un(x, 0) = 0 x ∈ R, ∂yun(x, 0) = ϕn(x) x ∈ R,

where ϕn(x) = n−1sin nx, for n = 1, 2, . . . . The solution is

un(x, y) = n−2sin nx sinh ny.

As n tends to infinity, the data sequence ϕn tends uniformly to zero. However,

for y > 0, we see that un(x, y) tends to infinity. Hence, the third condition in

the definition of well-posedness is not satisfied.

A common situation where ill-posed problems occur, is the following. Example 1.3. Let B be an invertible operator from one Hilbert space to an-other, and let y be given. If the inverse of B is not bounded, then the problem of finding x such that

Bx = y, (1.1)

is ill-posed.

It is well known from operator theory that a linear compact operator on a Hilbert space with infinite dimension, has no bounded inverse. Linear compact operators are common in applications. In fact, problems that involve integral equations belong to a large physical class. Let Ω be a bounded domain of Rn. From the theory of Fredholm, one knows that an integral equation of the first kind, i.e., the problem of finding f for a given h, related to each other by

Z

Ω

G(x, y)f (x) dx = h(y), ∀y ∈ Ω,

where G ∈ L2(Ω × Ω), is a compact operator equation of the form (1.1), see Groetsch [27]. Here, G is called a kernel.

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1.2 Outline of the thesis 3

The foundation of the theory of ill-posed problems was laid by A. N. Tikhonov in the sixties, see [56] and [57]. Later, F. John, M. M. Lavrente’v and V. K. Ivanov made important contributions to the subject. For a brief sur-vey, see the introduction to the monograph by V. A. Morozov [45].

One manner in which to solve an ill-posed problem of the form (1.1), is to use so-called regularizing methods. For example, one can modify problem (1.1), by adding an extra term, so that the resulting operator has a bounded inverse. Then, in the sense of some norm, one lets the extra term tend to zero. The extra term is designed so that when it tends to zero, the solution tends to a solution of (1.1). One such method is the method called Tikhonov regular-ization, see Tikhonov [56], [57]. Tikhonov regularization is an example of a non-iterative method. Such methods which are based on adding extra terms, are commonly denoted as methods of quasi reversibility. The standard reference is Lattice and Lions [42].

Another way to solve an ill-posed problem is to try to use iterative meth-ods. One example of a common iterative method is the method of Landweber see Landweber [41] and Fridman [24]. Another example are the conjugate gra-dient methods by Hestenes and Stiefel [32]. For an introduction to the subject of iterative regularizing methods, see Groetsch [28]. For a more thorough text, see Engl et al. [22].

Some difficulties are incurred when one uses methods of quasi reversibil-ity. The extra terms one adds are not always unique. The new problem that arises might be nonstandard, which implies that there is no standard numerical way to handle the perturbed problem. This observation led V. G. Maz’ya and V. A. Kozlov to study iterative methods for ill-posed problems based on partial differential equations, see [34] and [35]. In these methods, one tries to preserve the structure of the original equation. Later, G. Bastay used and developed these ideas further for some boundary value problems for the wave equation and heat equation. G. Bastay also showed some optimal convergence results for these methods, see [5].

1.2 Outline of the thesis

In this section, we present two examples. The first example is of a physical situation, whose problem type we study in this thesis. The second example shows that problems of this type are ill-posed. We then give an outline of the remainder of this thesis.

Example 1.4. Let Γ0and Γ1be two spheres in R3with radii r0and r1satisfying

0 < r1 < r0. Let Ω be the domain which lies between these two spheres and

assume that this region is filled with a fluid, for example water. Further, suppose that both spheres rotate. The object is to determine the velocity of the fluid at the inner sphere by taking measurements on Γ0. From fluid dynamics, a

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approximation to the fluid velocity, by solving the following problem        µ∆u + (b(x) · ∇)u − ∇p = f in Ω, divu = 0 in Ω, u = ϕ on Γ0, pν − µ(∇u + ∇uT_{)ν = ψ} _{on Γ} 0. (1.2)

Here u = (u1, u2, u3) is the velocity field of the fluid and p is the pressure. The

function f represents a body force acting on the fluid, ϕ is the velocity of the fluid at the outer sphere, ψ is the measured surface pressure and b is a function with divb = 0. We will soon specify which functions spaces these functions should belong to. The function ν is the outward unit normal to boundary of Ω. The number µ is the viscosity of the fluid. Since µ will not affect our calculations, we will assume that µ = 1 in this thesis.

From the following example it can be seen that problems of the above type are ill-posed.

Example 1.5. Define

Ω = {(x, y) : x ∈ R and y ∈ [0, 1]}. The boundary, Γ, is the union of the disjoint pieces

Γ0= {(x, y) ∈ R2and y = 0} and Γ1= {(x, y) ∈ R2 and y = 1}.

Consider the problem                                                ∆u1− ∂p ∂x = − 2 sin nx n in Ω, ∆u2− ∂p ∂y = 0 in Ω, divu = 0 in Ω, u1(x, 0) = sin nx n3 on Γ0, u2(x, 0) = 0 on Γ0, ∂u1(x, 0) ∂y + ∂u2(x, 0) ∂x = 0 on Γ0, p(x, 0) − 2∂u2(x, 0) ∂y = 0 on Γ0, (1.3)

The solution to this problem is

u(x, y) = sin nx cosh ny

n3 , − cos nx sinh ny n3 , p(x, y) = −2 cos nx n2 .

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Despite the fact that the input tends to zero uniformly when n → ∞, the solution u tends to infinity, for each y > 0. As in the examples in Section 1.1, we see that this problem is ill-posed in Hadamard’s sense.

Regularizing methods are needed if one wants to solve problems of the form (1.2) in a stable way.

In this report, we present and prove convergence of iterative procedures for problems of the form

       ∆u + (b(x) · ∇)u − ∇p = 0 in Ω, divu = 0 in Ω, u = ϕ on Γ0, pν − (∇u + ∇uT)ν = ψ on Γ0. (1.4)

This problem is called the generalized Stokes system. Here, u = (u1, . . . , un),

where n ≥ 2, is a vector-valued function and p is a scalar function. We assume that Ω is a bounded domain of Rn with boundary of class C2_{, such that the}

boundary is the union of two closed and disjoint pieces, Γ0and Γ1. The vector

ν is the outward unit normal to the boundary. The function b : Ω → Rn is as-sumed to have bounded first derivatives in Ω and satisfy divb = 0. Furthermore, we assume that the boundary functions ϕ and ψ are in L2_(Γ

0)n.

Recently, G. Bastay, V. A. Kozlov and B. O. Turesson considered a Cauchy problem for a scalar parabolic equation with non stationary coefficients, see [6]. To solve this problem, they proposed iterative procedures involving solutions of a series of well-posed problems. They proved convergence of these procedures and performed numerical experiments. They also showed that these methods work equally well for an elliptic Cauchy-problem.

In this thesis, we use their ideas and show how to apply them to problems of the form (1.4). Consider the following two problems

       ∆u + (b(x) · ∇)u − ∇p = 0 in Ω, divu = 0 in Ω, u = η, on Γ1, pν − (∇u + ∇uT)ν = ψ, on Γ0. (1.5) and        ∆v − (b(x) · ∇)v − ∇q = 0 in Ω, divv = 0 in Ω, v = 0 on Γ1, qν − (∇v + ∇vT_{)ν + (b · ν)v = ξ} _{on Γ} 0. (1.6)

In this thesis we prove that the following algorithm, will generate a solution to (1.4).

• Choose an arbitrary function η0∈ L2(Γ1)n.

• The first approximation u0 and p0 to the solution u and p is obtained

by solving problem (1.5) with the first boundary condition changed to u0= η0on Γ1.

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• Then we find v0and q0by solving problem (1.6) with the second boundary

condition changed to

q0ν − (∇v0+ ∇vT0)ν + (b · ν)v0= u0− ϕ on Γ0.

• When the solutions, uk−1 and pk−1, and vk−1 and qk−1, have been

con-structed, the approximation, uk and pk, is the solution to (1.5) with data

uk= ηk on Γ1, where

η_k= uk−1+ γ(qk−1ν − (∇vk−1+ ∇vTk−1)ν + (b · ν)vk−1) on Γ1,

and γ is a fixed positive number.

• Then vk and qk are the solution to (1.6) with data

qkν − (∇vk+ ∇vTk)ν + (b · ν)vk= uk− ϕ on Γ0.

We see that there are several problems that must be solved in order to prove that the above procedure converges. One must, among other things, show that the problems used in this algorithm are well-posed and that solutions to these problems have traces on the boundary.

Chapter 2 in this thesis contains some basic theory from analysis, inverse problems, and partial differential equations that are use in order to address these problems.

As mentioned above, in Chapter 3 we present some basic facts from fluid dynamics and show how equation (1.4) is derived.

The first problem we solve is that of whether (1.5) is well-posed for boundary data in L2. The proof that it is well-posed depends on the existence of a solution u ∈ H2(Ω)n and p ∈ H1(Ω) to problem (1.5) for boundary data with sufficient smoothness. To prove this, we examine, for the case η = 0, weak solutions to (1.5) that satisfy a system of the form

     Find (u, p) ∈ H1 Γ1(Ω) n_{× L}2_{(Ω) such that:} a(u, v) + b(p, v) = F (v), ∀v ∈ H1 Γ1(Ω) n_, b(q, u) = (g, q)L2_(Ω), ∀q ∈ L2(Ω). (1.7)

Here, F is a certain functional on H1 Γ1(Ω)

n_{, a( · , · ) and b( · , · ) are given}

contin-uous bilinear forms. For a definition of the space H1 Γ1(Ω)

n_{, see Section 2.3 and}

Section 5.2.1. A paper by Brezzi, see [11], deals with problems of the form (1.7) in more general spaces. In Chapter 4, some parts of this theory are presented for the case when H1and H2 are separable Hilbert spaces.

In Chapter 5, we show that there indeed exists a solution u ∈ H2_(Ω)n _and

p ∈ H1_{(Ω) to (1.5) for boundary data with sufficient smoothness. In fact, we}

prove more than this. We prove that the more general problem        ∆u + (b(x) · ∇)u − ∇p = f in Ω, divu = g in Ω, u = η on Γ1, pν − (∇u + ∇uT)ν = ψ on Γ0, (1.8)

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has a unique solution u ∈ H2_(Ω)n _{and p ∈ H}1_{(Ω), if f ∈ L}2_(Ω)n_{, g ∈ H}1_(Ω),

η ∈ H3/2_(Γ

1)n, and ψ ∈ H1/2(Γ0)n. To prove these facts, we investigate weak

solutions that satisfy (1.7). We then show that system (1.8) is elliptic in the sense of Agmon, Douglis and Nirenberg and that it satisfies the complement-ing condition. Then, uscomplement-ing a result of Agmon, Douglis and Nirenberg [2], the regularity of this solution is investigated.

In Chapter 5 similar results can be found for an adjoint problem to (1.8) of the form        ∆v − (b(x) · ∇)v − ∇q = 0 in Ω, divv = 0 in Ω, v = ζ on Γ1, qν − (∇v + ∇vT)ν + (b · ν)v = ξ on Γ0. (1.9)

In Chapter 6, problem (1.4) is investigated employing the results of Chapter 5. For the numerical applications we have in mind we consider the case where the data belong to certain L2-spaces. After specifying what we mean by a weak solution to (1.5) for such boundary data, the existence of a unique weak solution u ∈ L2(Ω)n and p ∈ (H1(Ω))∗ is proved. This result can be found in Lemma 6.3. Using a local estimate, it follows that this solution has traces on the boundary. Specifically, we prove that u|Γ0 ∈ H

1/2_(Γ

0)n, and

if η = 0, then u ∈ H3/2_(Γ

1)n and p ∈ H1/2(Γ1). Similar results are proved

for (1.9).

To prove that the above algorithm converges, we introduce two operators K : L2(Γ1)n→ L2(Γ0)n, where Kη = u|Γ0, (1.10)

and

K1: L2(Γ0)n → L2(Γ0)n, where K1ψ = v|Γ0. (1.11)

Here u is the solution to problem (1.5) with ψ = 0 and v is the solution to problem (1.5) with η = 0. To solve problem (1.4), is then equivalent to solving

Kη = ϕ − K1ψ, (1.12)

for η. We prove that K is a compact operator and that the kernel of K contains only zero. That the kernel of K contains only zero is proved in Section 6.3.1. This result follows from a theorem by Regbaoui [48] about strong unique contin-uation of solutions to the generalized Stokes system (1.4). Iterative regularizing methods are then used to solve problem (1.12). Since the most common itera-tive regularizing methods involve the adjoint operator, K∗_{, this operator is also}

examined in Chapter 6.

Finally, iterative methods from the theory of ill-posed problems are used to solve equation (1.12). We concentrate on two iterative methods, a one-step iterative method and a minimal error procedure. We prove that the above algo-rithm corresponds to the Landweber method (see Section 2.2.1 for a description of this method) for equation (1.12). Convergence for this one-step method is

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proved in Theorem 6.16. This proof follows from a result on convergence for the Landweber method. We also present another algorithm which corresponds to a conjugate gradient method for (1.12). This algorithm can also be found in Chapter 6 and the proof of convergence is found in Theorem 6.17.

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9

2 Mathematical preliminaries

In this chapter, we present the necessary mathematical tools to be employed in the following chapters, from analysis, from the theory of ill-posed problems, and from the theory of partial differential equations.

2.1 Basic notations

In this report, H will always denote a separable, real Hilbert space, endowed with an inner product ( · , · )H. The corresponding norm will be denoted by

|| · ||H. If we deal with more then one Hilbert space, we will index them with

subscripts, e.g. H1. A linear and continuous operator, mapping an element

from one Hilbert space, H1, to another Hilbert space, H2, will be denoted by T .

For the range of T , we write R(T ). The kernel of T is ker(T ). If ker(T ) = {0}, then we can define an inverse mapping from R(T ) to H1.

Let T be a bounded linear operator. We denote by T∗ the adjoint operator of T , i.e., the bounded and linear operator mapping H2into H1, such that

(T ϕ, ψ)H2 = (ϕ, T

∗_ψ)

H1, (2.1)

holds for all ϕ ∈ H1 and ψ ∈ H2.

Some further definitions. By a domain Ω in Rn, we mean an open and connected subset of Rn. The support of a function u defined in Ω, is the closure of the set

M = {x ∈ Ω : u(x) 6= 0}, and is denoted as supp(u).

Let m be a positive integer. A function belongs to Cm(Ω) if it has continuous derivatives of order m throughout Ω. If a function has compact support in Ω in addition to having continuous derivatives of order m, then the function is defined to be an element of Cm

0 (Ω). The space Cm(Ω) consists of those functions in

Cm_{(Ω) whose derivatives of order m = 0, 1, . . . , have continuous extentions to}

Ω. A function belongs to C∞(Ω), if it is infinitely differentiable. Finally, we say that a function is an element of the space C₀∞(Ω) if its support is a compact set of Ω, and if the function has derivatives of any order. These spaces can be seen as topological vector spaces, see Chapter 1 in Rudin [50] for a discussion of this topic.

We use a superscript N to indicate that a function is vector valued. As an example, a vector valued function u = (u1, . . . , uN), where N ≥ 2, is said to be

an element of Ck(Ω)N, if its components ui belong to Ck(Ω), for i = 1, . . . , N .

By Lp_{(Ω), where p is a positive number, we mean the set of all Lebesgue}

measurable functions u such that Z

Ω

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The notation of multi-indices will be used throughout this thesis along with the following conventions

∂_xα= ∂ |α| ∂xα1 1 ∂x α2 2 . . . ∂x αn n , xα= xα1 1 · · · x αn n , for x = (x1, . . . , xn) ∈ Rn, where α = (α1, . . . , αn) and |α| = α1+ . . . + αn.

If no confusion arises, we shall write ∂α_{instead of ∂}α x.

Finally, we point out that throughout this thesis the symbol C will denote an arbitrary positive constant. Unless otherwise specified, different constants occuring in the same series of calculations will be denoted by the same C.

2.2 Regularizing methods

Let T : H1 → H2 be a bounded linear operator, such that ker(T ) = {0}.

Consider the following problem. For a given y ∈ R(T ), find x ∈ H1such that

T x = y. (2.2)

Since ker(T ) = {0}, there exists an inverse mapping from R(T ) to H1. It is

well-known that if R(T ) is not closed, then the inverse mapping is not bounded, see chapter 3 in Birman and Solomjak [8]. So, if R(T ) is not closed, then (2.2) is an ill-posed problem.

Consider now the case when there is some error in y. That is, instead of y, we have yδ where

||y − yδ||H2 ≤ δ, (2.3)

with δ > 0. If R(T ) is not closed, we cannot guarantee that the inverse mapping of yδ _{(if y}δ _{∈ R(T )), will approximate x, since this mapping is not continuous.}

Instead, one shall attempt to find a family of continuous operators that in some sense approximate this inverse mapping.

Definition 2.1. Let T : H1 → H2, be a bounded linear operator between

Hilbert spaces, with ker(T ) = {0}. Also, let {Rk}∞k=1be a family of continuous

operators (not necessarily linear), where

Rk: H2→ H1, for k = 1, 2 . . . .

This family is said to be a regularizing family for equation (2.2) if, for each y ∈ R(T ), there exists a parameter choice rule

k : (0, ∞) × H2→ Z+,

and a function

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2.2 Regularizing methods 11

where ε(δ) → 0 as δ → 0, such that the inequality ||y − yδ_||

H2 ≤ δ,

implies the estimate

||Rk(δ,yδ₎yδ− x||_H₁ ≤ ε(δ).

For a specific y ∈ R(T ), a pair ({Rk}∞k=1, k( · , · )) is called a convergent

regular-izing method .

We note that k is defined for y ∈ R(T ), thus, k also depends on y. Observe that we approximate the inverse mapping of T with operators Rk, and measure

convergence in the strong operator norm.

In those equations to be studied in this thesis, ker(T ) = {0}. If ker T 6= {0}, then one can instead consider the so-called generalized inverse and obtain similar results, see Engl et al. [22].

2.2.1 The Landweber method

An iterative regularizing method for problem (2.2), is the the Landweber itera-tion. This method was first investigated by Landweber [40] and Fridman [24]. It can be expressed as

xk = xk−1+ ωT∗(y − T xk−1), (2.4)

for k ≥ 1. Here x0 is equal to some initial guess, which starts the procedure,

y ∈ R(T ) and 0 < ω < ||T ||−2. Assuming that x0= 0, it is not difficult to show

that

xk= ωPk−1(ωT∗T )T∗y,

where Pk−1 is a polynomial of degree k − 1. Hence, the Landweber method

generates a family of operators.

One can show that if there is some error in y (see (2.3)), and if one iterates according to the scheme

xδ_k= xδ_k−1+ ωT∗(yδ− T xδ k−1), with xδ 0=0, then ||xδ k− xk||H1 ≤ √ kδ. (2.5)

Furthermore, equality can occur, see Engl et al. [22, p. 156]. Let T x = y. From the triangle inequality, it follows that

||x − xδ

k||H1≤ ||x − xk||H1+ ||xk− x

δ k||H1.

We will see in the theorem below, that the first term on the right tends to zero. The second term can diverge because of the estimate in (2.5). This is typical of the difficulty found when trying to regularize ill-posed problems. Therefore, we must have some stopping criterion in the iteration procedure. For a general discussion, see Engl et al. [22]. For the case of a problem that stems from a partial differential equation, see Bastay [5] .

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Theorem 2.2. Let T : H1 → H2 be a bounded linear operator between Hilbert

spaces, with ker(T ) = {0}. Assume that T x = y and let xk and xδk be the k-th

iterate in the Landweber method corresponding to y and yδ _{respectively, where}

||y − yδ_||

H2 ≤ δ. If 0 < ω < ||T ||

−2_{, then we have:}

(i) xk→ x, as k → ∞, in the Landweber iteration.

(ii) There exists a function k : (0, ∞) × H2→ Z+, such that

sup δ→0 ||xδ k(δ,yδ)− x||H1 = 0, where xδ_k = xδ_k−1+ ωT∗(yδ− T xδ k−1). Moreover, k(δ, yδ_{) = O(δ}−2_).

2.2.2 The minimal error method

Another iterative regularizing procedure is the minimal error method . This method was first investigated by Craig [15] and Shamanskii [52]. It can be described as:

First, calculate

r0= y − T x0 and d0= T∗r0,

where x0is some initial guess. Then iterate according to:

αk = krkk2 kdkk2 , xk+1 = xk+ αkdk, rk+1 = rk− αkT dk, βk = ||rk+1||2 ||rk||2 , dk+1 = T∗rk+1+ βdk.

It is easy to prove that rk= y − T xk for k ≥ 0 in this algorithm.

In each iteration, the error norm ||x − xk||H1 is minimized over the k-th

Krylov space, Kk(T∗(y − T x0), T∗T ), where

Kk(x, T∗T ) = span{x, T∗T x, (T∗T )2x, . . . , (T∗T )k−1x},

see Hanke [31, p. 18].

The minimal error method, which is one of the so-called conjugate gradient methods (see Hestenes and Stiefel [32]), is known to be a more powerful method than that of Landweber. However, the price one pays is that the regularizing operators that this method generates, will be nonlinear. The proof of conver-gence for the minimal error method, is therefore more difficult and technical than that of Landweber. The proof of the following theorem can be found in Hanke [31, pp. 57–66].

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2.3 Sobolev spaces 13

Theorem 2.3. Let T : H1 → H2 be a bounded linear operator between Hilbert

spaces, with ker(T ) = {0}. Assume that T x = y and let xk and xδk be the

k-th iterate in the minimal error method corresponding to y and yδ _{respectively,}

where ||y − yδ_||

H2 ≤ δ. Then we have:

(i) xk→ x, as k → ∞, in the minimal error iteration.

(ii) There exists a function k : (0, ∞) × H2→ Z+, such that

sup

δ→0

||xδ

k(δ,yδ₎− x||H1 = 0,

where xδ

k is the k-th approximation in the minimal error procedure.

Moreover, if x = (T∗T )µ/2_{ω, for ||ω||} H1 ≤ ρ, then ||xδ k(δ,yδ₎− x||H1≤ Cρ 1/µ+1_δµ/µ+1_.

2.3 Sobolev spaces

We start with some definitions.

Definition 2.4. Let Ω be a domain in Rn. A function u ∈ L2_{(Ω) possesses a}

generalized derivative ∂α_{u := v in L}2_{(Ω), if v ∈ L}2_{(Ω) and if}

Z Ω vφ dx = (−1)|α| Z Ω u ∂αφ dx, holds for every φ ∈ C∞

0 (Ω).

Definition 2.5. Let k be a positive integer. We denote by Hk_{(Ω), the vector}

space consisting of functions u, whose generalized derivatives exist for |α| ≤ k and belong to L2_{(Ω). Here, we identify H}0_{(Ω) with L}2_{(Ω). The norm on H}k_(Ω)

is ||u||2 Hk(Ω)= X |α|≤k ||∂α_u||2 L2_(Ω).

The space Hk(Ω) is called a Sobolev space.

Definition 2.6. Let k be a positive integer. By Hk

0(Ω), we mean the closure of

C₀∞(Ω) in the Sobolev norm || · ||_Hk_(Ω).

Definition 2.7. Let k be a positive integer. The dual space of Hk

0(Ω), i.e., the

space of linear functionals on Hk

0(Ω), is denoted by H−k(Ω). The dual space of

Hk_{(Ω) is denoted by (H}k_(Ω))∗_.

The above definitions can be extended to the case where k is not an integer, see Chapter 7 in Adams [1].

A proof of the following theorem can be found in Renardy and Rogers [49, p. 222].

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Theorem 2.8. Let Ω be a bounded domain with ∂Ω of class Ck_{, where k is a}

positive integer. Then there exists a continuous linear operator

γk: Hk(Ω) → k−1 Y j=0 Hk−j−1/2(∂Ω). For φ ∈ Ck(Ω), we have γk(φ) = (φ, ∂φ ∂ν, . . . , ∂k−1_φ ∂νk−1),

where ν is the outward unit normal to ∂Ω. We call the operator γk the trace

operator. The kernel of γk is

ker(γk) = H0k(Ω).

For example, this means that if ∂Ω is of class C1_{, we can characterize H}1 0(Ω)

as those u ∈ H1_{(Ω) with γ}

1(u) = 0.

We now recall the following basic inequality from integration theory, a proof of which can be found in Rudin [51, p. 63].

Theorem 2.9. (Cauchy’s Inequality) Let X be a measure space and let f and g be measurable functions on X. Then

Z X |f g| dµ ≤ Z X |f |2_dµ1/2Z X |g|2_dµ1/2_.

In later chapters, we shall consider so-called bilinear forms and formulate our generalized Stokes system, as a system of bilinear forms. To guarantee that there exists a unique solution for such a system, there are some conditions that must be satisfied. In order to confirm that these conditions are satis-fied, we need the following two theorems. For a proof of the first theorem, see Braess [9, pp. 30–31].

Theorem 2.10. Let Ω be a bounded domain with boundary of class C2 and let Γ1 be a closed part of the boundary, with |Γ1| 6= 0. Assume that u ∈ H1(Ω) and

that γ1(u)|Γ1 = 0, where γ1 is the trace operator. Then there exists a constant

C > 0 such that

||u||H1_(Ω)≤ C||∇u||_L2_(Ω).

The next theorem is a variant of Korn’s inequality. For a proof of this, see Oleinik et al. [46, p. 21].

Theorem 2.11. Let Ω be a bounded domain with boundary of class C2_{. Let}

Γ1 be a closed part of the boundary, with |Γ1| 6= 0. Assume that u ∈ H1(Ω)n

satisfies γ1(u)|Γ1 = 0. Then there exists a constant C1> 0 such that

C1||u||2H1(Ω)n ≤ Z Ω n X i,j=1 ∂u_i ∂xj +∂uj ∂xi 2 dx.

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2.4 Elliptic partial differential equations 15

2.4 Elliptic partial differential equations

We say that L = L(x, ∂) is a linear partial differential operator , if L = L(x, ∂) = X

|α|≤m

aα(x)∂α,

where m is a positive integer. Here, x belongs to some domain Ω and aα(x) are

real valued functions on Ω. By the degree of a term aα(x)∂α in the above sum,

we mean |α|. If L is not identically zero in Ω, then the order of L is the integer m1= max{|α| : aα6≡ 0 on Ω}.

Let ξ ∈ Rn, the symbol of L(x, ∂) is defined to be L(x, ξ) = X

|α|≤m

aα(x)(iξ)α,

while the principal part is given by Lp(x, ξ) = X

|α|=m1

aα(x)(iξ)α.

As stated in the preceding chapter, we shall be concerned with systems of linear partial differential equations. A system of linear partial differential equations has the form

N

X

j=1

Lij(x, ∂)uj = fi, i = 1, . . . , N, (2.6)

where each Lijis a partial differential operator defined as above, and N is a

pos-itive integer. In this report, the order of each Lij is at most two. In particular,

we will deal with a so-called elliptic system. For systems of partial differential equations, one can define ellipticity in many ways. We use a definition first suggested by A. Douglis and L. Nirenberg [19].

Definition 2.12. Assign weights si ∈ Z to each equation and weights tj ∈ Z

to each dependent variable in (2.6), such that the order of each Lij does not

exceed si+ tj. Define Lpij to be the principal part of Lij if the order of Lij

equals si+ tj, otherwise let Lpij = 0. The principal part of the system is the

matrix Lp= Lp_ij

1≤i, j≤N. If we can find weights such that

det Lp(x, ξ) = det    Lp₁₁(x, ξ) . . . Lp_1N(x, ξ) .. . . .. ... Lp_{N 1}(x, ξ) . . . Lp_{N N}(x, ξ)   6= 0, (2.7)

for every x ∈ Ω and for every nonzero ξ ∈ Rn, then the system is called elliptic (in Ω). The total order of the system is the degree of det Lp. If N = 2, then we additionally assume that the polynomial P (τ ) = det Lp(x, ξ + τ ζ) has half of its roots in the upper half plane Im τ > 0.

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In the remainder of this report, an elliptic system will be defined according to this definition. We assume that the weights are chosen such that si≤ 0 and

tj ≥ 0. This can always be done since we can subtract a constant from each si

and add this constant to each tj.

It is not obvious how to determine if there exist weights for a given system, such that (2.7) holds. However, there exist equivalent definitions of ellipticity, see Volevic [58]. For a discussion on these different definitions of ellipticity, we refer to Cosner [14].

Assume that system (2.6) is elliptic and that the total order is 2M , where M > 0. A set of boundary conditions corresponding to system (2.6), can then be written in the form

N

X

j=1

Blj(x, ∂)uj = gl, x ∈ ∂Ω and l = 1, . . . , M, (2.8)

where each Blj is a linear partial differential operator.

To say something about regularity of a solution to an elliptic system, i.e., how smooth a solution is in Ω, we must impose some restrictions to (2.8). Definition 2.13. Assume that we have an elliptic system of order 2M , where M > 0 is an integer. Let siand tj be weights chosen according to the definition

of ellipticity (see Definition 2.12). Assign weights rl, where rl≤ 0 is an integer

for l = 1, . . . , M , to each boundary condition such that the order of each Blj is

bounded by rl+ tj. Define B p

lj to be the principal part of Blj if the order of

Blj equals rl+ tj, otherwise let B_ljp = 0. Let Bp= B_ljp_{1≤l≤M, 1≤j≤N}. We say

that the complementing condition holds at x0∈ ∂Ω, if the system

Lp_(x

0, ξ − iν(x0)∂t) u(t) = 0 for t > 0,

Bp((x0, ξ − iν(x0)∂t)u(t) = 0 for t = 0,

has no nontrivial integrable solutions. Here ξ is an arbitrary tangential vector to ∂Ω at x0and ν(x0) is the outward unit normal to ∂Ω.

The following result can be found in Renardy and Rogers [49, p. 308]. Theorem 2.14. Let {si}Ni=1, {tj}Nj=1, and {rl}Ml=1 be the weights prescribed by

Definitions 2.12 and 2.13. Let k ≥ k1, where k1 = max(0, maxl(rl+ 1)), be

a positive integer and let t0 = maxjtj. Assume that u = (u1, . . . , uN), where

uj∈ Hk1+tj(Ω) for j = 1, . . . , N , is a solution to the elliptic system (2.6), with

the boundary conditions (2.8), which satisfies the complementing condition. As-sume that ∂Ω is of class Ck+t0_{, that the coefficients of L}

ij are in Ck−si(Ω), and

that the coefficients of Blj are in Ck−rl(∂Ω). Assume also that fi∈ Hk−si(Ω),

and that gl ∈ Hk−rl−1/2(∂Ω). Then uj is in Hk+tj(Ω) and the following

in-equality holds N X j=1 ||uj||_Hk+tj_(Ω)≤ C _XN i=1 ||fi||Hk−si(Ω)+ M X l=1 ||gl||Hk−rl−1/2(∂Ω)+ N X j=1 ||uj||L2_(Ω) .

If the solution is unique, then the term PN

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17

3 Fluid dynamics

It is not our intention to give a complete and rigorous derivation of the equations of fluid motion, nor to discuss the merits of the physical assumptions that are involved in such a derivation. The available literature adequately performs this task, and we refer the reader to some of the classic references to this extensive subject Batchelor [7], Lamb [39], Landau and Lifshitz [40], and Meyer [43]. However, it is convenient to give a brief presentation of the subject, to introduce the reader to the most important, physical points involved.

3.1 The continuum hypothesis and other assumptions

In this thesis the term ”fluid” refers to any amorphous material which can-not withstand any tendency to deform it under the action of applied forces. The fluid’s microstructure, i.e., particle size, particle motion and interactions, is reflected by bulk properties such as density, ρ, and viscosity, µ. In a math-ematical study of the dynamics of fluids we consider a mathmath-ematical n-volume (denoted dx) to be sufficiently small so as to ensure well-defined mathematical limiting processes involved in the concepts of spatial derivatives, while at the same time sufficiently large to justify the definitions of the local density and viscosity, and even fluid velocity, u, and pressure, p. This reasoning is called the continuum hypothesis. The term amorphous suggests a material which is both homogeneous (independent of position within the region of interest) and isotropic (independent of direction), thus justifying in part the reference to the above macroscopic scalar quantities.

In this thesis we assume that the fluid under consideration is Newtonian in character. This implies the existence of a linear relationship between the action of an external force (stresses) and the responding rates of displacement.

Later, we shall assume the fluid is assumed to be incompressible, that is, the fluid density is constant, independent of any fluid motion. However, in the initial derivation we take the density to be a function of both position and time.

3.2 A formal derivation of the equations of continuity and

momentum

For the rest of this chapter, we assume that all functions and derivatives we are using, are continuous. All functions which we differentiate are assumed to be at least of class C1_{. The surfaces and domains appearing lie in R}n_{. We adopt the}

so-called summation convention in this chapter, i.e., the convention that when an index (subscript or superscript) is repeated in an expression one is to sum over that index from 1 to n. Physical applications arise in cases when n = 2 or n = 3.

In the derivation of the equations of fluid motion we take into consideration temporal changes to two important physical properties of a fluid: mass and momentum. Consider an arbitrary, mathematical surface, ∂Ω, with outward directed unit normal vector, ν, enclosing a volume, Ω, fixed in the body of the

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liquid. We are interested in obtaining expressions for the rates at which the mass and the momentum of a fluid contained within the mathematical surface, ∂Ω, changes with time. We first consider the changes to the mass.

At any time, t, the amount of material, Mt, found within ∂Ω is given by the

integral of the density ρ(x, t) over the volume Ω Mt=

Z

Ω

ρ dx. (3.1)

The rate at which the mass within ∂Ω changes is determined partly by the rate at which mass is transported into Ω by the fluid flow itself, that is, the flux of material in through ∂Ω, and partly by the rate at which material is created within the region V . This balance can be expressed by the relation

∂ ∂t Z Ω ρ dx = − Z ∂Ω ρu · ν dS + Z Ω smdx. (3.2)

The second term on the right-hand side represents a source or sink term describ-ing the creation or destruction of fluid within the control volume, Ω. Applydescrib-ing Gauss’ theorem to the flux term, we are lead to the expression

Z

Ω

∂ρ

∂t + ∇(ρu) − sm dx = 0. (3.3)

As the region Ω bounded by the surface ∂Ω was chosen arbitrarily, the integrand itself vanishes locally. For source free regions, sm = 0, this gives rise to the

so-called equation of mass conservation, often referred to as the equation of continuity,

∂ρ

∂t + ∇(ρu) = 0. (3.4)

For an incompressible, constant density fluid this equation reduces to

divu = 0. (3.5)

In a similar manner, we consider the total momentum of the fluid contained within ∂Ω. This is given by the integralR

Ωρu dx.

To determine the rate at which the momentum of the fluid within Ω changes, we rely primarily on Newton’s second law. The momentum of the fluid within the arbitrarily chosen surface is influenced partly by the flux of momentum through the surface, partly by the forces that act on the fluid at any given time, and partly by the creation or destruction of momentum occuring within the volume. This balance of forces and momentum can be expressed mathematically as ∂ ∂t Z Ω ρu dx = − Z ∂Ω (ρu)u · ν dS + F + Z Ω spdx. (3.6)

For convenience we assume no creation or destruction of momentum occurs (sp= 0). The externally applied force, F , can be partitioned into a total body

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3.2 A formal derivation of the equations of continuity and momentum 19

or volume force acting locally throughout Ω, and a total surface or contact force acting on the surface S. Let f denote the sum of all local body forces per unit volume, such as gravity and electromagnetic forces, etc., acting in the bulk of the liquid, and let T represent the local surface forces per unit area. With these definitions we can write F as

F = Z ∂Ω T dS + Z Ω f dx. (3.7)

For the given surface, the surface force, T , can be written as a vector with one component parallel to the surface normal, and components perpendicular to this normal. Furthermore, T can be expressed in terms of a second order tensor, σij,

Ti= σijνj, (3.8)

where νj is the jth Cartesian component of the surface normal vector, ν. The

σij is traditionally called the Cauchy stress tensor.

Employing once again the divergence theorem, the surface force contribution to F becomes Z ∂Ω TidS = Z ∂Ω σijνjdS = Z Ω ∂σij ∂xj dx. (3.9)

As stated in (3.1), we restrict our study to those fluids most familiar to us, such as water, gases, simple molecular oils, fuels, etc.. For these cases the stress tensor is, to a good approximation, proportional to the relative rate of shear that exists between neighboring layers of fluid. In fact, it can be argued that for an isotropic fluid, the following relationship is valid,

σij = −pδij+ dijkl ∂uk ∂xl = −pδij+ 2µ(eij− 1 3ekkδij). (3.10) where p is the dynamic fluid pressure, and

eij = 1 2( ∂ui ∂xj +∂uj ∂xi ),

is the rate of strain tensor, see Lamb [39]. For an incompressible fluid, the trace of the latter tensor, ekk, vanishes.

Thus, from the derivations above and the assumption of an arbitrary mathe-matical volume, Ω, the momentum equation for an incompressible fluid reduces to the local force balance

ρ(∂u

∂t + u ∇u) = f − ∇p + µ∆u. (3.11)

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3.3 Boundary conditions

The solution space of vector valued functions, satisfying equation (3.5) and equation (3.11) diminishes upon application of conditions to be satisfied on the boundary of the fluid domain. The two most natural boundary conditions relevant to solid surfaces involve either the fluid velocity or the dynamic force applied to surfaces.

The motion of a body through a viscous liquid extends a constraint on the fluid velocity of the fluid in contact with the body. A (piecewise) regular surface described by the equation F (x, t)=0 must satisfy the equation

∂F

∂t + u · ∇F = 0, (3.12)

where u = x0(t). This equation can be re-expressed as

(u − v) · ν = 0, (3.13)

where v denotes the velocity of the surface in the direction of its normal. This condition is valid for both viscous and inviscid fluids. For viscous fluids, the more restrictive, physical condition

u = v, (3.14)

applies. That is, the fluid velocity in contact with a solid surface is equal to the velocity of the (moving) solid boundary. This condition is often referred to as the stick boundary condition.

Another, purely physical situation arises when, for one reason or another, the force, ψ, exerted on a surface by the motion of the fluid adjacent to it is given (i.e., measured). This force is related to both the dynamic pressure and the shear stress via the so-called traction boundary condition,

− pδij+ µ ∂ui ∂xj +∂uj ∂xi νj = ψi. (3.15)

In this thesis both conditions (3.14) and (3.15) are employed.

3.4 Reduction of Navier-Stokes equation

Equation (3.1) along with equation (3.12) above are complicated and difficult to solve or even analyze in a general situation. However, some physical situations allow one to make simplifying assumptions. These lead to a more manageable, though still difficult, system to solve. Introduce two representative dimensional scales, U and L, which are respectively, a representative speed and a represen-tative length scale for the problem under consideration. From these one derives representative time, L/U , and pressure, µ/L, scales. Normalizing both indepen-dent and depenindepen-dent variables in the Navier-Stokes equations w.r.t. these scales results in the system:

∂u

∂t + u ∇u = 1

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3.4 Reduction of Navier-Stokes equation 21

in which all variables are now dimensionless and of order unity. The normaliza-tion also produces a dimensionless number R defined as

R = ρU L

µ , (3.17)

called the Reynolds number. The number R is used as an overall measure of the ratio of inertial terms to viscous and pressure terms. For small R values, appro-priate for low representative flow speeds, finite domains, or large viscosity fluids, the inertial terms are negligible compared to the viscous and pressure gradient terms. In this limit the Navier-Stokes system of equations for an incompressible fluid reduce to

∆u − ∇p = −f in Ω,

divu = 0 in Ω. (3.18)

This system of linear equations is referred to as em the stationary Stokes system. One can also linearize the Navier-Stokes equation in another way. Linearizing the term u∇u leads to a system of the form

∆u + (b(x) · ∇)u − ∇p = f in Ω,

divu = 0 in Ω, (3.19)

where divb = 0. Observe that f need not be the same function as appear in (3.18). This system is known as the generalized Stokes System or Oseen system. It is this system with a mixed boundary condition that we investigate in this thesis.

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23

4 Bilinear forms

In this chapter we present some facts about bilinear forms relevant to the re-mainder of this thesis. The first section below deals with the definition of a bilinear form and with an existence theory concerning solutions of an equa-tion involving a bilinear form. In the second secequa-tion, we prove existence and uniqueness of a solution to a problem involving a system of bilinear forms. The main theorem of this chapter, Theorem 4.7 and its Corollary, are found on pages 28–30.

4.1 Solutions to equations involving a bilinear form

In the next chapter, we show that in proving existence of solutions to the gener-alized Stokes system, one deals with equations where bilinear forms are involved. As stated in Chapter 1, the facts that we present here on bilinear forms are not new. However, we have designed the proofs to suit our purposes, which are to prove existence and uniqueness of solutions to the generalized Stokes system. For complementary information to that provided in the first section, we refer to Chapter 7 of Dautray [16] and for more facts on systems of bilinear forms, we refer to Brezzi [11] and Babuˇska [4].

Let H1and H2 be two real Hilbert spaces. If

b( · , · ) : H1× H2→ R,

is linear in both arguments, and if there exists a constant C, such that |b(u, v)| ≤ C||u||H1||v||H2,

holds for every u ∈ H1 and v ∈ H2, then b( · , · ) is a continuous bilinear form.

Throughout this chapter, we shall only consider continuous bilinear forms that assumes at least one nonzero value. Also, the spaces H1 and H2 are supposed

to be real Hilbert spaces.

A continuous bilinear form induces a continuous linear operator

B : H1→ H2, (4.1)

defined by

(Bu, v)H2= b(u, v) ∀v ∈ H2.

In the next chapter, we need to know that there exists a unique solution to a problem of the following form

Find u ∈ H1, such that

b(u, v) = (f, v)H2 ∀v ∈ H2.

(4.2) Here f ∈ H2 is given. Using the operator introduced in (4.1), we have the

equivalent problem

Find u ∈ H1, such that

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The question of what conditions should apply so that (4.3) has a unique solution, corresponds to the question of when ker(B) = {0}.

Definition 4.1. Let b( · , · ) : H1× H2 → R be a continuous bilinear form. If

there exists a C > 0 such that inf u∈H1\{0} sup v∈H2\{0} b(u, v) ||u||H1||v||H2 ≥ C, then we say that the bilinear form satisfies the inf-sup condition.

We now determine which bilinear forms satisfy the inf-sup condition. Lemma 4.2. Let b( · , · ) : H1× H2 → R, be a continuous bilinear form. The

operator B in (4.1) associated with this form is injective and has closed range if and only if b( · , · ) satisfies the inf-sup condition.

Proof. Assume that B is injective and has closed range. Since B is a continuous operator, it follows that the inverse mapping, B−1 : R(B) → H1 is linear and

bounded. Take u ∈ H1. There exists an f ∈ R(B) such that

u = B−1f. We then have

||u||H1 = ||B

−1_{f ||}

H1 ≤ C||f ||H2, (4.4)

because of the continuity of B−1. The norm of f can be expressed as ||f ||H2= sup

v∈H2\{0}

(f, v)H2

||v||H2

. (4.5)

Using this and the definition of f and B, it follows that ||f ||H2 = sup v∈H2\{0} (f, v)H2 ||v||H2 = sup v∈H2\{0} (Bu, v)H2 ||v||H2 = sup v∈H2\{0} b(u, v) ||v||H2 . (4.6) Using (4.4) and (4.6), we find that there exists a constant C > 0 such that

C||u||H1 ≤ sup

v∈H2\{0}

b(u, v) ||v||H2

. So, b( · , · ) satisfies the inf-sup condition.

Now, suppose that b( · , · ) satisfies the inf-sup condition. Using this fact and (4.5), we have ||Bu||H2= sup v∈H2\{0} (Bu, v)H2 ||v||H2 ≥ C||u||H1. (4.7)

This implies that ker(B) = {0}, hence one can define B−1 to be the mapping from R(B) into H1 such that B−1y = x, if Bx = y. Substituting u = B−1v,

where v ∈ R(B), in (4.7), we have

||v||H2 ≥ C||B

−1_v|| H1.

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4.1 Solutions to equations involving a bilinear form 25

This can be rewritten as

||B−1v||H1 ≤ C||v||H2.

Hence, the operator B−1 _{is bounded and R(B) is closed.}

To characterize the range of B, we have the following lemma, a proof of which can be found in Chapter 3 of Birman and Solomjak [8].

Lemma 4.3. Let B : H1 → H2 be a bounded and linear operator with closed

range. Then

R(B) = (ker(B∗))⊥.

We now consider problem (4.2) in the case when H1= H2.

Definition 4.4. Let H be a Hilbert space and let a( · , · ) : H × H → R be a continuous bilinear form. The bilinear form a( · , · ) is called coercive if there exists a constant C > 0, with

a(u, u) ≥ C||u||2_H, ∀u ∈ H.

The following lemma is a variant of the classical Lax-Milgram lemma. Lemma 4.5. Let H be a real Hilbert space and let a( · , · ) : H × H → R be a coercive and continuous bilinear form. Then for a given f ∈ H the problem

Find u ∈ H, such that

a(u, v) = (f, v)H ∀v ∈ H,

(4.8)

has a unique solution.

Proof. Since a( · , · ) is coercive, we have

sup v∈H\{0} a(u, v) ||v||H ≥ a(u, u) ||u||H ≥ C||u||H,

where C is a positive constant. Thus, the conditions in Lemma 4.2 are satisfied, so the operator A associated with this bilinear form is injective and has closed range. We now show that A is bijective. Suppose that there existed a v ∈ H \{0} such that (u, A∗v)H = 0, for every u ∈ H, i.e., suppose that ker(A∗) 6= {0}.

From the definition of A∗we have (u, A∗v)H = a(u, v). Since a( · , · ) is coercive,

it follows that

(v, A∗v)H= a(v, v) ≥ C||v||H> 0.

Hence, v = 0. This means that ker(A∗) = {0}, a contradiction. By Lemma 4.3, the operator A is bijective. Therefore, problem (4.8) has a unique solution.

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4.2 Solutions for a system of bilinear forms

We shall see in the next chapter that a weak formulation of the generalized Stokes system corresponds to a certain system of bilinear forms. The problem we are interested in can be expressed as

  

 

Find (u, λ) ∈ H1× H2 such that:

a(u, v) + b(λ, v) = (f, v)H1, ∀v ∈ H1,

b(µ, u) = (g, µ)H2, ∀µ ∈ H2.

(4.9)

In (4.9) and for the rest of this chapter, f ∈ H1 and g ∈ H2 are given and

a( · , · ) : H1× H1→ R and b( · , · ) : H2× H1→ R are continuous bilinear forms.

We also identify a Hilbert space with its dual space in the usual way. Thus, we can associate operators with each of the above bilinear forms. Define

A : H1→ H1, by (Au, v)H1 = a(u, v), ∀v ∈ H1, (4.10)

B : H2→ H1, by (Bλ, v)H1 = b(λ, v), ∀v ∈ H1. (4.11)

Using these operators and the definition of the adjoint operator, see (2.1), prob-lem (4.9) can be rewritten as

 



Au + Bλ = f, B∗u = g.

(4.12)

Define

V = {v ∈ H1 : b(µ, v) = 0 for every µ ∈ H2}, (4.13)

that is, V is the null space of B∗. We now sketch one way of solving (4.9). Let R(B∗) = H2 and let B∗ be one-to-one on the orthogonal complement to the

null space of B∗. Define u1 = (B∗|V⊥)−1g, where B∗|_V⊥ is the restriction of

B∗ to V⊥, and introduce u2= u − u1. Then (4.12) becomes the problem

 



Find (u2, λ) ∈ H1× H2 such that:

Au2+ Bλ = f − Au1,

B∗u2= 0.

First, we try to find a u2∈ V that satisfies

(Au2, v)H1 = (f − Au1, v)H1, ∀v ∈ V . (4.14)

If A corresponded to a coercive form on V , it would follow from Lemma 4.5 that such a u2 exists and is unique. We then try to find a λ with Bλ ∈ V⊥, such

that

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4.2 Solutions for a system of bilinear forms 27

holds. We observe that if the range of B is closed then R(B) = V⊥. There-fore, such a λ would exists. Thus, for this case we have found a solution that satisfies (4.9).

We shall now formulate the above idea in a more precise mathematical way and prove that this solution is unique.

First, we show that the inf-sup condition guarantees that B∗ is one to one on V⊥.

Lemma 4.6. The following two assertions are equivalent for a continuous bi-linear form b( · , · ) : H2× H1→ R :

(i) The bilinear form b( · , · ) satisfies the inf-sup condition.

(ii) The restriction of the adjoint operator B∗, where B is defined in (4.11), to V⊥, where V = ker(B∗), is injective and R(B∗) = H2.

Proof. From Lemma 4.2, it is clear that B has a closed range and is injective if (i) holds, and from Lemma 4.3, we see that R(B) = ker(B∗)⊥= V⊥. Therefore, for u ∈ V⊥, define f by f (v) = (u, v)H1. Observe that f (v) = 0, if v ∈ V . Since

B is bijective between H2and V⊥, there exists a λ ∈ H2 such that

b(λ, v) = (u, v)H1 ∀v ∈ H1.

Observing that ||u||H1 = ||Bλ||H1 ≥ C||λ||H2, the estimate

sup µ∈H2\{0} b(µ, u) ||µ||H2 ≥b(λ, u) ||λ||H2 = (u, u)H1 ||λ||H2 ≥ C||u||H1,

follows. From Lemma 4.2, we see that B∗_|

V⊥ has closed range and is injective.

Since R(B∗) = ker(B)⊥, it follows that R(B∗|V⊥) = H₂.

Assume that (ii) holds. Let g ∈ H2. The norm of g can be written as

||g||H2 = sup

µ∈H2\{0}

(g, µ)H2

||µ||H2

.

By assumptions, B∗ is a bijective mapping between V⊥ and H2. This implies

that ||g||H2= sup u∈V⊥_\{0} (g, B∗_u) H2 ||B∗_u|| H2 .

Since B∗|V⊥ is invertible, we have that ||B∗u||H2 ≥ C||u||H1, for every u ∈ V

⊥_.

Using this fact and the definition of B∗, we have ||g||H2 = sup u∈V⊥_\{0} (g, B∗u)H2 ||B∗_u|| H2 = sup u∈V⊥_\{0} b(g, u) ||B∗_u|| H2 ≤ sup u∈V⊥_\{0} b(g, u) C||u||H1 . Since sup u∈V⊥_\{0} b(g, u) C||u||H1 = sup u∈H1\{0} b(g, u) C||u||H1 ,

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it follows that condition (i) is satisfied.

We now state a theorem which guarantees that the problem 





Au + Bλ = f, B∗u = g.

(4.16)

has a unique solution. This theorem was first proved by Brezzi [11].

Theorem 4.7. Let a( · , · ) : H1× H1 → R and b( · , · ) : H2 × H1 → R be

continuous bilinear forms. Let V be as in (4.13) and let P be the orthogonal projection of H1 onto V . Then there exists a unique solution to (4.16) if and

only if the operator P A|V : V → V , where A is defined in (4.10), is bijective

and b( · , · ) satisfies the inf-sup condition. Moreover, the operator

T : H1× H2→ H1× H2, defined by (u, λ) → (Au + Bλ, B∗u),

is an isomorphism.

Proof. If we assume that the conditions on a( · , · ) and b( · , · ) are satisfied, then the proof follows in the same way as outlined in the beginning of this section. From Lemma 4.6 we know that B∗ has an inverse on V⊥, since b( · , · ) satisfies the inf-sup condition. Hence, we can find a u1∈ V⊥ such that B∗u1= g. Let

u2= u − u1. Then problem (4.16) can be restated as

 



Find (u2, λ) ∈ H1× H2 such that:

Au2+ Bλ = f − Au1,

B∗u2= 0.

Since P A|V is a bijective operator, there exists a u2∈ V with

(Au2, v)H1 = (f − Au1, v)H1, ∀v ∈ V.

The operator P A|V has a continuous inverse on V . Thus there exists a constant

C with which we can bound the norm of u2 from above

||u2||H1 ≤ C(||f ||H1+ ||u1||H1). (4.17)

What remains is to find a λ ∈ H2 with Bλ ∈ V⊥, such that the relation

(Bλ, v)H1 = (f − Au1− Au2, v)H1, holds for v ∈ H1. Lemma 4.6 guarantees

that the operator B is bijective between H2 and V⊥. Therefore, there exists

such a λ. We write λ = B−1(f − Au1− Au2), and from this relation an estimate

for λ follows

||λ||H2 ≤ C(||f ||H1+ ||u1||H1+ ||u2||H1). (4.18)

Hence, u and λ satisfy

Au + Bλ = f,

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4.2 Solutions for a system of bilinear forms 29

To see that the solution is unique, it is necessary and sufficient to show that if f = g = 0, then the only solution is the trivial solution u = 0 and λ = 0. We therefore must consider the problem

  

 

(Au, v)H1+ (Bλ, v)H1 = 0, ∀v ∈ H1,

(B∗u, µ)H2 = 0, ∀µ ∈ H2.

(4.20)

Since a solution u ∈ H1to this problem must satisfy (B∗u, µ) = 0, for every

µ ∈ H2, we see that u must belong to V . Using that (Bλ, v)H1 = 0 if v ∈ V , it

follows from the first equation in (4.20) that (Au, v) = 0 for every v ∈ V . Since P A|V is a bijective operator, we deduce that u = 0. From the first equation

in (4.20), using that u = 0, we have that sup

v∈H1\{0}

|b(λ, v)| = 0.

Hence, λ = 0, because b( · , · ) satisfies the inf-sup condition by the assumptions. Consequently, T is bijective.

Using the inequalities (4.17) and (4.18) and the inequality ||u1||H1 ≤ C||g||H2,

we get that

||u||H1 ≤ C(||f ||H1+ ||g||H2) and ||λ||H2≤ C(||f ||H1+ ||g||H2).

Hence, T is an isomorphism.

Now, we consider the case when the operator T is an isomorphism. That T is an isomorphism means in particular that for every (f, g) ∈ H1× H2, there

exists a unique solution to (4.16). Suppose that (f, 0) ∈ V × H2. Denote the

corresponding solution by (u, λ) ∈ H1× H2. Since b(µ, u) = 0, for every µ ∈ H2,

we deduce that u ∈ V . Furthermore, since

Au + Bλ = f, (4.21)

and since (Bλ, v)H1 = 0 for v ∈ V , we have

(Au, v)H1= (f, v)H1 ∀v ∈ V.

So, for every f ∈ V , this problem has a solution , i.e., P A|V is onto on V . Since

T is assumed to be an isomorphism, this solution must be unique. So, P A|V is

a bijective operator.

To prove that b( · , · ) satisfies the inf-sup condition, let g ∈ H2and let (u, λ)

be a solution to (4.16) with f = 0 and g as data. Since T is an isomorphism, it follows that ||u||H1 ≤ C||g||H2. Let u

⊥ _{be the projection of u onto V}⊥_{. Then}

B∗u⊥= g. Hence

B∗: V⊥→ H2,

is bijective and has closed range. Lemma 4.6 then implies that b( · , · ) satisfies the inf-sup condition.

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Corollary 4.8. Let a( · , · ) : H1× H1 → R and b( · , · ) : H2 × H1 → R be

continuous bilinear forms. Let V be as in (4.13). Then there exists a unique solution to (4.16) if a( · , · ) is coercive on V and if b( · , · ) satisfies the inf-sup condition.

Proof. Arguments similar to those we used when we proved Lemma 4.5, show that P A|V is bijective on V . Using Theorem 4.7, the corollary follows.

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31

5 Weak solutions

In this chapter, we introduce and motivate the concept of weak solutions. We specify a weak solution of our generalized Stokes system and show that there exists a unique solution in a suitable Sobolev space. To do this, we use the theory outlined in the previous chapter. We then examine the regularity of this solution and in the last section show that these results hold for an adjoint problem. These results will be used in the next chapter.

5.1 Introduction

Instead of only searching for a classical solution to a partial differential equation, one can enlarge the concept of a solution to invoke Sobolev spaces. That is, functions that do not necessarily have derivatives in the classical sense.

Suppose that we want to solve an equation such as L(x, ∂x)u = f , where u

and f are smooth and L(x, ∂x) is partial differential operator. If we can extend

the domain of L(x, ∂x), then it would be easier to prove existence of solutions.

With this as motivation we extend the domain to include a suitable Sobolev space.

Example 5.1. Consider the Laplace operator, L(x, ∂x)u = ∆u. Assume that

u ∈ H2_{(Ω). Then, for every v ∈ H}1

0(Ω), we have, by using integration by parts

Z Ω v∆u dx = − Z Ω ∇v∇u dx. Let f ∈ L2_{(Ω) be given. We call u ∈ H}1

0(Ω) a weak solution of ∆u = f in Ω, u = 0 on ∂Ω, (5.1) if − Z Ω ∇u∇v dx = Z Ω f v dx (5.2)

holds for every v ∈ H1 0(Ω).

Using integration by parts, it is easy to see that if there exists a solution u ∈ H2(Ω) ∩ H01(Ω) to (5.2), then this solution also satisfies (5.1).

From Example 5.1, we can define a bilinear form a( · , · ) as a(u, v) = −

Z

Ω

∇u∇v dx,

and call u a weak solution of (5.1) if this bilinear form equals the right-hand side of (5.2) for every v ∈ H01(Ω). We can then use the theory in the previous

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Having shown that a solution exists, one can investigate its regularity. If there exists a solution u ∈ H1

0(Ω) to (5.2), then it follows that ∆u ∈ L2(Ω).

From this, if the boundary of Ω is sufficiently regular, it can be deduced that u ∈ H2_{(Ω). Such ideas concerning regularity can be generalized to more general}

elliptic partial differential equations, see Agmon et al. [2].

5.2 Weak solutions to the generalized Stokes system

Using the above ideas and the theory in the previous chapter, we show that there exists a unique weak solution to our generalized Stokes system.

5.2.1 Formulation of the boundary value problem

Let Ω be a bounded domain in Rn, where n ≥ 2, with boundary Γ = ∂Ω of class C2_{. We assume that the boundary is the union of two closed and disjoint}

pieces, Γ0 and Γ1. Let b(x) = (b1(x), . . . , bn(x)), be a function defined on Ω,

such that ∇bi is bounded on Ω, for i = 1, . . . , n. Moreover, we assume that

divb = 0. Define Liui = Li(x, ∂x)ui= n X j=1 ∂2u_i ∂x2 j + bj(x) ∂ui ∂xj , Niui = Ni(x, ∂x)ui= n X j=1 ∂u_i ∂xj +∂uj ∂xi νj,

where ν is the outward unit normal to Γ. Furthermore, we put Lu = (Liui)ni=1

and N u = (Niui)ni=1.

For convenience in what follows, we define the following spaces: H_Γk₁(Ω)n = { v ∈ Hk(Ω)n: v|Γ1 = 0 },

where k > 0 is an integer.

Consider the following problem        Lu − ∇p = f in Ω, divu = g in Ω, u = η on Γ1, pν − N u = ψ on Γ0. (5.3)

We assume that f ∈ L2(Ω)n and g ∈ H1(Ω). For the boundary functions, we assume that η ∈ H3/2(Γ1)n and ψ ∈ H1/2(Γ0)n. The aim of this chapter is to

prove that there exists a unique solution u ∈ H2(Ω)n and p ∈ H1(Ω) to this problem.

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5.2 Weak solutions to the generalized Stokes system 33

5.2.2 Definition of a weak solution

As a first step to solving (5.3), we consider the problem        Lu − ∇p = f in Ω, divu = 0 in Ω, u = 0 on Γ1, pν − N u = ψ on Γ0, (5.4) where f ∈ L2_(Ω)n _{and ψ ∈ L}2_(Γ 0)n.

Introduce the two bilinear forms:

a(u, v) = Z Ω n X i,j=1 ∂u_i ∂xj +∂uj ∂xi ∂v_i ∂xj − bj ∂ui ∂xj vi dx, (5.5) b(q, v) = Z Ω q divv dx. (5.6) Definition 5.2. We call u ∈ H1 Γ1(Ω)

n _{and p ∈ L}2_{(Ω), a weak solution to}

problem (5.4) if ( a(u, v) − b(p, v) = F (v) ∀v ∈ H1 Γ1(Ω) n_, b(q, u) = 0, ∀q ∈ L2_(Ω), (5.7) hold. Here F (v) = −(f , v)L2_(Ω)n− (ψ, v)_L2_(Γ₀₎n, for every v ∈ H1 Γ1(Ω) n_.

Sometimes, we shall consider the problem of finding solutions to (5.7) when F belong to a more general class. Precisely, F should be an arbitrary functional in (H1

Γ1(Ω)

n₎∗_.

Weak solution to (5.4) having additional smoothness will solve problem (5.4) in the strong sense. This fact is proved in the next lemma.

Lemma 5.3. Let f ∈ L2_(Ω)n _{and ψ ∈ L}2_(Γ

0)n. Assume that u ∈ HΓ21(Ω)

n

and p ∈ H1(Ω) is a weak solution to (5.4). Then u and p solve (5.4) in the strong sense and ψ ∈ H1/2(Γ0)n.

Proof. From the condition that b(q, u) = 0, for every q ∈ L2(Ω), it follows that

divu = 0. (5.8)

We then observe that

Liui= n X j=1 ∂ ∂xj ∂u_i ∂xj +∂uj ∂xi − ∂ 2_u j ∂xj∂xi + bj(x) ∂ui ∂xj , (5.9)

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for i = 1, . . . , n. Differentiating the divergence condition with respect to xi, gives ∂2_u 1 ∂xi∂x1 + . . . + ∂ 2_u n ∂xi∂xn = 0. Using this fact and that

∂2_w ∂xi∂xj = ∂ 2_w ∂xj∂xi ,

for every function w ∈ H2(Ω), it follows from (5.9) that Liui can be put in the

form Liui= n X j=1 ∂ ∂xj ∂u_i ∂xj +∂uj ∂xi + bj(x) ∂ui ∂xj . (5.10) Let v ∈ H1 Γ1(Ω)

n_{. Integration by parts in the term b(p, v) in (5.7), the expression}

at the first line in (5.7) can be written as Z Ω n X i,j=1 _∂u i ∂xj +∂uj ∂xi ∂v_i ∂xj − bi ∂ui ∂xj vi dx + Z Ω v · ∇p dx + Z Γ0 (−pν + ψ) · v dS = − Z Ω f · v dx.

Again, integration by parts and using (5.10) we arrive at Z Ω (Lu − ∇p − f ) · v dx + Z Γ0 (pν − N u − ψ) · v dS = 0. (5.11) Taking first v ∈ H1

0(Ω)n, we derive from (5.11) that

Lu − ∇p = f in Ω. (5.12)

Using this fact, it follows again from (5.11) that Z

Γ0

(pν − N u − ψ) · v dS = 0. (5.13)

Hence, pν − N u − ψ = 0, so ψ ∈ H1/2_(Γ

0)n. From this and the fact that

u ∈ H2 Γ1(Ω)

n_{, we see that the following boundary conditions are satisfied}

u = 0 on Γ1,

pν − N u = ψ on Γ0.

(5.14)

Reconstruction of a stationary flow from boundary data

Link¨

oping Studies in Science and Technology. Theses

No. 853

Reconstuction of a Stationary Flow

From

Boundary Data

Tomas Johansson

Department of Science and Technology,

Campus Norrk¨

oping, Link¨

opings University, S-601 74 Norrk¨

oping, Sweden

Norrk¨

oping 2000

Link¨

oping Studies in Science and Technology. Theses

No. 853

Reconstruction of a Stationary Flow

from

Boundary Data

Tomas Johansson

Department of Science and Technology,

Campus Norrk¨

oping, Link¨

oping University, S-601 74 Norrk¨

oping, Sweden

Norrk¨

oping 2000

LiU-TEK-LIC-2000:29

Abstract

Contents

1

Introduction

1.1

Ill-posed problems

1.2

Outline of the thesis

2

Mathematical preliminaries

2.1

Basic notations

2.2

Regularizing methods

2.3

Sobolev spaces

2.4

Elliptic partial differential equations

3

Fluid dynamics

3.1

The continuum hypothesis and other assumptions

3.2

A formal derivation of the equations of continuity and

momentum

3.3

Boundary conditions

3.4

Reduction of Navier-Stokes equation

4

Bilinear forms

4.1

Solutions to equations involving a bilinear form

4.2

Solutions for a system of bilinear forms

5

Weak solutions

5.1

Introduction

5.2

Weak solutions to the generalized Stokes system