Linear and Quasilinear Parabolic Systems: Sobolev Space Theory 2020036612, 9781470461614, 9781470463205

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Mathematical Surveys and Monographs Volume 251

Linear and Quasilinear Parabolic Systems Sobolev Space Theory

David Hoff

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Linear and Quasilinear Parabolic Systems Sobolev Space Theory

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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10.1090/surv/251

Mathematical Surveys and Monographs Volume 251

Linear and Quasilinear Parabolic Systems Sobolev Space Theory

David Hoff

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EDITORIAL COMMITTEE Robert Guralnick, Chair Natasa Sesum Bryna Kra Constantin Teleman Melanie Matchett Wood 2010 Mathematics Subject Classification. Primary 35K51, 35K61.

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Library of Congress Cataloging-in-Publication Data Names: Hoff, David Charles, 1948– author. Title: Linear and quasilinear parabolic systems : sobolev space theory / David Hoff. Description: Providence, Rhode Island : American Mathematical Society, [2021] Series: Mathematical surveys and monographs, 0076-5376 ; volume 251 | Includes bibliographical references and index. Identifiers: LCCN 2020036612 | ISBN 9781470461614 (paperback) | ISBN 9781470463205 (ebook) Subjects: LCSH: Differential equations, Partial. | Differential equations, Parabolic. | AMS: Partial differential equations – Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] – Initial-boundary value problems for second-order parabolic systems. | Partial differential equations – Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] – Nonlinear initial-boundary value problems for nonlinear parabolic equations. Classification: LCC QA377 .H62 2021 | DDC 515/.3534–dc23 LC record available at https://lccn.loc.gov/2020036612

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10.1090/surv/251/01

To my family–may they all thrive!

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Contents Preface

ix

Chapter 1. Introduction 1.1 Weak forms: An Example 1.2 Basic Notations and Definitions 1.3 Weak Measurability 1.4 Vector-Valued Lp Spaces 1.5 Geometric Properties of Open Sets in Rn

1 1 2 4 5 6

Chapter 2. Differential Equations in Hilbert Space 2.1 Basic Existence and Uniqueness 2.2 Weak Absolute Continuity 2.3 Higher Order Regularity 2.4 Proof of Theorem 2.7 2.5 Incompatible Data and Initial Layer Regularization 2.6 Systems with Symmetry 2.7 Spectral Representations

11 11 16 24 29 40 43 49

Chapter 3. Linear Parabolic Systems: Basic Theory 3.1 Linear Parabolic Systems and Their Weak Forms 3.2 Preliminaries 3.3 The Basic Existence Theorem 3.4 Parabolic Systems with Symmetry

59 60 63 67 72

Chapter 4. Elliptic Systems: Higher Order Regularity 4.1 Statements of Results 4.2 Interior Regularity: Proof of Theorem 4.1 4.3 Global Regularity: Proof of Theorem 4.2

77 78 83 93

Chapter 5. Parabolic Systems: Higher Order Regularity 5.1 Boundary Conditions, Coefficients and Inhomogeneities 5.2 Higher Order Regularity for Parabolic Systems 5.3 Incompatible Data and Initial Layer Regularization

99 99 111 122

Chapter 6. Applications to Quasilinear Systems 6.1 A Quasilinear System in Hilbert Space

127 127

vii

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viii

CONTENTS

6.2 6.3 6.4 6.5 6.6 6.7

Navier–Stokes Equations of Incompressible Flow Nonlinearities with Polynomial Growth Higher Order Regularity for Quasilinear Systems Global Existence and Stability of Steady-States Invariant Regions for Quasilinear Systems A Global Existence Result for Systems in R2 and R3

135 139 148 166 172 177

Appendix. Selected Topics in Analysis A.1 Density and Imbedding Theorems A.2 Weak Solutions of Ordinary Differential Equations A.3 Applications of Spectral Measures A.4 Boundary Measures: Proofs of Theorems 3.1–3.3 A.5 The Spaces Lp,q A.6 A Representation Theorem for Boundary Integrals

187 187 190 193 197 201 214

Notes and References

219

Bibliography

221

Index

225

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Preface In this monograph I present a systematic theory of weak solutions in HilbertSobolev spaces of initial-boundary value problems for parabolic systems of partial differential equations. The goal is to provide a practical reference for fundamental results often considered well–known but often difficult to access in the desired generality, as well as to provide instructional material for students preparing for research in applied analysis. The development begins in Chapter 2 with an abstract theory in which H0 and H1 are separable Hilbert spaces with H1 dense and continuously included in H0 and A is a mapping of the time axis into the set of bilinear forms on H1 × H1 . The problem is to construct a mapping u for which the equation Z t
hu(t), viH0 = hς0 , viH0 + − A(s)(u(s), v) + F (s) · v ds 0

holds for all test functions v ∈ H1 , for all time t and for given ς0 ∈ H0 and F mapping time into the dual H1∗ . This abstract theory is then applied in Chapters 3 and 5 to initial–boundary value problems for linear parabolic systems with H0 a closed subspace of [L2 (Ω)]N for a given open set Ω ⊂ Rn (not assumed to be bounded) and H1 a closed subspace of [H 1 (Ω)]N whose elements satisfy essential boundary conditions. The basic theory of Chapter 3 addresses questions of measurability, continuity and energy conservation in the absence of regularity, and includes special considerations for problems with symmetry and representations of solutions in terms of spectral measures. Higher-order regularity of solutions is discussed in Chapter 5 in three related but categorically different formulations: the dynamical systems (Bochner space) formulation in which solutions are mappings from the time axis into various Sobolev spaces, the weak derivative formulation in which solutions are locally integrable on the space-time cross product, and the continuous derivative formulation familiar from calculus. Chapter 5 also includes a discussion of instantaneous regularization of solutions whose data fail to satisfy compatibility conditions. (Chapter 4 is a brief excursion into higher order regularity theory for elliptic systems, required for the application in Chapter 5.) This linear theory is then applied in chapter 6 to quasilinear systems in which A and F may depend on u. A basic theory of solutions with minimal regularity is given, corresponding to the basic theory for linear problems. This includes the theorems of Leray and Hopf for the Navier-Stokes equations as well as results for more general problems in which nonlinearities have specific, controlled growth as |u| → ∞. Results for more highly regular solutions of quasilinear systems are then derived by iteration from the corresponding regularity theory for linear problems. This includes local-in-time existence with large compatible data, global existence for data near an attracting rest point, invariant regions for quasilinear systems, and ix

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x

PREFACE

a large data, global existence result in two and three space dimensions for certain systems with symmetry. This book was written in a continuous, linear process in which the material itself dictated formulation, order of topics and level of detail. This choice was made to insure an internal consistency of notation, terminology and perspective that would not have been achieved by collating results from the literature (no claim is made to mathematical originality, however). Maximal generality was an important goal, and this has occasionally resulted in considerable but unavoidable technicality. I have therefore attempted to present proofs in such a way that the experienced reader can easily grasp the flow of the arguments without reading every detail, but at the same time the intrepid student will succeed in checking these details without undue difficulty. The following are the major theorems of the book, listed here for convenient reference: • Basic existence and uniqueness: Theorem 3.6 for general linear systems and Theorem 3.7 for systems which are symmetric to leading order. • Spectral representations: Theorem 3.8. • Regularity for linear elliptic system: Theorems 4.1 and 4.2. • Higher order regularity for linear parabolic systems: Theorem 5.5 for compatible data, Theorem 5.8 for incompatible data. • Theorems of Leray and Hopf for the Navier-Stokes system: Theorem 6.5. • Local existence of minimally regular solutions of quasilinear systems with nonlinearities of polynomial growth: Theorem 6.6. • Local existence of smooth solutions of quasilinear systems: Theorem 6.10. • Global existence for dissipative quasilinear systems with small initial data, Theorem 6.12. • Invariant regions for quasilinear systems, Theorem 6.13. • Global existence for certain symmetric quasilinear systems with large data in two and three space dimensions, Theorem 6.14. Few readers will read the book cover-to-cover. The following suggestions for reading plans may therefore be considered (the first three should include the basic material in Chapter 1 and sections 2.1., 2.2 and 3.1–3.3): • Any – – –

of the following extensions of the basic theory: systems with symmetry (sections 2.6, 2.7 and 3.4) Navier-Stokes equations (sections 6.1 and 6.2) quasilinear systems with polynomial growth (sections 6.1 and 6.3).

• Higher order regularity for linear parabolic systems (sections 2.2–2.4, 5.1 and 5.2) with the possible addition of initial layer regularization (sections 2.5 and 5.3). • Higher order regularity for quasilinear parabolic systems (sections 2.2–2.4, 5.1 and 5.2 are prerequisite, 6.4 is essential, and any of 6.5, 6.6 or 6.7 may be included independently). • Chapter 4 is a self-contained treatment of higher-order regularity for elliptic systems.

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PREFACE

xi

• Section A.4 is a self-contained treatment of calculus on Lipschitz hypersurfaces, including the construction of the Radon measure corresponding to surface area and the normal vector field, the trace theorem W 1,p (Ω) → Lp (∂Ω) and the divergence theorem for W 1,p integrands. • Section A.5 is a self-contained treatment of certain spaces Lp,q (Ω × I) where I is an interval in R and Ω is an open set in Rn . The elements of these spaces are in canonical one-to-one correspondence with those of the Bochner spaces Lp (I; Lq (Ω)) but are measurable on the cross product. This correspondence finds application throughout the book in navigating different formulations of regularity. In all cases students should be familiar with the basic facts of measure theory and functional analysis, including weak derivatives and Sobolev spaces. Spectral measures appear twice in the book but their use can be side-stepped with minimal loss by appeal to the spectral theorem for compact self-adjoint operators. The Hilbert space framework chosen here has the virtues of relative simplicity and broad applicability. Other approaches, including semigroup theory, Green’s functions, and the method of continuation (Schauder theory) have important uses as well, however, but could not be included. For these the reader may consult standard references such as Ladyzhenskaya et. al. [26], Friedman [15] and [16] and Evans [12]. Many of the perspectives and formulations in this book reflect the influences of collaborators, colleagues and students, too numerous to mention, with whom I have had the good fortune to interact over many years. My thanks to them all. Of course, responsibility for any shortcomings, deficiencies or errors lies solely with me. David Hoff

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10.1090/surv/251/01

CHAPTER 1

Introduction Mathematicians invent a setting with a handful of characters...[and] a few rules of interaction, and watch how the plot unfolds. Kelsey Houston-Edwards [24]

We begin with an example in which an elementary initial–boundary value problem is converted to its corresponding weak form, which is then generalized to an abstract Hilbert space framework. A brief review of basic concepts and notations from functional analysis follows and we conclude with a discussion of geometric properties of open sets in Rn .

§1.1

Weak forms: An Example

Consider the following elementary problem: let Ω be a proper open subset of Rn and ς0 a given function on Ω, and let aij , 1 ≤ i, j ≤ n, and g be given functions on Ω × (0, T ) where T ∈ (0, ∞]. A solution u : Ω × [0, T ) → R to the initial–boundary value problem   ut = (aij uxi )xj + g, (x, t) ∈ Ω × (0, T ), u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ), (1.1)  u(x, 0) = ς0 (x), x ∈ Ω is then to be constructed (subscripts denote partial derivatives and summation over repeated indices is understood), apparently satisfying fairly strong differentiability conditions. These conditions may be overly restrictive, however, and may have little significance for the underlying application. It is therefore of interest that a weaker form is available, both for this problem and for the broader class to which it belongs, which is often more convenient and which may in fact be closer to the physical principles from which the system is derived. To obtain this weak form for the above example we multiply the differential equation by a function v of x which is zero on ∂Ω and formally integrate over Ω × [0, t]. Applying the divergence theorem, we find that Z

Z u(x, t)v(x) dx =

(1.2)

Ω

ς0 (x)v(x) dx Z tZ
+ − aij uxi vxj + gv dx ds. Ω

0

Ω

An appropriate weak formulation is now apparent: assume that for every t ∈ (0, T ) aij is bounded and measurable on Ω × (0, t), that f ∈ L2 (Ω × (0, t)) and that 1

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2

1. INTRODUCTION

ς0 ∈ L2 (Ω). We then say that u is a weak solution of (1.1) if u is weakly continuous from [0, T ) into L2 (Ω) (which means that the inner product on the left above is continuous in t for every v), that u(0) = ς0 , that the Lebesgue equivalence class of u on (0, T ) is an element of L2 ((0, t); H01 (Ω)) (the precise meaning of which will be given below), and which satisfies (1.2) for every v ∈ H01 (Ω) and t ∈ (0, T ). It is clear that sufficiently smooth solutions of (1.1) are weak solutions in this sense, but a proof of the converse requires greater regularity. Little is lost, however, if weak solutions are shown to be unique, and the weak formulation has the clear advantage that broader classes of problems and solutions are allowed. A compact Hilbert space formulation can be given and generalized as follows. If we define A(t) : H01 (Ω) × H01 (Ω) → R and F (t) : H01 (Ω) → R for this example by Z   (1.3) A(t)(z, v) = aij (x, t)zxi (x)vxj (x) dx Ω

and

Z F (t) · v =

g(x, t)v(x)dx Ω

for specific representatives of aij and g, then the requirement (1.2) is that Z t 2 2 (1.4) hu(·, t), viL (Ω) = hς0 , viL (Ω) + [−A(s)(u(·, s), v) + F (s) · v] ds 0

for every v ∈ H01 (Ω) and t ∈ [0, T ). In Chapter 2 we study the abstract version of (1.4) in which the spaces L2 (Ω) and H01 (Ω) are replaced by general separable Hilbert spaces H0 and H1 with H1 continuously contained in H0 , A is a fairly general mapping from [0, T ) into the set of bilinear forms on H1 × H1 and F is a mapping from [0, T ) into the dual of H1 . This abstract theory is then applied in Chapter 3 to generalizations of (1.1) in which systems of partial differential equations with quite general essential and natural boundary conditions are considered. Higher order regularity of solutions is examined in Chapter 5 and applications to quasilinear systems are discussed in Chapter 6.

§1.2

Basic Notations and Definitions

Here and throughout the book Ω will be an open set in Rn which need not k be bounded unless boundedness is explicitly assumed. We denote by Cbdd (Ω) the Banach space of real-valued functions on Ω having continuous, bounded derivatives on Ω up to order k ≥ 0 with norm α |ϕ|Cbdd k (Ω) = max sup |D ϕ(x)|. |α|≤k x∈Ω

k,λ For k ≥ 0 and λ ∈ (0, 1], Cbdd (Ω) will denote the Banach space of elements ϕ in k Cbdd (Ω) whose derivatives are H¨ older continuous with exponent λ ∈ (0, 1]; that is, k,λ whose Cbdd norm, given by

|ϕ|C k,λ (Ω) ≡ |ϕ|Cbdd k (Ω) + sup bdd

sup |

|α|≤k x6=y ∈ Ω

|Dα ϕ(y) − Dα ϕ(x)| , |y − x|λ

k,λ is finite. Observe that if Ω is a proper subset of Rn and ϕ ∈ Cbdd (Ω), then ϕ and its derivatives up to order k have continuous extensions to ∂Ω and that the suprema in the above two norms may then be taken over Ω. If k = 0 then no

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10.1090/surv/251/02

1. BASIC NOTATIONS AND DEFINITIONS

3

differentiability is required, so that the domain space need not be open; thus for any 0,λ set W ⊆ Rn , Cbdd (W ) will denote the space of real–valued functions on W which are bounded and H¨older continuous with exponent λ. Finally, if Q is a subset of 0,(λ,λ0 ) Rn × R and λ, λ0 ∈ (0, 1], then Cbdd (Q) will denote the set of functions ψ which are continuous and bounded on Q and for which |ψ(x2 , t2 ) − ψ(x1 , t1 )| λ λ0 (x1 ,t1 )6=(x2 ,t2 )∈Q |x2 − x1 | + |t2 − t1 | sup

0,(λ,λ0 )

is finite. Elements of Cbdd (Q) have continuous, bounded extensions to the closure of Q. k We denote by Cck (Ω) the set of elements in Cbdd (Ω) having compact support ∞ in Ω and by Cc (Ω) the space of infinitely differentiable functions having compact support in Ω. Next recall that if u is a locally integrable function on an open set Ω ⊆ Rn and α is a multi-index, then the α–derivative of u in Zthe sense of distributions is the linear map from Cc∞ (Ω) to R given by ϕ 7→ (−1)|α| u Dα ϕ dx. Ω Z Z If there is a locally integrable function v such that (−1)|α| u Dα ϕ dx = v ϕ dx Ω

Ω

for all ϕ ∈ Cc∞ (Ω) then v is the weak α–derivative of u on Ω and we write Dα u = v. Thus the weak derivative, if it exists, gives a representation of the distribution derivative, which always exists. The subspace of elements of Lp (Ω) whose weak derivatives in Ω up to order k exist and are also elements of Lp (Ω) is the Sobolev space W k,p (Ω), in which the
1/p P α norm of an element u is given by |u|W k,p (Ω) = if p < ∞ |α|≤k |D u|Lp (Ω) and by |u|W k,∞ (Ω) = max|α|≤k |Dα u|L∞ (Ω) if p = ∞; H k (Ω) is the Hilbert space W k,2 (Ω); and H0k (Ω) is the closure with respect to the H k (Ω) norm of the space Cc∞ (Ω). Concise accounts of the relevant facts concerning Sobolev spaces are given in [46], [12] and [19], and a more complete exposition in [1]. Much of our work will concern systems of differential equations whose solutions will be elements of the cross product [W k,p (Ω)]N where N ≥ 1. The norm of an P 1/p j p element u = (u1 , . . . , uN ) in the latter Banach space is given by |u | k,p j W (Ω) if p < ∞ and by maxj |uj |W k,∞ (Ω) if p = ∞. The space [W k,2 (Ω)]N = [H k (Ω)]N is P a Hilbert space with inner product hu, vi[H k (Ω)]N = j huj , v j iH k (Ω) . More generally, the norm of an element u in a normed vector space X is denoted by |u|X (the subscript is omitted when X = Rn ) and the inner product of two elements u and v in a Hilbert space H by hu, viH . The dual of a Banach space X is denoted by X ∗ , the action of an element F ∈ X ∗ on an element v ∈ X by F · v, and the vector space of bounded linear transformations from one Banach space X1 into another Banach space X2 by L(X1 , X2 ). The norm of such a transformation S is denoted by kSkX1 ,X2 , or in the case that X1 = X2 = X, simply by kSkX . A mapping A from X1 × X2 to R is bilinear if A(u1 , ·) is linear on X2 for every fixed u1 ∈ X1 and A(·, u2 ) is linear on X1 for every fixed u2 ∈ X2 . The norm kAkX1 ×X2 of such a map is sup |A(u1 , u2 )| taken over unit vectors uj ∈ Xj , and the vector space of bilinear forms with finite norm is denoted by B(X1 , X2 ). If H and H0 are Hilbert spaces and A : H × H0 → R is bilinear, then A ∈ B(H, H0 ) if and only if there is an operator S ∈ L(H, H0 ) such that A(u, u0 ) = hSu, u0 iH0 for all u ∈ H and u0 ∈ H0 ; if so, then S is unique and kSkH,H0 = kAkH×H0 .

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4

1. INTRODUCTION

If I ⊂ R is a nontrivial interval and X is a normed vector space, then C(I; X), C λ (I; X) and Lip(I; X) are the vector spaces of mappings u from I into X which are respectively continuous, H¨older continuous with exponent λ ∈ (0, 1), or Lipschitz continuous. Thus u ∈ C λ (I; X) if there is a constant C such that (1.5)

|u(t2 ) − u(t1 )|X ≤ C|t2 − t1 |λ

for all t1 , t2 ∈ I; and u ∈ Lip(I; X) if (1.5) holds with λ = 1. The vector space λ Cloc (I; X) is the space of mappings which are in C λ (I 0 ; X) for every compact interval I 0 ⊆ I, and similarly for Liploc (I; X). Finally, if H is a Hilbert space then C(I; Hweak ) (or C(I; Hweak ) if notationally convenient) is the vector space of weakly continuous mappings from I into H; that is, the set of functions u : I → H such that t → hu(t), viH is continuous on I for every fixed v ∈ H.

§1.3

Weak Measurability

The following notions of weak measurability of mappings into the vector spaces of section 1.2 will be applied throughout:

Definitions 1.1. Let I ⊆ R be a nontrivial interval. A mapping u from I into a Banach space X is weakly measurable if for every F ∈ X ∗ the mapping t → F ·u(t) is (Lebesgue) measurable on I. A mapping F from I into the dual of a Banach space X is weak-∗ measurable if for every u ∈ X the mapping t → F (t) · u is measurable on I. A mapping A from I into B(X1 × X2 ), where X1 and X2 are Banach spaces, is weakly measurable if for every pair (u1 , u2 ) ∈ X1 × X2 the mapping t → A(t)(u1 , u2 ) is measurable on I; and a mapping S from I into L(X1 , X2 ) is weakly measurable if for all u ∈ X1 the mapping t → S(t)u is weakly measurable into X2 . It is easy to check that if H and H0 are Hilbert spaces, A : I → B(H, H0 ) and A(t)(u, u0 ) = hS(t)u, u0 )iH0 for all u ∈ H and u0 ∈ H0 , then A is weakly measurable if and only if S is weakly measurable. In general, norms and inner products of weakly measurable mappings into separable spaces are measurable: Theorem 1.2 Let I ⊆ R be a nontrivial interval, X a separable Banach space and H and H0 separable Hilbert spaces. (a) If F : I → X ∗ is weak-∗ measurable on I, then the mapping t → |F (t)|X ∗ is measurable on I. (b) If F : I → H∗ is weak-∗ measurable on I and u : I → H is weakly measurable on I, then the mapping t → F (t) · u(t) is measurable on I. (c) If A → B(H × H0 ) is weakly measurable on I then the mapping t → kAkH×H0 is measurable on I; and if u and v are weakly measurable mappings from I into H and H0 respectively, then the mapping t → A(t)(u(t), v(t)) is measurable on I. PROOF. If F is as in (a) and {uj }∞ 1 is countable and dense in X, then each aj (·) ≡ |F (·) · uj |/|uj |X is measurable on I, hence so is |F (·)|X ∗ = supj aj (·). The proof of the first assertion in (c) is similar, and for the second we let

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1. VECTOR-VALUED Lp SPACES

5

∞ 0 {ϕj }∞ 1 and {ψk }1 be orthonormal bases for H and H respectively and u and v as in (c); then

A(u, v) =

X hu, ϕj iH hv, ψk iH0 A(ϕj , ψk ). j,k

Each summand here is the product of measurable functions and so is measurable, hence so is the converging sum. The proof of (b) is similar.  Notice that both (b) and (c) show in particular that hu(·), v(·)iH is measurable when u and v are weakly measurable. Also, it is easy to check that as a consequence of (c), if S : I → L(H, H0 ) is weakly measurable on I, then the mapping t → kSkH,H0 is measurable on I, and if u is weakly measurable from I into H, then the mapping t → S(t)u(t) is measurable from I into H0 . It is of interest that, in the presence of separability, weak measurability generally implies measurability in the usual sense; see Exercise 1.2.

§1.4

Vector-Valued Lp Spaces

Two mappings from an interval I ⊆ R into a nonempty set V are Lebesgue equivalent if they agree for almost all t ∈ I, and the equivalence class of such a map F is denoted by [F ]I,V , or if the context makes clear, simply by [F ]. If V is the dual X ∗ of a separable Banach space X and if F : I → X ∗ is weak-∗ measurable, then the mapping t → |F (t)|X ∗ is measurable by the above theorem and we can therefore consider whether its equivalence class over I belongs to Lp (I). Similar considerations apply to mappings from I into vector spaces B(H × H0 ) and L(H, H0 ) of bilinear forms and linear operators: Definitions 1.3. Let I ⊆ R be a nontrivial interval and F a weak-∗ measurable mapping from I into the dual X ∗ of a separable Banach space X. Then for p ∈ [1, ∞] the vector space of equivalence classes [F ]I,X ∗ of weak-∗ measurable mappings F : I → X ∗ such that |F |X ∗ ∈ Lp (I) is denoted p ∗ by |Lp (I;X ∗ ) = L (I; X ) and the norm of an element F is defined by |F |F (·)|X ∗ p . The vector space of mappings which are in Lp (I 0 ; X ∗ ) for L (I) every compact interval I 0 ⊆ I is denoted by Lploc (I; X ∗ ). The spaces Lp (I; B(H × H0 )), Lploc (I; B(H × H0 )), Lp (I; L(H, H0 )) and p Lloc (I; L(H, H0 )), where H and H0 are separable Hilbert spaces, are defined in a similar way. Observe the important distinction between, for example, Lploc ([0, T );X ∗ ) and Lploc ((0, T );X ∗ ). In nearly every case occurring in this book the Banach space X will be either a separable Hilbert space, in which case weak and weak-∗ measurability are equivalent, or X ∗ will be Lq (Ω) for q ∈ (1, ∞] and Ω open in Rn , in

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6

1. INTRODUCTION 0

which case the predual X is the separable space Lq (Ω) where q 0 ∈ [1, ∞) is the H¨older conjugate. It is easy to see that Lp (I; X ∗ ) and Lp (I; B(H × H0 )) are normed linear spaces but we do not consider here their completeness or characterizations of their duals; see [9] for a more complete discussion. One exception is the case that X is a separable Hilbert space and p = 2: Lemma 1.4. Let I ⊆ R be a nontrivial interval and H a separable Hilbert space. Then L2 (I; H) is a Hilbert space with inner product Z hu, viL2 (I;H) = hu(t), v(t)iH dt. I

PROOF. The proof of completeness is nearly identical to the proof of completeness of L2 -spaces of real-valued functions, mutatis mutandi. See [13], Theorem 6.5 or [34], Theorem 6.3.5 for example.  Integrals of Hilbert space–valued functions of t will appear frequently and are defined as follows: Definition 1.5. Let H be a separable Hilbert space, I a nontrivial interval in R and u ∈ L1 (I; H). Then since Z hu(t), viH dt ≤ |u|L1 (I;H) |v|H I

Z for all v ∈ H, there is a unique element of H denoted by

u(t)dt such that I

Z I

Z hu(t), viH dt = h u(t) dt, viH I

for all v ∈ H.

§1.5

Geometric Properties of Open Sets in Rn

Standard notions of boundary regularity for open sets in Rn will be applied throughout the text. These follow closely those in Adams [1], section 4.4: Definition 1.6. Let Ω be an open set in Rn , n ≥ 2, not necessarily bounded. (a) Ω satisfies the segment condition if either Ω = Rn or if for every x ∈ ∂Ω there is a neighborhood U of x and a nonzero vector y such that z +ty ∈ Ω for every z ∈ Ω ∩ U and t ∈ (0, 1). (b) Ω satisfies the cone condition if there is a cone K, that is, a set of the form K = {x ∈ Rn : |x| ≤ r and x · ω ≥ δ|x|} for some r > 0, δ ∈ (0, 1) and unit vector ω, such that every x ∈ Ω is the vertex of a cone contained in Ω and congruent to K.

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1. GEOMETRIC PROPERTIES OF OPEN SETS IN Rn

7

The segment condition is required for the proof that the set of restrictions to Ω of elements of Cc∞ (Rn ) is dense in W k,p (Ω) for p ∈ [1, ∞) (Theorem A.2 in the Appendix) and the cone condition is required for various results concerning continuous and compact imbeddings of Sobolev spaces into Lebesgue spaces (Theorems A.3–A.5). Neither of these conditions implies the other; see Exercise 1.6. Further regularity properties of a given set are formulated in terms of the regularity properties of functions whose graphs comprise its boundary: Definition 1.7. Let Ω be a nonempty, proper open subset of Rn , n ≥ 2, not necessarily bounded. Assume that there is given an at most countable ensemble {(Ui , Ri , ri , si , Bi , ψi )}i satisfying the following: • Ui = Ri (Bi × (0, si )) where Bi is a ball in Rn−1 of positive radius ri , si > 0, and Ri is a rigid motion (that is, Ri (z) = ai + Qi z where ai is a constant vector and Qi is an orthogonal matrix with positive determinant) • {Ui } is a locally finite open cover of ∂Ω • ψi is a mapping from Bi to a compact subset of (0, si ) • Ω ∩ Ui is the image under Ri of the set {(y, s) : y ∈ Bi and 0 < s < ψi (y)}. Then: (a) Ω is a Lipschitz domain if each ψi is Lipschitz continuous on Bi . (b) Ω is a uniformly Lipschitz domain if there are positive constants r, ε0 and L such that for all i, ri ≥ r, ψi ≥ ε0 on Bi and the Lipschitz constant for ψi is bounded by L. k (resp. C k,λ ) domain, where k is a nonnegative integer and (c) Ω is a Cbdd bdd λ ∈ (0, 1], if there are positive constants r, s < s, ε0 < 2r ∧ s/2 (the smaller of 2r and s/2), M and i0 such that • for all i, s ≤ si ≤ s, ri ≥ r and ψi (Bi ) ⊂ [ε0 , s − ε0 ] k (resp. C k,λ ) norm of each ψ is bounded by M • the Cbdd i bdd • the sets
Ri Bi0 × (ε0 /2, s − ε0 /2) , where Bi0 is the ball of radius ri − ε0 /2 concentric with Bi , also cover ∂Ω • every subcollection of i0 + 1 of the sets Ui has empty intersection.

In all cases we define Ti : Bi → ∂Ω ∩ Ui by (1.6)

Ti (y) = Ri (y, ψi (y)) k Cbdd

and for upper bound

k,λ (resp. Cbdd ) domains −1 −1 for r , s , s, ε−1 0 , M

Ω we denote by MΩk (resp. MΩk,λ ) an and i0 .

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8

1. INTRODUCTION

k domains with k ≥ 1 are C 0,1 domains, The reader can easily check that Cbdd 0,1 C domains are uniformly Lipschitz, and uniformly Lipschitz domains k and C k,λ satisfy the segment and cone conditions. We reiterate that Cbdd bdd domains need not be bounded. Boundary measures corresponding to surface area are constructed in Theorem 3.1 for Lipschitz domains and the trace and divergence theorems are stated in Theorems 3.2 and 3.3 for uniformly Lipschitz domains; proofs are given in section A.4 in the Appendix. These domains provide a convenient balance between generality, allowing for boundaries with corners and edges, and simplicity of analysis. In particular, the basic existence and uniqueness theory developed in Chapter 3 will apply to problems (1.4) posed on uniformly Lipschitz domains. Higher-order regularity of solutions will require more highly regular regions in which they are posed, however, and these are the k domains defined above. Cbdd

We conclude with a construction of specialized partitions of unity subordinate to covers such as those in Definition 1.7. The following result applies k domains but is much more in particular to covers of the boundaries of Cbdd general: Theorem 1.8. Let {Ui }i be an at most countable, locally finite open cover of a nonempty proper subset X of Rn . Assume that there is a positive number δ such that the diameter of each Ui is at least 2δ and that if Vi = {x ∈ Ui : dist(x, ∂Ui ) > δ} then the sets Vi also cover X. Then there is a neighborhood V of X and a collection of functions {ϕi }i such that • dist(X, ∂V) ≥ δ/3 • ϕi ∈ Cc∞ (Ui ) and dist(supp ϕi , ∂Ui ) ≥ δ/6 • 0 ≤ ϕi (x) ≤ 1 for all x ∈ Rn and all i P • i ϕi (x) = 1 for all x ∈ V • for every multi-index α there is a number C depending on δ and α such that supi maxx∈Ui |Dα ϕi (x)| ≤ C. PROOF. Let V1i and V2i be the sets of points in Ui whose distance to ∂Ui is greater than 2δ/3 and δ/3 respectively, and let V1 = ∪i V1i and V2 = ∪i V2i . Then dist(X, ∂V1 ) and dist(V1 , ∂V2 ) are bounded below by δ/3. (To prove the first of these let x ∈ V1i with dist(x, ∂Ui ) = δ 0 > δ and check that if ε is small then for z ∈ ∂V 1 the inequality |x − z| < δ/3 + ε leads to a contradiction). Thus V ≡ V1 satisfies the first conclusion above and there is a function χ ∈ Cc∞ (V2 ) which is identically one on V1 and which satisfies the third and fifth conditions. Also, there are functions ϕ0i ∈ Cc∞ (Ui ) which are identically onePon V2i and which satisfy the second, third, and fifth conditions. Then i ϕ0i (x) ≥ 1 for x ∈ V2 and since the collection {V2i }i

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1. GEOMETRIC PROPERTIES OF OPEN SETS IN Rn

9

∞ 2 is locally finite,
reciprocal of this function is C on V . The functions P the 0 0 ϕi ≡ χϕi / i ϕi are then easily seen to satisfy all the required conditions. 

Exercises 1.1 Construct a function from a nontrivial interval in R into a separable Hilbert space which is weakly continuous but not continuous. 1.2 Prove that a weakly measurable mapping of a nontrivial interval in R into a separable Hilbert space is measurable in the usual sense that inverse images of open sets are measurable. Suggestion: prove that the inverse image of a ball is measurable, then prove that an arbitrary open set is the countable union of balls. 1.3 Let H and H0 be separable Hilbert spaces with H0 continuously contained in and dense in H. Show that a mapping from a nontrivial interval in R into H0 is weakly measurable as a mapping into H if and only if it is weakly measurable as a mapping into H0 . Suggestion: apply Theorem A.10(b) in the Appendix. 1.4 Let u be a Lipschitz mapping of an open interval I ⊆ R into a separable Hilbert space H. Prove that there exists a bounded, weakly measurable mapping u0 : I → H such that for all t1 , t2 ∈ I, Z t2 u(t2 ) − u(t1 ) = u0 (t)dt t1

h−1

  and for almost all t ∈ I, u(t + h) − u(t) * u0 (t) weakly in H as h → 0. (See also Exercise 2.7 for a related result.) 1.5 Consider the formal problem: find u : Rn × [0, T ) → R such that   ut = −∆2 u − u + g, (x, t) ∈ Rn × (0, T ), u(x, t) → 0 as |x| → ∞, t > 0,  u(x, 0) = ς0 (x), x ∈ Rn , where ∆2 is the “biharmonic” operator, which is the Laplacean composed with itself. Derive the weak form of this problem as in section 1.1, identifying the bilinear form A and the spaces H0 and H1 . Then show that there is a positive constant θ such that A(u, u) ≥ θ|u|2H1 for all u ∈ H1 . Suggestion: make use of the Fourier transform. 1.6 Construct examples showing that neither the segment condition nor the cone condition implies the other. 1.7 Simplify the conditions in Definition 1.7(c) for the case that Ω is bounded and the collection {Ui } is finite.

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10.1090/surv/251/02

CHAPTER 2

Differential Equations in Hilbert Space Separable Hilbert spaces we can understand; beyond these lies only mysticism. Anonymous

[The space of almost periodic functions belies the epigraph ([35] pg. 327); still, all the spaces occurring here will be separable, and this property will serve us well.] In this chapter we develop an existence and regularity theory for solutions of the abstract Hilbert space problem introduced in section 1.1. The most basic result, essential for all that follows, is given in Theorem 2.1. In section 2.2 the concept of weak absolute continuity of solutions is defined, and corresponding higher order regularity results are proved in subsequent sections in Theorems 2.5, 2.7 and 2.10. The first two of these require a certain compatibility condition between the initial data and the inhomogeneous terms, and the third shows how instantaneous regularization in positive time occurs when this compatibility condition fails. An alternative regularity framework is developed in Theorems 2.11 and 2.12 for problems with symmetry, and in Theorem 2.13 a spectral representation is derived for solutions of problems in which the leading-order terms are symmetric and constant in t, generalizing familiar representations based on Fourier series and the Duhamel integral.

§2.1

Basic Existence and Uniqueness

The problem of interest is formulated as follows. Assume: (2.1) • T ∈ (0, ∞] • H0 and H1 are separable Hilbert spaces with H1 continuously contained in and dense in H0 (but see Remark 1 below) • A ∈ L2loc ([0, T ); B(H1 × H1 )) 11

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12

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

• there is a function λ ∈ L1loc ([0, T )) and a nonincreasing function θ : [0, T ) → (0, ∞) such that for almost all t ∈ [0, T ), A(t)(z, z) ≥ θ(t)|z|2H1 − λ(t)|z|2H0 for all z ∈ H1 . Then given (2.2)

ς0 ∈ H0 and F ∈ L2loc ([0, T ); H1∗ )

the problem is to find u ∈ C([0, T ); H0weak ) whose equivalence class on [0, T ) is in L2loc ([0, T ); H1 ) and which satisfies Z t (2.3) hu(t), viH0 = hς0 , viH0 + [−A(s)(u(s), v) + F (s) · v] ds 0

for all v ∈ H1 and all t ∈ [0, T ). The following is our most basic theorem on existence and uniqueness of solutions: Theorem 2.1. Let T, H0 , H1 and A be as in (2.1) and ς0 and F as in (2.2). (a) There is an element u ∈ C([0, T ); H0weak ) whose Lebesgue equivalence class on [0, T ) is in L2loc ([0, T ); H1 ) which satisfies (2.3) for all v ∈ H1 and for which the bound Z t¯ R t¯ 2 ¯ e t 2λ(s)ds |u(t)|2H1 dt sup |u(t)|H0 + θ(t) 0 0≤t≤t¯ (2.4) Z ¯ R R ≤e

t¯ 0 2λ(t)dt |ς0 |2 H

0

t

+ θ(t¯)−1

e

t¯ t 2λ(s)ds |F (t)|2 H∗ dt 1

0

holds for all t¯ ∈ [0, T ). (b) If in addition A ∈ L∞ loc ([0, T ); B(H1 × H1 )) then u ∈ C([0, T ); H0 ) (that is, H0 with the norm topology) and the energy equality Z t Z t 2 2 1 1 (2.5) A(s)(u(s), u(s))ds = 2 |ς0 |H0 + F (s) · u(s) ds 2 |u(t)|H0 + 0

0

holds for all t ∈ [0, T ). Solutions of (2.3) with ς0 and F as in (2.1) and A ∈ L∞ loc ([0, T ); B(H1 × H1 )) are unique in the regularity class weak C([0, T ); H0 ) ∩ L2loc ([0, T ); H1 ). The proof of Theorem 2.1 is given below following several remarks: Remarks: (1) There is no loss of generality in assuming that H1 is dense in H0 : we need only replace H0 by the completion H00 of H1 with respect to the H0 norm and ς0 by its H0 -projection onto H00 . The solution u of the amended problem will then be a solution of the original problem.

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2. BASIC EXISTENCE AND UNIQUENESS

13

(2) There is no assumption in (2.1) about the sign of λ. In particular, if λ is negative then (2.4) may give important information about the large-time behavior of the solution u. (3) The application of Theorem 2.1 to the example of section 1.1 is immediate: ∞ ¯ If aij ∈ L∞ loc ([0, T ); L (Ω)) and if for all t ∈ [0, T ) there is a positive number θ(t¯) such that for almost all (x, t) ∈ [0, t¯] × Ω, aij (x, t)ξ i ξ j ≥ θ(t¯)|ξ|2 for all ξ ∈ Rn , then given ς0 ∈ L2 (Ω) and f ∈ L2loc ([0, T ); L2 (Ω)) there is a unique u ∈ C([0, T ); L2 (Ω)) with [0, T )–equivalence class in L2loc ([0, T ); H01 (Ω)) satisfying (1.2) for every v ∈ H01 (Ω). (4) Equality in (2.5) is important in certain applications in which the square of the H0 norm represents an energy and exact energy balance is of interest. More generally, (2.5) provides rigorous justification for a standard formal “energy estimate” argument familiar in more elementary discussions. For example, for the initial-boundary value problem discussed in section 1.1, the argument consists in multiplying the differential equation in (1.1) by u then integrating over Ω × [0, t] and applying the divergence theorem and the boundary condition, all formally. The result is that Z tZ 2 1 aij uxi uxj dxdt 2 |u(·, t)|L2 (Ω) + 0 Ω Z tZ = 21 |ς0 |2L2 (Ω) + g(x, s)u(x, s)dxds, 0

Ω

which is the specific realization of (2.5) for this example. The two terms on the left may be interpreted as representing respectively the energy in a physical system at time t and the cumulative energy lost to dissipation up to time t, and the two terms on the right as the initial energy and the mechanical work done on the system by an external force g up to time t. (5) Uniqueness, strong continuity into H0 and the energy equality (2.5) are not proved for the case that A ∈ (L2 − L∞ )(B(H1 × H1 )); indeed, the integral on the left in (2.5) may not be defined in this case. These deficits are remedied under certain additional symmetry conditions, however; see Theorem 2.11 below. Notice that, at the very least, the solution u in (a) is strongly continuous from the right at t = 0 because by (2.4) and the weak continuity, lim supt→0+ |u(t)|H0 ≤ |ς0 |H0 ≤ lim inf t→0+ |u(t)|H0 . PROOF of Theorem 2.1. Let {ϕk } be the special orthonormal basis for H0 constructed in Theorem A.10 in the Appendix: ϕk ∈ H1 for all k and if v ∈ H1 and v K is its H0 -orthogonal projection onto HK ≡ span{ϕ1 , . . . , ϕK }, then |v K |H1 ≤ 2|v|H1 for all K and v K → v in the H1 norm. An approximate solution uK : [0, T ) → HK is then constructed by requiring that Z t   (2.6) huK (t), v K iH0 = hς0 , v K iH0 + −A(s)(uK (s), v K ) + F (s) · v K ds 0

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14

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

for all t ∈ [0, T ) and all v K ∈ HK . It suffices that (2.6) holds with v K = ϕk P for all k ≤ K, and therefore that uK (t) = K 1 ak (t)ϕk where a = (a1 , . . . , aK ) satisfies Z t (2.7) ak (t) = ak (0) + [−A(s)(ϕi , ϕk )ai + F (s) · ϕk ] 0

for k = 1, . . . , K and t ∈ [0, T ), and where ak (0) = hς0 , ϕk iH0 (summation over i is understood). The existence of an absolutely continuous solution a : [0, T ) → RK of (2.7) is guaranteed by Theorem A.9 in the Appendix. Now recall that an absolutely continuous real or RK -valued function is differentiable almost everywhere and is the antiderivative of its derivative (see [34] pp. 106–107 or [13] Corollary 3.33 for example). It follows that, for almost all t, a˙ k = −A(ϕi , ϕk )ai + F · ϕk and therefore that

(2.8)

d K2 |u |H0 = −2A(uK , uK ) + 2F · uK dt ≤ −2θ|uK |2H1 + 2λ|uK |2H0 + 2F · uK

≤ −θ|uK |2H1 + 2λ|uK |2H0 + θ−1 |F |2H1∗ .  R  t Multiplying by the integrating factor exp −2 0 λ(s)ds and applying the absolute continuity, we conclude that, for t¯ ∈ [0, T ), Z t¯ R t¯ K 2 sup |u (t)|H0 +θ(t¯) e t 2λ(s)ds |uK (t)|2H1 dt 0 0≤t≤t¯ (2.9) Z ¯ R R t¯ 0 2λ(t)dt |ς0 |2 H

≤e

0

t

+ θ(t¯)−1

e

t¯ t 2λ(s)ds |F (t)|2 H∗ dt

0

1

and therefore that (2.10)

K

sup |u 0≤t≤t¯

(t)|2H0

Z + 0

t¯

|uK (t)|2H1 ≤ C(t¯)

where the constant C depends on A, F, ς0 and t¯ but is independent of K. In particular, the sequence uK is uniformly bounded in L2 ([0, t¯]; H1 ) for every t¯ ∈ (0, T ) (modulo Lebesgue equivalence class on [0, T )) and so has a weakly converging subsequence. A diagonal process then shows that there is a single subsequence Kl → ∞ and an element u ∈ L2loc ([0, T ); H1 ) such that [uKl ][0,t¯],H1 * u weakly in L2 ([0, t¯]; H1 ) for every t¯ ∈ (0, T ). We claim that for this same subsequence, uKl (t) converges weakly in H0 for every t ∈ [0, T ). To see this we fix t and observe that, by the bound in (2.10), uKl (t) has weak-H0 subsequential limits; it therefore suffices to show that all such subsequential limits agree. Assume therefore that a further subsequence uKli (t) converges weakly in H0 , say to w. Let v ∈ H1 and let v M be its H0 projection onto HM . Then v M ∈ HKli if Kli ≥ M and so Z t   huKli (t), v M iH0 = hς0 , v M iH0 + −A(s)(uKli (s), v M ) + F (s) · v M ds 0

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2. BASIC EXISTENCE AND UNIQUENESS

15

by (2.6). We first let i → ∞ using the weak convergence uKli (t) * w in H0 and uKli * u in L2 ([0, t]; H1 ) (and the fact that there is an element S ∈ L2loc ([0, T ); L(H1 )) such that for all z, w ∈ H1 and almost all t, A(t)(z, w) = hz, S(t)wiH1 ), then let M → ∞ using the strong convergence v M → v in H1 . The result is that Z t (2.11) hw, viH0 = hς0 , viH0 + [−A(s)(u(s), v) + F (s) · v] ds. 0

The right side here depends on u and on the sequence Kl but not on w or the subsequence Kli . Thus if w ˜ is another such weak-H0 subsequential limit, then hw ˜ − w, viH0 = 0 for all v ∈ H1 . It follows that w ˜ = w because H1 is dense in H0 . This proves that uKl (t) converges weakly in H0 , say to w(t), for all t ∈ [0, T ). Observe also that since the right side of (2.11) is continuous in t and H1 is dense in H0 , the limit w is weakly continuous from [0, T ) into H0 . Recall that u ∈ L2loc ([0, T ); H1 ) is an equivalence class whereas w ∈ C([0, T ); H0weak ) is a function. We will show that u is in fact the Lebesgue equivalence class of w on [0, T ). First note that, if f ∈ H0 is fixed, then the map z ∈ H1 → hz, f iH0 is an element of H1∗ , so there exists (a different) S ∈ L(H0 , H1 ) such that hz, Sf iH1 = hz, f iH0 for all z ∈ H1 . Then since u(t) ∈ H1 for almost all t and uKl (t) ∈ H1 for all t, we have that for t ∈ [0, T ) and f ∈ L2 ([0, t]; H0 ), Z

t

Z

t

Z

t

hu, Sf iH1 = lim huKl , Sf iH1 0 0 Z t Z t = lim huKl , f iH0 = hw, f iH0 ,

hu, f iH0 = 0

0

0

the last step following from the dominated convergence theorem and the bound (2.10). It follows that [w][0,T ),H0 = u, as claimed. This regularity together with (2.11) shows that u is a solution of (2.3) in the required sense, and (2.4) follows from (2.9). This proves (a). We now set aside this particular u and instead let u ∈ C([0, T ); H0weak ) with [0, T )-equivalence class in L2loc ([0, T ); H1 ) be any solution of (2.3), and then derive the energy equality (2.5) as an a priori consequence of (2.3) under the assumption that A ∈ L∞ loc ([0, T ); B(H1 × H1 )). Uniqueness of solutions will then follow by the application of (2.5) to the difference between two such solutions, which is a solution of the corresponding problem with ς0 = 0 and F = 0. Also, (2.5), once proved, will show that |u(t)|H0 is continuous in t, so that u ∈ C([0, T ); H0 ), again as an a priori consequence of (2.3). To prove (2.5) we let uK (t) and ς0K denote the H0 -orthogonal projections of u(t) and ς0 onto HK . Then repeating the computation in the first line of (2.8) we obtain that |uK (t)|H0 is absolutely continuous in t and that for

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16

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

almost all t, d K |u (t)|2H0 = −2A(u, uK ) + 2F · uK dt and therefore that 2 1 K 2 |u (t)|H0 +

(2.12)

Z

t

A(u, u)ds 0

= 12 |ς0 K |2H0 +

Z

t

 A(u, u − uK ) + F · uK ds.

0

Recall that, since u(t) ∈ H1 for almost all t, uK (t) → u(t) a.e. strongly in H1 by our choice of basis. The integrand on the right in (2.12) therefore converges to F · u for almost all t and is bounded by C(|u|2H1 + |u|H1 |F |H1∗ ), where C is independent of K, which is integrable on [0, t] (it is here that the hypothesis that A ∈ L∞ loc is applied). The energy equality (2.5) therefore follows in the limit as K → ∞, and the bound in (2.4) is derived as in (2.8). This completes the proof of (b). 

§2.2

Weak Absolute Continuity

Higher order regularity of solutions typically entails differentiability, and while differentiability of mappings from an interval in R into a Banach space can be defined in various ways, our purposes will require that solutions be the antiderivatives in a particular sense of their derivatives. The reader will recall in this regard the familiar fact that a real-valued function of bounded variation is differentiable in the usual sense almost everywhere but need not be the antiderivative of its derivative. The relevant concept for us will be weak absolute continuity: Definition 2.2. Let I ⊆ R be a nontrivial interval. (a) A mapping F from I into the dual X ∗ of a separable Banach space X is weakly absolutely continuous with derivative F 0 ∈ L1loc (I; X ∗ ) if for each v ∈ X and all t1 , t2 ∈ I, Z t2 F (t2 ) · v − F (t1 ) · v = F 0 (t) · v dt. t1

The vector space of weakly absolutely continuous functions is denoted by wAC(I; X ∗ ) and the set of such functions with derivatives in Lploc (I; X ∗ ) by wAC p (I; X ∗ ). (b) Let H and H0 be separable Hilbert spaces. A mapping A : I → B(H × H0 ) is weakly absolutely continuous with derivative A0 ∈ L1loc (I; B(H × H0 )) if for each (z, w) ∈ H × H0 and all t1 , t2 ∈ I, Z t2 A(t2 )(z, w) − A(t1 )(z, w) = A0 (t)(z, w)dt. t1

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2. WEAK ABSOLUTE CONTINUITY

17

The vector space of such weakly absolutely continuous mappings is denoted by wAC(I; B(H × H0 )) and the space of such maps with A0 ∈ Lploc (I; B(H × H0 )) by wAC p (I; B(H × H0 )). Remarks: (1) Since Hilbert spaces can be identified with their duals, part (a) of the definition includes the following statement: if H is a separable Hilbert space then u ∈ wAC p (I; H) with derivative u0 ∈ Lploc (I; H) if and only if Z t2 hu(t2 ), viH − hu(t1 ), viH = hu0 (t), viH dt t1

for all v ∈ H and t1 , t2 ∈ I. (2) The notations F 0 and A0 do not reference the underlying Banach or Hilbert spaces, so that some care must be taken when two or more spaces are involved, as in Theorem 2.1. It is an easy exercise to show that if H and H0 are separable Hilbert spaces with H0 continuously included in H and if u ∈ wAC(I; H0 ) with derivative u0 , then u ∈ wAC(I; H) with the same derivative u0 . A partial converse is given in Lemma 2.3 below. (3) If F ∈ wAC(I; X ∗ ) then F ∈ C(I; X ∗ ), and if F ∈ wAC p (I; X ∗ ) for some p ∈ (1, ∞] then F ∈ C λ (I; X ∗ ) with λ = 1 − 1/p, since for v ∈ X and t1 , t2 ∈ I, Z t2 |F (t2 ) · v − F (t1 ) · v| ≤ |F 0 (t)|X ∗ dt |v|X t1

which implies that |F (t2 ) − F (t1 )|X ∗ ≤ |F 0 |Lp ([t1 ,t2 ];X ∗ ) |t2 − t1 |λ . Similar remarks apply to mappings in wAC(I; B(H × H0 )). (4) Observe that if u is a solution of (2.3) as described in Theorem 2.1 and if u∗ : [0, T ) → H1∗ is defined by u∗ (t) · v = hu(t), viH0 , then u∗ ∈ wAC([0, T ); H1∗ ) in case (a) and u∗ ∈ wAC 2 ([0, T ); H1∗ ) in case (b), with derivative (u∗ )0 · v = −A(u, v) + F · v in either case. (5) If H is a separable Hilbert space and if u, v ∈ wAC(I; H), then the inner product hu, viH is absolutely continuous on I with derivative hu0 , viH + hu, v 0 iH . Consequently |u|2H is absolutely continuous with derivative 2hu0 , uiH . See Exercise 2.6. The following result, pursuant to Remark 2 above, will be applied in later sections: Lemma 2.3. Let I ⊆ R be a nontrivial interval and H and H0 separable Hilbert spaces with H0 continuously included in H. Assume that

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18

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

u ∈ wAC 2 (I; H) with derivative u0 ∈ L2loc (I; H0 ) and that u(t) ∈ H0 for all t ∈ I. Then u ∈ wAC 2 (I; H0 ) with the same derivative u0 . PROOF. First, since the hypotheses hold with H replaced by the H-closure of H0 , we can assume that H0 is dense in H. Theorem A.10 in the Appendix then applies to show that there is an orthonormal basis {ϕk }k for H with the following properties: each ϕk is in H0 and if z ∈ H0 and z K is its H0 orthogonal projection onto the span of {ϕ1 , . . . , ϕK }, then |z K |H0 ≤ 2|z|H0 for all K and z K → z strongly in H0 . Thus for v ∈ H0 and t1 , t2 ∈ I, K X t t t hu t21 , viH0 = limhuK t21 , viH0 = lim hu t21 , ϕk iH hϕk , viH0 1

= lim

K Z t2 X t1

1

Z

hu0 (s), ϕk iH hϕk , viH0 ds

t2

0 K

Z

t2

h(u ) (s), viH0 ds =

= lim t1

hu0 (s), viH0 ds,

t1

where the dominated convergence theorem and the fact that |(u0 )K |H0 ≤ 2|u0 |H0 a.e. have been applied in the last step.  Compositions of mappings such as the integrand t → A(t)(u(t), ·) in (2.3) will arise frequently and it will therefore be necessary to consider their absolute continuity. The following abstract product rule will be applied in later sections: Lemma 2.4. Let I ⊆ R be a nontrivial interval and H and H0 separable Hilbert spaces with H0 continuously included in H, and let X be a third separable Hilbert space. Assume that • u ∈ wAC 2 (I; H) with derivative u0 ∈ L2loc (I; H) • u(t) ∈ H0 for every t ∈ I and |u(·)|H0 ∈ L2loc (I) • B ∈ L2loc (I; B(H × X)) • the restriction B H0 ×X of B to H0 × X is in wAC 2 (I; B(H0 × X)) with derivative B 0 . Then the map t ∈ I → B(t)(u(t), ·) ∈ X ∗ is in wAC(I; X ∗ ) with derivative B(t)(u0 (t), ·) + B 0 (t)(u(t), ·). ˜ PROOF. First, we can assume that H0 is dense in H. To prove this let H 0 ⊥ ˜ for all t. It follows that if v ∈ H ˜ be the H-closure of H so that u(t) ∈ H 0 then hu (t), viH = 0 for almost all t. Applying separability, we then find ˜ for almost all t depending on the specific representative. The that u0 (t) ∈ H

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2. WEAK ABSOLUTE CONTINUITY

19

˜ which hypotheses of the lemma are therefore satisfied with H replaced by H, 0 we rename H and in which H is dense, as required. Now let {ϕk }k ⊂ H0 be the H-orthonormal basis of Theorem A.10(b) and for general v ∈ H let v K denote the H-orthogonal projection of v onto the span of {ϕ1 , . . . , ϕK }. Then u(t)K → u(t) strongly in H0 for all t and u0 (t)K → u0 (t) strongly in H for almost all t. Thus for v ∈ X and t1 , t2 ∈ I, K t2 t2 t2 X B(·)(u(·), v) = lim B(·)(uK (·), v) = lim hu(·), ϕk iH B(·)(ϕk , v) t1

t1

= lim

K Z X



t1

 hu0 , ϕk iH B(ϕk , v) + hu, ϕk iH B 0 (ϕk , v) dt

t1

1

Z

t2

1

t2

= lim



 B((u0 )K , v) + B 0 (uK , v) dt

t1

Z

t2

=

  B(u0 , v) + B 0 (u, v) dt

t1

as required. We have applied here the absolute continuity of the product of two absolutely continuous functions of t in the third equality and the dominated convergence theorem in the last. To justify the latter, observe that the integrand in the second-last integral is bounded a.e. by
kBkH×X |u0 |H + kB 0 kH0 ×X |u|H0 |v|X , which is independent of K and whose summands are products of elements of L2 ([t1 , t2 ]).  A Leibnitz rule for higher order absolute continuity can be derived by iterating the product rule. We give the statement for the case that the three Hilbert spaces in Lemma 2.4 are the same: Corollary. Let I ⊆ R be a nontrivial interval, H a separable Hilbert space and m ≥ 1. Suppose that B = B (0) , . . . , B (m−1) ∈ wAC 2 (I; B(H × H)) are given with respective derivatives the equivalence classes of B (1) , . . . , B (m) , and u = u(0) , . . . , u(m−1) ∈ wAC 2 (I; H) are given with respective derivatives the equivalence classes of u(1) , . . . , u(m) (thus B (m) and u(m) are themselves equivalence classes). Let v ∈ H and define h(t) = B(t)(u(t), v) and j   X j (j) h (t) = B (i) (t)(u(j−i) (t), v) i i=0

h(0) , . . . , h(m−1)

for j = 0, . . . , m. Then are absolutely continuous on I with respective derivatives the equivalence classes of h(1) , . . . , h(m) , and if m ≥ 2 then h(0) , . . . , h(m−2) are continuously differentiable on I (with one-sided derivatives at endpoints in I, if any).

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20

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

PROOF. The m = 1 case is included in Lemma 2.4 and the general case is proved by induction in the usual way. Continuous differentiability follows from the second remark following Definition 2.2.  We now consider the conditions under which the solution u of the problem (2.3) described in Theorem 2.1 is in wAC 2 ([0, T ); H0 ), say with derivative u0 ∈ L2loc ([0, T ); H0 ). If so, then higher order regularity can be considered by induction provided that u0 has a specific representative u(1) ∈ C([0, T ); H0 ) with equivalence class in L2loc ([0, T ); H1 ) solving an initial value problem of the same type with u(1) (0) ≡ ς1 , which is to be determined. To explore this we proceed formally from (2.3), anticipating that for all v ∈ H1 , (2.13)

hu(1) (t), viH0 = −A(t)(u(t), v) + F (t) · v,

which may be considered the “strong form” of (2.3). This suggests that u(1) (0) = ς1 where (2.14)

hς1 , viH0 = −A(0)(ς0 , v) + F (0) · v,

which in the applications of interest expresses a necessary compatibility between the initial value ς0 and boundary conditions satisfied by elements of H1 (see the detailed discussion in section 3.1). Formally differentiating (2.13) and applying the product rule and integrating, we arrive at Z t  (1) hu (t), viH0 = hς1 , viH0 + − A(s)(u(1) (s), v) (2.15) 0  − A0 (u(s), v) + F 0 (s) · v ds, which is of the type considered in (2.3) with ς0 and F (t) replaced by ς1 and F 0 (t) − A0 (t)(u(t), ·). We will see in Theorem 2.5 below that if • A ∈ wAC 2 ([0, T ); B(H1 × H1 )) • F ∈ wAC 2 ([0, T ); H1∗ ) • (ς0 , ς1 ) ∈ H1 × H0 satisfies (2.14) then indeed u ∈ wAC 2 ([0, T ); H0 ) with derivative u0 having a representative u(1) ∈ C([0, T ); H0 ) satisfying (2.13)–(2.15). This is the m = 1 case of the following theorem. We point out, however, that, while satisfactory in applications to differential equations whose coefficients are highly regular, this theorem is suboptimal because the absolute continuity hypotheses on A are overly strong, as we will see in the next section. The more satisfactory higher-order result is proved in Theorem 2.7, the following being a necessary starting point: Theorem 2.5. Let T, H0 , H1 and A satisfy the hypotheses in (2.1) and let m ≥ 1. Assume that A = A(0) , . . . , A(m−1) ∈ wAC 2 ([0, T ); B(H1 × H1 )) are

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2. WEAK ABSOLUTE CONTINUITY

21

given with respective derivatives [A(1) ], . . . , [A(m−1) ], A(m) ∈ L2loc ([0, T ); B(H1 × H1 )) ([·] denotes equivalence class on [0, T )). Then given F = F (0) , . . . , F (m−1) ∈ wAC 2 ([0, T ); H1∗ ) with respective derivatives [F (1) ], . . . , [F (m−1) ], F (m) ∈ L2loc ([0, T ); H1∗ ) and given (ς0 , . . . , ςm ) ∈ H1 × · · · × H1 × H0 satisfying j   X j (2.16) hςj+1 , viH0 = − A(i) (0)(ςj−i , v) + F (j) (0) · v i i=0

for j = 0, . . . , m − 1 and all v ∈ H1 , the corresponding solution u of the problem (2.3) described in Theorem 2.1(b) satisfies the following: (a) u is m times continuously differentiable in the following sense: there are mappings u = u(0) , . . . , u(m−1) ∈ wAC 2 ([0, T ); H1 ) and u(m) ∈ C([0, T ); H0 ) such that the derivative of u(j) is [u(j+1) ] for j ≤ m − 1. (b) The equivalence classes of the linear functionals j   X j (2.17) Gj ≡ − A(i) (u(j−i) , ·) + F (j) i i=1

are in L2loc ([0, T ); H1∗ ) for j = 1, . . . , m; and if m ≥ 2 then G1 , . . . , Gm−1 ∈ wAC 2 ([0, T ); H1∗ ) with respective derivatives the equivalence classes of G2 , . . . , Gm . (c) For 1 ≤ j ≤ m, u(j) is the unique solution of the problem Z th i (j) (2.18) hu (t), viH0 = hςj , viH0 + −A(s)(u(j) (s), v) + Gj (s) · v ds 0

in the sense of Theorem 2.1(b); and for 0 ≤ j ≤ m − 1 the strong forms hu(j+1) (t), viH0 = −A(t)(u(j) (t), v) + Gj (t) · v

(2.19)

hold for all v ∈ H1 and all t ∈ [0, T ) (where G0 = F ). (d) Given t¯ ∈ (0, T ) there is a constant C depending on upper bounds for m, t¯, θ(t¯)−1 , the norms of λ and the A(j) on [0, t¯] occurring in the above hypotheses and on the norm of the inclusion H1 ⊆ H0 , but independent of F and of (ς0 , . . . , ςm ), such that Z t¯ m−1 X (j) 2 (m) 2 sup |u (t)|H1 + sup |u (t)|H0 + |u(m) |2H1 dt ¯ j=0 0≤t≤t

≤ C(t¯)

0≤t≤t¯

m X j=0

|ςj |2H0 +

0

m−1 X

sup |F

¯ j=0 0≤t≤t

(j)

(t)|2H1∗

Z + 0

t¯

 |F (m) |2H1∗ dt .

PROOF. The proof is by induction on m. First note that the fourth hypothesis in (2.1) can be combined with the absolute continuity assumption on A

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22

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

to show that there are positive functions θ0 and λ0 on [0, T ) such that for t¯ ∈ [0, T ), (2.20)

A(t)(z, z) ≥ θ0 (t¯)|z|2H1 − λ0 (t¯)|z|2H0

for all z ∈ H1 and t ∈ [0, t¯] (see Exercise 2.9). For the case m = 1 we let HK and uK be as in the first half of the proof of Theorem 2.1 except that uK (0) ≡ u ˜K 0 is chosen differently. Specifically, K since A˜ ≡ A(0) + λ0 (0)h·, ·iH0 is coercive on H1 × H1 , we can define u ˜K 0 ∈H by the requirement that (2.21)

˜ uK , v K ) = −hς1 , v K iH + λ0 hΠK ς0 , v K iH + F (0) · v K A(˜ 0 0 0

for all v K ∈ HK , where ΠK is the H0 -orthogonal projection onto HK . We claim that uK 0 → ς0 in H1 as K → ∞. To prove this we subtract (2.14) from ˜ uK − ς0 , v K ) = 0 for all v K ∈ HK . Then taking (2.21) to obtain that A(˜ 0 ˜K v K = Π K ς0 − u 0 we compute that for a generic constant C independent of K, 2 K ˜ uK C −1 |˜ uK ˜K 0 − ς0 |H1 ≤ A(˜ 0 − ς0 , u 0 − ς0 + v ) K K = A(˜ uK uK 0 − ς0 , Π ς0 − ς0 ) ≤ C|˜ 0 − ς0 |H1 |Π ς0 − ς0 |H1 , K so that |˜ uK 0 − ς0 |H1 ≤ C|Π ς0 − ς0 |H1 , which goes to zero as K → ∞ by Theorem A.10(b), as claimed. Next we prove that uK t (0) converges to ς1 in K H0 as K → ∞. First note that u is continuously differentiable on [0, T ) by (2.7) and our hypotheses on A and F . Thus

(2.22)

K K K K huK t (t), v iH0 = −A(t)(u (t), v ) + F (t) · v

for all v K ∈ HK and t ∈ [0, T ). Setting t = 0 and substituting from (2.21) we find that K K K K huK uK t (0), v iH0 = hς1 , v iH0 + λ0 h˜ 0 − Π ς0 , v iH0 . K K This shows that uK uK t (0) = Π ς1 + λ0 (˜ 0 − Π ς0 ), which converges to ς1 in H0 as K → ∞, as claimed. Now recall from the first half of the proof of Theorem 2.1 that (2.6)–(2.10) hold, that for a particular subsequence Kl , uKl (t) converges to u(t) weakly in H0 for every t and that [uKl ] converges to [u] weakly in L2loc ([0, T ); H1 ). We will prove convergence of the derivatives uK t , bounds for which are obtained as follows. Since the right side of (2.22) is absolutely continuous in t, we can apply the product rule Lemma 2.4 to obtain that

(2.23)

K K K (1) K K (1) huK · vK tt (t), v iH0 = −A(ut , v ) − A (u , v ) + F

K for such t and applying the for almost all t. Putting v K = uK t (t) ∈ H absolute continuity, we get that Z t   2 2 K (1) K K (1) 1 K 1 K |u (t)| = |u (0)| + − A(uK · uK H0 H0 t , ut ) − A (u , ut ) + F t . 2 t 2 t 0

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2. WEAK ABSOLUTE CONTINUITY

23

Applying (2.10) and Gronwall’s inequality we then obtain Z t¯ h 2 K 2 ¯ sup |uK (t)| + |u | dt ≤ C( t ) |ς0 |2H0 + |ς1 |2H0 t H0 t H1 0≤t≤t¯

0

Z

t¯

+ 0


i kA(1) k2H1 ×H1 |uK |2H1 + |F (1) |2H1∗ dt

for t¯ ∈ (0, T ), where C(t¯) is a generic constant as described in (d). We also have from (2.20) and (2.22) that h i 2 2 |uK (t)|2H1 ≤ C(t¯) |uK |2H0 + |uK | + |F (t)| ∗ t H0 H1 for every t ∈ [0, t¯]. Combining the last two inequalities and applying the bound in (2.10) and Gronwall’s inequality again, we conclude that for t¯ ∈ ˜ t¯) depending on the quantities listed in (d) and (0, T ) there is a constant C( also on ς0 , ς1 and F , but which is independent of K, such that Z t¯ K 2 K 2 2 ˜ ¯ (2.24) sup |u (t)|H1 + sup |ut (t)|H0 + |uK t |H1 dt ≤ C(t). 0≤t≤t¯

0≤t≤t¯

0

We can now repeat the compactness argument following (2.10) in the proof of Theorem 2.1 to show that there is a further subsequence, still denoted by Kl , and there is a function z ∈ C([0, T ); H0weak ) whose equivalence class [z] is l in L2loc ([0, T ), H1 ) such that uK t (t) * z(t) weakly in H0 for every t ∈ [0, T ) 2 l and [uK t ] * [z] weakly in Lloc ([0, T ), H1 ). Observe also that the uniform pointwise bound in (2.24) improves the weak H0 -convergence uK (t) * u(t) to weak H1 -convergence. Taking limits in the equation Z t Kl Kl l hu (t), viH1 = hς0 , viH1 + huK t (s), viH1 ds, v ∈ H1 , 0

we then obtain that u ∈ wAC 2 ([0, T ), H1 ) with derivative the equivalence class of z, which we therefore designate u(1) . This proves the regularity statement in (a) for u. The statement for u(1) will follow by the application of Theorem 2.1 to (2.18) once we prove (b) and (c). First, we have that G1 (t) = −A(1) (u(t), ·) + F (1) (t), whose equivalence class is in L2loc ([0, T ), H1∗ ) by our hypotheses on A and F and the above regularity of u; this proves (b). To prove (c) we integrate (2.23) to get Z t  K K K K huK (t), v i = hu (0), v i + − A(uK H0 H0 t t t ,v ) 0  − A(1) (uK , v K ) + F (1) · v K ds, then take the limit as K = Kl → ∞ to obtain that Z th i (2.25) hu(1) , viH0 = hς1 , viH0 + −A(u(1) , v) + G1 · v ds. 0

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24

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

To justify the last step, say for the A(1) term, we observe that A(1) (t)(uKl (t), v Kl ) → A(1) (t)(u(t), v) for almost all t because uKl (t) * u(t) weakly in H1 for all t, v Kl → v strongly in H1 , and A(1) (t) ∈ B(H1 × H1 ) for almost all t. Also, ˜ t¯)kA(1) kH ×H |v|H , |A(1) (uKl , v Kl )| ≤ kA(1) kH1 ×H1 |uKl |H1 |v Kl |H1 ≤ C( 1 1 1 which is integrable on [0, t] and is independent of Kl ; convergence can therefore be passed under the integral. This proves that u(1) is a solution of (2.25) in the sense of Theorem 2.1(b), which therefore applies to show that u(1) ∈ C([0, T ); H0 ), thus completing the proof of (a) and the first assertion in (c). Finally we take the limit as K = Kl → ∞ in (2.22) to obtain the other assertion in (c), and (d) follows immediately from (2.24). This completes the proof for the case m = 1. Now assume that m ≥ 2 and that the theorem holds with m replaced by m − 1. We have only to show that the m = 1 result applies to the problem (2.18) satisfied by u(m−1) . To do this we need to check the compatibility condition hςm , ·iH0 = −A(0)(ςm−1 , ·) + Gm−1 (0) and the regularity conditions Gm−1 ∈ wAC 2 ([0, T ), H1∗ ) with (Gm−1 )0 = Gm ∈ L2loc ([0, T ), H1∗ ). The first is immediate because u(l) (0) = ςl for 1 ≤ l ≤ m − 1. For the second we check only the absolute continuity of terms A(i) (u(m−1−i) , ·) for 1 ≤ i ≤ m − 1, everything else being similar. For such i, A(i) ∈ wAC 2 ([0, T ); B(H1 × H1 )) by hypothesis, and since m − i − 1 ≤ m − 2, u(m−i−1) ∈ wAC 2 ([0, T ); H1 ) by induction. Lemma 2.4 (with H = H0 = X = H1 ) therefore applies to show that the term in question is in wAC([0, T ); H1∗ ) with derivative A(i+1) (u(m−1−i) , ·) + A(i) (u(m−i) , ·), whose H1∗ norm is bounded locally on [0, T ) by kA(i+1) kH1 ×H1 |u(m−1−i) |H1 + kA(i) kH1 ×H1 |u(m−i) |H1 ∈ L2 · L∞ + L∞ · L2 ⊆ L2 . This shows that A(i) (u(m−1−i) , ·) ∈ wAC 2 ([0, T ); B(H1 × H1 )), and the relation (Gm−1 )0 = Gm then follows easily. This completes the induction step and the proof of the theorem. 

§2.3

Higher Order Regularity

In this section we improve the result in Theorem 2.5 by giving more refined regularity information about the solution u and by weakening the hypotheses on the bilinear form A. The complete statement is given in Theorem 2.7 below and its proof in section 2.4. The following preliminary discussion will motivate the rather lengthy technical analysis to follow. First recall the Lax-Milgram theorem: if H is a Hilbert space and B is a bounded,

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2. HIGHER ORDER REGULARITY

25

coercive bilinear form on H, that is, there is a positive constant θ such that B(z, z) ≥ θ|z|2H for all z ∈ H, then there is an S ∈ L(H∗ , H) such that B(SG, v) = G · v for all v ∈ H and G ∈ H∗ . The operator S is injective, so if V0 is a proper subspace of H∗ then S(V0 ) is a proper subspace of H. If S(V0 ) is contained in a proper subspace H2 of H and inclusion in H2 is regarded as a regularity restriction, then the obvious statement that G ∈ V0 implies SG ∈ H2 can be interpreted as a regularity theorem. To apply this observation to the problem at hand, suppose that H0 , H1 , A, F and u are as in the m = 1 case of Theorem 2.5 and that A(t) is coercive for every t, so that the operators S(t) ∈ L(H1∗ , H1 ) are defined as above. Assume further that S(t)(V0 ) ⊆ H2 for a particular subspace V0 ⊆ H1∗ containing the set {f ∗ ≡ hf, ·iH0 : f ∈ H0 } and for a particular subspace H2 of H1 . We can then deduce from the equation A(u(t), v) = F (t) · v − hu(1) (t), viH0 , which is the j = 0 case of (2.19), that u(t) ∈ H2 provided that F (t) ∈ V0 . While significant in itself, the above result has an important implication for our hypotheses on A. Recall that the analysis of Theorem 2.5 required the weak absolute continuity of A(t)(u(t), ·) as a mapping into H1∗ , and this in turn required the weak absolute continuity of A as a mapping into B(H1 × H1 ). Now, if it is known that u(t) ∈ H2 for every t, then the milder condition that the restriction A H2 ×H1 be weakly absolutely continuous as a mapping into B(H2 × H1 ) should suffice. Let us examine this point in the context of the example of section 1.1 in which H0 = L2 (Ω) and H1 = H01 (Ω) for a bounded open set Ω ⊂ Rn and A is given by (1.3). We can reasonably anticipate that if A is to be weakly absolutely continuous as a mapping into B(H01 × H01 ) (abbreviating H01 (Ω) = H01 , etc.), then its derivative A(1) should be given by Z (1) A (t)(u, v) = (Dt aij )uxi vxj dx Ω

L2loc ([0, T ); L∞ ).

with Dt aij ∈ On the other hand, we also expect on the basis of standard elliptic theory (see Theorem 4.2) that under reasonable hypotheses on aij and with V0 = {f ∗ : f ∈ L2 }, we can take H2 = H 2 ∩ H01 . If so, then u(t) ∈ H 2 for all t, so that uxi (t) ∈ Lp for all p ∈ [2, p∗ ] (if n ≥ 3) for a certain p∗ determined by n (Theorem A.3(a)). The weaker condition that the restriction of A be weakly absolutely continuous as a map into B((H 2 ∩ H01 ) × H01 )) will therefore hold if Dt aij ∈ L2loc ([0, T ); Lq ) for any particular q satisfying q −1 + p−1 = 12 for some p ∈ [2, p∗ ]. This is a clear improvement on the above assumption required for the application of Theorem 2.5 that Dt aij ∈ L2loc ([0, T ); L∞ ). The reader will no doubt have detected a circularity in the above reasoning, however: for the specific example discussed above, the weak absolute continuity of the restricted bilinear form will suffice for the proof of Theorem 2.5 only if it is known that u(t) ∈ H 2 ∩ H01 for every t. But this fact

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26

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

was derived from (2.19), which is a conclusion of Theorem 2.5. We will circumvent this difficulty by replacing A by approximate bilinear forms Aε for small ε > 0, applying Theorem 2.5 to obtain corresponding approximate solutions uε with the desired regularity properties, and then showing that this regularity persists in the limit as ε → 0. The analysis is straightforward but rather technical. For example, the compatibility conditions (2.16) for Aε and F will differ from those for A and F ; it will therefore be necessary to construct nearby compatible initial data for the perturbed problem. Before stating the main theorem we introduce approximate bilinear forms, constructed by a standard mollifying procedure: Definition 2.6. Fix a C ∞ function h : R → [0, ∞) whose support is contained in [−1, 1] and whose integral over R equals one, and let hε (t) = ε−1 h(t/ε) for ε > 0. Let H and H0 be separable Hilbert spaces, T ∈ (0, ∞] and B ∈ L1loc ([0, T ); B(H × H0 )). Then for ε > 0 and t ∈ [0, T − 2ε) define the approximation Bε by Z Bε (t)(z, w) = hε (t − s)B(s + ε)(z, w)ds |s−t| 0, λB ≥ 0 and δB ∈ (0, 1] such that the following hold: (a) B(z, z) ≥ θB |z|2H1 − λB |z|2H0 for all z ∈ H1 . (b) For j = −1, . . . , 2m − 1 and λ0 ≥ λB : if for some z ∈ H1 and G ∈ Vj , B(z, v) + λ0 hz, viH0 = G · v for all v ∈ H1 , then z ∈ Hj+2 and |z|Hj+2 ≤ CB |G|Vj . (c) If m ≥ 2 then for i = 1, . . . , m − 1 there are bilinear forms B (i) ∈ B(H2i+1 × H1 ) such that if 0 ≤ l ≤ 2m − 2i − 1 and z ∈ H2i+l+2 , then B (i) (z, ·) ∈ Vl and B |B (i) (z, ·)|Vl ≤ CB |z|δHB2i+l |z|1−δ H2i+l+2 .

Let σ be either 0 or 1. Then given f = (f0 . . . , fm−1 ) ∈ Yσ ≡ V2m−2+σ × · · · × Vσ and wm ∈ Hσ , there is a unique w = (w0 , . . . , wm−1 ) ∈ Xσ ≡ H2m+σ × · · · × H2+σ such that for j = 0, . . . , m − 1, (2.31)

hwj+1 , viH0

j   X j + B (i) (wj−i , v) = fj · v i i=0

for all v ∈ H1 , where B (0) = B. Furthermore, there is a constant C depending −1 on upper bounds for m, CB , θB , λB , δB −1 and the norms of the inclusions and the ∗-maps in the Hilbert space array in the statement of Theorem 2.7 but independent of wm and the fj , such that (2.32)

m−1 X j=0

h

|wj |H2m−2j+σ ≤ C |wm |Hσ +

m−1 X

i |fj |V2m−2j−2+σ .

j=0

Finally, if every B (i) ∈ B(H1 × H1 ), then the solution w is unique in the space [H1 ]m .

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2. PROOF OF THEOREM 2.7

33

PROOF. First, it suffices to prove that there is a number λ ≥ λB determined by the same quantities as the constant C in (2.32) such that the conclusions of the lemma hold with B replaced by Bλ ≡ B + λh·, ·iH0 and B (1) , . . . , B (m−1) unchanged. The proof of this assertion is a discrete version of the change of variables in Exercise 2.1. Specifically, we define for given λ ∈ R a map Tλ : Xσ → Xσ by j   X j i (Tλ (w))j = λ wj−i i i=0

for j = 0, . . . , m − 1. Abusing notation slightly, we also denote by Tλ the map from Yσ to Yσ defined by exactly the same formula. We then claim first that the composition Tλ ◦ T−λ is the identity map and second that w satisfies the equations (2.31) if and only T−λ w satisfies the corresponding system m   X m in which f is replaced by T−λ f , wm by (−λ)i wm−i , B by Bλ , and i i=0

B (1) , . . . , B (m−1) unchanged. The proofs of these assertions involve little more than algebraic manipulations of binomial coefficients and so are omitted. Once proven, these facts allow us to change variables, solve the amended problem, and then invert the change of variables to recover a solution of the original system (2.31). Observe that hypotheses (b) and (c) will hold for the amended problem with the same constant CB independently of λ ≥ λB and that (2.33)

Bλ (z, z) ≥ θB |z|2H1 + λ|z|2H0 , z ∈ H1 ,

(we have taken the now inconsequential constant λB to be zero). In what follows λ will be a free parameter to be chosen later and C will denote a generic positive constant as described in the conclusion of the lemma but which is independent of λ. Given wm and f as in the statement we proceed to solve (2.31) by iteration. Thus define S : Xσ → Xσ by z = Sw, where for j = 0, . . . , m − 1 and all v ∈ H1 , j   X j hzj+1 , viH0 + Bλ (zj , v) = − B (i) (wj−i , v) + fj · v, i i=1

where zm = wm and the sum is zero for j = 0. The existence of a unique solution z ∈ Xσ follows easily by induction, starting with j = m − 1, from hypothesis (b) and the observation that B (i) (wj−i , ·) and fj are in V2m−2j−2+σ by (c) and our assumptions on fj . The next step is to construct a metric on Xσ in which S is contractive. To do this we let w0 , w00 ∈ Xσ , z 0 = S(w0 ) and z 00 = S(w00 ), and w = w00 − w0 and z = z 00 − z 0 . Then for j = 0, . . . , m − 1, j   X j (2.34) hzj+1 , ·iH0 + Bλ (zj , ·) = − B (i) (wj−i , ·) i i=1

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34

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

where zm = 0 and the right side is zero if j = 0. Two preliminary bounds are required. We claim first that m−1 X

(2.35)

|zi |Hσ ≤ Cλ(σ−1)/2 M

i=0

where M=

m−1 X

|wi |H2m−2i

i=0

(which hints at contractiveness for σ = 0 and λ large). To prove this we apply (2.33) in (2.34) to get that (2.36)

θB |zj |2H1

+

λ|zj |2H0

j   X j ≤ |hzj+1 , zj iH0 | + |B (i) (wj−i , zj )|. i i=1

For the B (i) term on the right we have 1 ≤ i ≤ j ≤ m − 1 so that if l = 2m − 2j − 2, then 0 ≤ l ≤ 2m − 2i − 1 as in hypothesis (c), and 2i + l + 2 = 2m − 2(j − i). Therefore |B (i) (wj−i , zj )| ≤ C|B (i) (wj−i , ·)|Vl |zj |H1 ≤ C|wj−i |H2m−2(j−i) |zj |H1 ≤ Cη −1 M 2 + η|zj |2H1 for arbitrary η > 0. Applying this bound in (2.36) we thus obtain
|zj |2H1 + λ|zj |2H0 ≤ C |zj+1 |2H0 + M 2 . The σ = 0 case of (2.35) now follows by induction on j starting with j = m−1 and the fact that zm = 0, and the σ = 1 case follows immediately from the σ = 0 case. This proves (2.35). Next we define norms hvi(j) σ

≡

j∧m−1 X

|vi |H2j−2i+σ

i=0

for j = 0, . . . , m and v = (v0 , . . . , vm−1) ∈ Xσ (the upper limit here is the smaller of j and m − 1). We claim that for j = 1, . . . , m and z and w as in (2.34), h i (σ−1)/2 (j−1) δB (j) 1−δB (2.37) hzi(j) ≤ C λ M + (hwi ) (hwi ) σ σ σ for a new constant C, still independent of λ. The proof is a double induction, (−1) as follows. First, as a notational convenience set hwiσ = 0 so that (2.37) holds for j = 0 by (2.35). Next fix j ≥ 1 and assume that (2.37) holds with j − 1 in place of j. We will show that |zj−k |H2k+σ is bounded by the right side of (2.37) by induction on k = 0, . . . , j. This is true for k = 0 by (2.35),

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2. PROOF OF THEOREM 2.7

35

and if it holds for a given k ≤ j − 1, then by hypotheses (b) and (c) applied in (2.34) with j replaced by j − (k + 1) and with l = 2k + σ, j−(k+1) h i X |zj−(k+1) |H2(k+1)+σ ≤ C |zj−k |H2k+σ + |B (i) (wj−(k+1)−i , ·)|V2k+σ i=1

  B ≤ C |zj−k |H2k+σ + |w(j−1)−(k+i) |δHB2(k+i)+σ |wj−(k+1+i) |1−δ H2(k+1+i)+σ , which by the induction hypothesis on k is bounded by the right side of (2.37). (j) This completes the induction on k and proves that hziσ is bounded by the quantity on the right side of (2.37); and this in turn completes the induction on j, which proves (2.37) for all j. We now fix the constant C in (2.37), which we recall is independent of λ, and construct a norm on Xσ in which S is contractive. Given any positive (−1) numbers c0 , . . . , cm we have from (2.37) and our convention that hwiσ = 0 that for the same z and w, m X

(σ−1)/2 cj hzi(j) M σ ≤ Cλ

j=0

X

cj

j m  X
δB
1−δB cj δB +C cj−1 hwi(j−1) cj hwi(j) σ σ cj−1 j=1



≤ C λ(σ−1)/2 c−1 m

X j

m  c δ B X  j (m) cj cm hwi0 + C max cj hwi(j) σ j cj−1 j=0

(m)

since M = hwi0 . The coefficient of the second sum on the right can be made arbitrarily small by choosing the ratios cj /cj−1 sufficiently small depending on C and δB , which are independent of λ. Then for the case (m) σ = 0 the coefficient of the term cm hwi0 can be made arbitrarily small by choosing λ sufficiently large. We can therefore choose the cj ’s and λ so that if kzkσ is defined to be the sum on the left above, then kzk0 ≤ 12 kwk0 and kzk1 ≤ Ckwk0 + 12 kwk1 for another constant C. The first bound here shows that S is a contraction on X0 , which proves the existence and uniqueness assertion for σ = 0; and the two bounds taken together show that S is a contraction on X1 in the norm k · k0 + ηk · k1 for η sufficiently small, which proves existence and uniqueness for σ = 1. To prove the bound in (2.32) we let z˜ = S(0), so that h˜ zj+1 , ·iH0 + Bλ (˜ zj , ·) = fj for j = 0, . . . , m − 1, where z˜m = wm . It follows easily by induction on j starting with j = m − 1 that each |˜ zj |H2m−2j+σ is bounded by the right side of (2.32), hence so is k˜ z kσ . Then if w is the fixed point of S and k · k is either

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36

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

of the contractive norms constructed above, kwk ≤ kS(w) − S(0)k + k˜ z k, so that kwk is bounded by Ck˜ z k and therefore also by the right side of (2.32). Finally, to prove the last statement of the lemma we consider the Hilbert space array in which each Vj is replaced by H1∗ and Hj is replaced by H1 for j ≥ 1 and then check that the other hypotheses of the lemma still hold. The above analysis therefore applies to show that solutions in [H1 ]m are unique.  We can now apply the results of Theorem 2.5 and Lemmas 2.8 and 2.9 to prove Theorem 2.7. Briefly, we replace A by the approximations Aε of Definition 2.6 and the initial data ς0 by corresponding approximate data ς0ε to obtain approximate solutions uε via Theorem 2.5. Lemma 2.9 will then show that the uε satisfy the required regularity independent of ε, and Lemma 2.8(b) will enable us to extract a limiting solution which retains this regularity. PROOF of Theorem 2.7. We let Aε be the approximations to A defined in Definition 2.6 and leave to the reader to check that Lemma 2.8 guarantees first that each Aε satisfies the hypotheses of Theorem 2.5 and second that (i) the hypotheses of Lemma 2.9 hold with B (i) = Aε (0). Consequently the given data ςm and F (0) (0), . . . , F (m−1) (0) uniquely determine approximate ε data (ς0ε , . . . , ςm−1 ) ∈ H2m × . . . × H2 satisfying the system of equations j   X j ε ε (j) (2.38) hςj+1 , viH0 = − A(i) (0) · v ε (0)(ςj−i , v) + F i i=0

ε = ς . Lemma 2.9 also shows for j = 0, . . . , m − 1 and all v ∈ H1 , where ςm m ε ε that (ς0 , . . . , ςm−1 ) is bounded in H2m × . . . × H2 independently of ε. There is therefore a sequence ε → 0 such that ςjε converges weakly in H2m−2j , say to ς˜j , for each j = 0, . . . , m − 1. We claim that ς˜j = ςj for these j. To prove this let ε → 0 in (2.38) above: convergence of the term on the left is obvious and for the summand on the right we have that at t = 0, ε (i) (i) ε (i) ε A(i) ε (ςj−i , v) = (Aε − A )(ςj−i , v) + A (ςj−i , v). ε The second term on the right converges to A(i) (˜ ςj−i , v) because ςj−i converges (i) weakly in H2m−2j+2i ⊆ H2i+2 and A (0) ∈ B(H2i+2 × H1 ); and the first (i) ε | term on the right is bounded by kAε (0) − A(i) (0)kH2i+2 ×H1 |ςj−i H2i+2 , which (i) goes to zero because A ∈ C([0, T ); B(H2i+2 × H1 )). This proves that (˜ ς0 , . . . , ς˜m−1 ) satisfies (2.16), which is the compatibility system satisfied by (ς0 , . . . , ςm−1 ). The uniqueness assertion in Lemma 2.9 therefore applies to show that ς˜j = ςj for each j, as claimed. Thus

(2.39)

ςjε * ςj weakly in H2m−2j

for j = 0, . . . , m − 1.

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2. PROOF OF THEOREM 2.7

37

We have now checked the hypotheses of Theorem 2.5 for the problem huε (t), viH0 = hς0ε , viH0 +

t

Z

[−Aε (uε (s), v) + F (s) · v] ds; 0

there is therefore a unique solution uε satisfying its conclusions. In partic(0) (m−1) ular, uε = uε , . . . , uε ∈ wAC 2 ([0, T ); H1 ) with respective derivatives (1) (m) [uε ], . . . , [uε ] and

hu(j+1) (t), viH0 ε

j   X j (j−i) =− A(i) (t), v) + F (j) (t) · v ε (t)(uε i i=0

for each t ∈ [0, T ) and j = 0, . . . , m − 1. This is a special case of the system (m) (2.31) of Lemma 2.9, and since uε (t) is in Hσ and (F (0) (t), . . . , F (m−1) )(t) ∈ (V2m−2+σ × . . . × Vσ ) for all t if σ = 0 and almost all t if σ = 1, we (m−1) conclude from Lemma 2.9 that uε , . . . , uε ∈ H2m × . . . , H2 for all t and in H2m+1 × . . . H3 for almost all t. Also, (2.32) shows that for each t¯ ∈ (0, T ) there is a positive constant C(t¯) as described in Theorem 2.7(f) which is independent of ε such that m−1 Xh

(2.40)

j=0

2 sup |u(j) ε (t)|H2m−2j +

Z

0≤t≤t¯

0

t¯

i 2 |u(j) | dt ε H2m−2j+1

Z t¯ h i (m) 2 2 ¯ ¯ ≤ C(t) sup |uε (t)|H0 + |u(m) ε |H1 dt + MF,{ςj } (t) 0≤t≤t¯

0

where MF,{ςj } (t¯) is the quantity in brackets on the right side of (2.28). To (m)

bound the uε terms on the right side of (2.40) we apply the conclusion (2.4) of Theorem 2.1 to the problem (2.18) with j = m and each A(i) replaced (i) by Aε , and then apply hypothesis Theorem 2.7(ci ), which by Lemma 2.8(a) (i) is satisfied by Aε : 2 sup |u(m) ε (t)|H0 0≤t≤t¯

Z + 0

t¯ 2 |u(m) ε |H1

m Z t¯ h i X (m−i) 2 ≤ C(t¯) MF,{ςj } (t¯) + |A(i) (u , ·)| )dt ε ε V−1 i=1

0

m Z t¯ h i X 2δ (m−i) 2(1−δ) ≤ C(t¯) MF,{ςj } (t¯) + (βε |u(m−i) | ) |u | dt . H2i ε H2i+1 i=1

0

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38

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

Substituting this into (2.40) we then obtain that, for arbitrary η > 0, m h X j=0

sup 0≤t≤t¯

2 |u(j) ε (t)|H2m−2j

h ≤ C MF,{ςj } (t¯) +

+ 0

m Z X i=1

t¯

Z

0

t¯

i 2 |u(j) ε |H2m−2j+1 dt


i 2 (m−i) 2 η −1 βε2 |u(m−i) | + η|u | ε H2i ε H2i+1 dt

(observe that the j = m term is now included on the left). The second term in the integrand on the right can be absorbed into the left side by taking η sufficiently small, and the first term can then be eliminated by Gronwall’s inequality. The result is that for t¯ ∈ (0, T − ε) there is a constant C(t¯) as described in Theorem 2.7(f) which is independent of β and ε such that m h X

(2.41)

sup 0≤t≤t¯

j=0

2 |u(j) ε (t)|H2m−2j

Z + 0

t¯ 2 |u(j) ε |H2m−2j+1 dt

≤ C(t¯)e

R t¯+ε 0

βε2

i

MF,{ςj } (t¯).

It follows that there is a further subsequence εl → 0 such that for (j) j = 0, . . . , m, [uεl ] converges weakly to some zj in L2 ([0, t¯]; H2m−2j+1 ) for every t¯ ∈ [0, T ). We claim that, for these j, for all t, and for this same (j) subsequence, uεl (t) converges weakly in H2m−2j , say to z˜j (t). The argument is essentially the same as in the proof of Theorem 2.1: the pointwise bound (j) in (2.41) shows that, for given t fixed, the sequence uεl (t) has weak H2m−2j subsequential limits, and all such limits are shown to be the same. The latter follows because for v ∈ H1 ,

(2.42)

εl hu(j) εl (t), viH0 = hςj , viH0 Z th j   i X j (j) (j−i) (j) + − Aεl (t)(uεl (t), v) − A(i) (u , v) + F · v ds εl εl i 0 i=1

(the sum is absent if j = 0). The right side here converges as εl → 0 to hςj , viH0 +

Z th 0

− A(t)(zj (t), v) −

j   i X j A(i) (zj−i , v) + F (j) · v ds i i=1

by (2.39) and Lemma 2.8(b), and this therefore holds for every further subsequence εli as well. Consequently the difference w between any two such subsequential limits satisfies hw, viH0 = 0 for all v ∈ H1 , and therefore that (j) w = 0 by the density of H1 in H0 . This proves that uεl (t) converges weakly in H2m−2j , say to z˜j (t), as claimed. We can then show exactly as in the

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2. PROOF OF THEOREM 2.7

39

proof of Theorem 2.1 that [˜ zj ] = zj , so that from (2.42) Z th hzj (t), viH0 = hςj , viH0 + − A(t)(zj (t), v) (2.43)

0 j   i X j A(i) (zj−i , v) + F (j) · v ds − i i=1

for all v ∈ H1 , t ∈ [0, T ) and j = 0, . . . , m (again, the sum is absent if j = 0). From this we conclude three things: first that z0 = u is the unique solution of the problem (2.3); second that zm ∈ C([0, T ); H0 ); and third that zj∗ ≡ hzj , ·iH0 ∈ C([0, T ); H1∗ ), which, together with the uniform H2m−2j weak for j = 0, . . . , m − 1. bound in (2.41) shows that zj is continuous into H2m−2j On the other hand, we can also take limits in the equation Z t εl hu(j) (t), vi = hς , vi + hu(j+1) , viH1 ds H1 H1 εl εl j 0

wAC 2 ([0, T ); H

to find that zj ∈ 1 ) with derivative zj+1 , and then by Lemma 2.3 that zj ∈ wAC 2 ([0, T ); H2m−2j−1 ) with derivative zj+1 . We then weak ) (with strong continuity for relabel zj = u(j) so that u(j) ∈ C([0, T ); H2m−2j j = m) and [u(j) ] ∈ L2loc ([0, T ); H2m−2j+1 ), and leave to the reader the task of checking the remaining details for the assertions in (d)–(f). Finally we prove (g) by induction on j, starting with j = m. The strong continuity u(m) ∈ C([0, T ); H0 ) has already been proved. Assume therefore that u(j+1) ∈ C([0, T ); H2m−2j−2 ) for a fixed j ∈ {0, . . . , m − 1}, let t0 , t ∈ [0, T ) and write A(t)(u(j) (t) − u(j) (t0 ), ·) = −hu(j+1) (t) − u(j+1) (t0 ), ·iH0 j    (j)  X j (j) + F (t) − F (t0 ) − A(i) (t)(u(j−i) (t) − u(j−i) (t0 ), ·) i i=1 j   X
j − A(i) (t) − A(i) (t0 ) (u(j−i) )(t0 ), ·). i i=0

Let l = 2m − 2j − 2, so that for i as in the above sums, 0 ≤ l ≤ 2m − 2i − 2, consistent with the hypotheses in (civ ) and (g). The first two terms on the right then go to zero in Vl as t → t0 by induction on j and by the assumption on F (j) . The summand in the third term on the right goes to zero in Vl by hypothesis (civ ) and by the conclusions already proved in (d) that u(j−i) is a Lipschitz map into H2m−2j+2i−2 = H2i+l and is a bounded map into H2m−2j+2i = H2i+l+2 . Finally, the summand in the last sum on the right goes to zero by the second hypothesis in (g) and the latter fact that u(j−i) (t0 ) ∈ H2i+l+2 . This shows that the right side above goes to zero in Vl as t → t0 and therefore that u(j) (t) → u(j) (t0 ) in Hl+2 = H2m−2j by the ellipticity property of A(t) assumed in (g). This

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40

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

proves that u(j) ∈ C([0, T ); H2m−2j ), thereby completing the induction, the proof of (g) and the proof of Theorem 2.7. 

§2.5

Incompatible Data and Initial Layer Regularization

In this section we consider problems (2.3) in which the regularity conditions on A in Theorem 2.7 are satisfied but the compatibility conditions (2.16) are not. It is reasonable to ask in this situation whether solutions attain the regularity described in Theorem 2.7 instantaneously in time, so that the conclusions of the theorem hold locally in (0, T ) if not in [0, T ). Consider the case m = 1 in which A and F satisfy the m = 1 hypotheses of Theorem 2.7, ς0 ∈ H0 , but no assumption is made concerning compatibility. There is then a unique solution u ∈ C([0, T ); H0 ) to the problem (2.3) as described in Theorem 2.1, but Theorem 2.7 does not apply. However, for the new unknown function w(t) = tu(t) we find that, for all v ∈ H1 , Z t   − A(s)(w(s), v) + F˜ (s) · v ds (2.44) hw(t), viH0 = 0

where F˜ (t) = tF (t) + hu, ·iH0 . The m = 1 compatibility condition (2.16) for this system is then satisfied with (ς0 , ς1 ) replaced by (0, ς0 ), and the conclusions of Theorem 2.7 therefore apply to w. Corresponding regularity statements for u then follow from the relation u(t) = w(t)/t. Observe that the above argument requires only that F satisfy the hypotheses of Theorem 2.1 and that tF (t), not F (t), satisfy those of Theorem 2.7. In fact, any algebraic blowup of the norms of F at t = 0+ can be accommodated by taking w(t) = t1+δ u(t) for δ sufficiently large. (We will require either that δ = 0 or δ > 21 , this stipulation insuring that tδ hu(t), ·iH0 is in wAC 2 ([0, T ); V−1 ) as required in Theorem 2.7(c).) The equation (2.44) for w will then hold with F˜ (t) = t1+δ F (t) + (1 + δ)tδ hu(t), ·iH0 , and the argument can be repeated under the following assumptions (in which (0, T ) and [0, T ) are carefully distinguished): • F ∈ wAC 2 ((0, T ); V−1 ) with F (1) ∈ L2loc ((0, T ); V−1 ) • t1+δ F (1) ∈ L2loc ([0, T ); V−1 ) 2 • t1+δ [F ] ∈ L∞ loc ([0, T ); V0 ) ∩ Lloc ([0, T ); V1 ) • there is a γ1 ∈ H0 such for all v ∈ H1 ,

(2.45)

lim t1+δ F (t) · v = hγ1 , viH0 .

t→0+

The second and third of these are the hypotheses (2.26) and (2.27) of Theorem 2.7 imposed on F˜ , and the last shows that the compatibility condition for (2.44) is satisfied with (ς0 , ς1 ) replaced by (0, γ), where γ is either γ1 or γ1 + ς0 according as δ > 12 or δ = 0. Theorem 2.7 therefore applies with

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2. INCOMPATIBLE DATA AND INITIAL LAYER REGULARIZATION

41

m = 1 and shows in particular that w(t) * 0 weakly in H2 and w(1) (t) → γ strongly in H0 as t → 0+ . Then since w(1) (t) = t1+δ u(1) (t) + (1 + δ)tδ u(t), • t1+δ u(t) * 0 weakly in H2 • t1+δ u(1) (t) → γ1 strongly in H0 , which give controlled rates of blowup of |u(t)|H2 and |u(1) (t)|H0 as t → 0+ . These results are generalized to arbitrary order in the following theorem in which the four itemized hypotheses above correspond to the rather technical hypotheses (b) and (c) with m = 1 and the two itemized conclusions above are generalized in (e). Theorem 2.10. Let m ≥ 1 and Hj , Vj and A satisfy the hypotheses of Theorem 2.7. Assume the following: (a) F ∈ L2loc ([0, T ); V−1 ) and ς0 ∈ H0 (as in Theorem 2.1). (b) F (0) = F, . . . , F (m−1) ∈ wAC 2 ((0, T ); V−1 ) are given with respective derivatives [F (1) ](0,T ) , . . . , [F (m−1) ](0,T ) , F (m) ∈ L2loc ((0, T ); V−1 ). (c) There is a δ ∈ {0} ∪ ( 12 , ∞), such that: (i ) for j = 0, . . . , m and k = j, . . . , m, tk+δ F (j) ∈ L2loc ([0, T ); V2k−2j−1 ); (ii ) for j = 0, . . . , m − 1 and k = j + 1, . . . , m, tk+δ F (j) ∈ L∞ loc ([0, T ); V2k−2j−2 ); (iii ) for j = 0, . . . , m − 1, lim tj+1+δ F (j) (t) · v = hγj+1 , viH0

t→0

for some γj+1 ∈ H0 and all v ∈ H1 . Then the unique solution u of (2.3) satisfies the following, in addition to the conclusions of Theorem 2.1: (d) The regularity statements in Theorem 2.7(d) and the strong forms (2.19) hold with [0, T ) replaced by (0, T ); and the weak forms (2.18) hold with [0, t] ⊂ [0, T ) replaced by [t1 , t2 ] ⊂ (0, T ). (e) For k = 1, . . . , m, ( *0 k+δ (j) t u (t) → γj

weakly in H2k−2j , k = j + 1, . . . , m strongly in H0 , k = j

as t → 0+ .

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42

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

(f) Given t¯ ∈ [0, T ) there is a constant C(t¯) as described in Theorem 2.7(f) such that Z t¯ m X k h i X 2(k+δ) (j) 2 sup t |u (t)|H2k−2j + t2(k+δ) |u(j) (t)|2H2k−2j+1 dt k=0 j=0

0≤t≤t¯

0

Z + 0

≤ C(t¯)e

R t¯ h 2 0β

|ς0 |2H0 +

m X

|γj |2H0 + MF

t¯

t2(m+δ )|((u(m) )∗ )0 |2V−1 dt

i

j=1

where β is as in hypothesis (ci ) of Theorem 2.7 and MF bounds the sum of the squares of the norms of tk+δ F (j) on [0, t¯] in hypotheses (ci ) and (cii ) of the present theorem. weak ) asserted in (d) (g) The regularity statement that u(j) ∈ C((0, T ); H2m−2j above can be strengthened to u(j) ∈ C((0, T ); H2m−2j ), that is, continuity in the norm topology, under the additional hypotheses that F (j) ∈ C((0, T ); VH2m−2j−2 ) for j = 0, . . . , m − 1 and that A satisfies the additional hypotheses described in Theorem 2.7(g).

PROOF. We leave to the reader to check the details of the discussion preceding the statement of the theorem for the case that m = 1. Assuming then that the theorem holds with m − 1 ≥ 1 in place of m we let w(t) ≡ tm+δ u(t) and compute that Z t   (2.46) hw(t), viH0 = − A(w, v) + F˜ · v ds 0

for all v ∈ H1 , where F˜ (t) = tm+δ F (t) + (m + δ)tm+δ−1 u(t)∗ and u(t)∗ = hu(t), ·iH0 . We will check that Theorem 2.7 applies to w. First, the hypotheses in (ci) and (cii) and the induction hypotheses on u show that the F˜ (j) satisfy the L2 and L∞ bounds required in (2.26) and (2.27). Next, to show that the absolute continuity in hypothesis (b) extends to t = 0 and to identify F˜ (j) (0) we let 0 < τ ≤ t < T , j ≤ m − 1 and v ∈ H1 . Then by (b), Z t
(j) (j) ˜ ˜ F (t) − F (τ ) · v = F˜ (j+1) (s) · v ds τ

for all v ∈ H1 . Hypothesis (ci) with k = m shows that the integral on the right converges as τ → 0. Concerning the left side we note that F˜ (j) (τ ) is a linear combination of terms τ m+δ−j+i−1 (u(i) (τ ))∗ and τ m+δ−j+i F (i) (τ ) for i = 0 . . . , j. Hypothesis (ciii) and the induction hypotheses on u show that for j < m − 1 these terms all go to zero in V0 as τ → 0+ and that F˜ (m−1) (τ ) * γ ∗ ≡ hγ, ·iH0 weak-∗ in H1∗ for some γ ∈ H0 determined by ς0 and the γj , but which need not be identified explicitly. Thus F˜ (j) ∈ wAC 2 ([0, T ); V−1 ) with value F˜ (j) (0) = 0 for j < m − 1 and F˜ (m−1) (0) = γ ∗ .

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2. SYSTEMS WITH SYMMETRY

43

It follows that the compatibility conditions (2.16) for (2.46) are satisfied with (ς0 , . . . , ςm ) replaced by (0, . . . , γ). Theorem 2.7 therefore applies to w and most of the corresponding conclusions for u(t) = t−(m+δ) w(t) then follow easily. One exception is the second conclusion in (e) that tm+δ u(m) (t) → γm strongly in H0 as t → 0+ . To prove this we note first that w(m) (t) = (tm+δ u(t))(m) → γ strongly in H0 as t → 0+ by Theorem 2.7. Since terms ti+δ u(i) (t) for i < m in its Leibniz expansion have strong H0 limits by the induction hypothesis, so too does the term tm+δ u(m) (t). To identify its limit we recall from the paragraph above that F˜ (m−1) (t) and w(m) (t)∗ = hw(m) (t), ·iH0 have the same limit γ ∗ in V−1 as t → 0. Since  (m−1) w(m) (t) = tm+δ u(1) (t) + (m + δ)tm+δ−1 u(t) and  (m−1) F˜ (m−1) (t) = tm+δ F (t) + (m + δ)tm+δ−1 u(t)∗ 
(m−1) we conclude that tm+δ u(1) (t)∗ − F (t) goes to zero weak-∗ in V−1  i+1+δ (i+1) ∗
 + as t → 0 . But terms t u (t) − F (i) (t) in its Leibnitz expansion go to zero by the induction hypothesis for i ≤ m − 2, hence so too does the term with i = m − 1. This together with hypothesis (ciii) shows that htm+δ u(m) (t) − γm , viH0 → 0 for all v ∈ H1 and therefore that tm+δ u(m) (t) → γm strongly in H0 , since the strong limit is known to exist and H1 is dense in H0 . 

§2.6

Systems with Symmetry

In this section we examine problems in which the bilinear form A in (2.3) is symmetric or nearly so; that is, for which A(u, v) − A(v, u) is either zero for all u and v or is of lower order in a certain sense. This symmetry leads to a regularity result not included in either Theorem 2.1 or Theorem 2.7: for the problem Z t (2.47) hu(t), viH0 = hς0 , viH0 + [−A(s)(u(s), v) + hg(s), viH0 ] ds 0

L2loc ([0, T ); H0 )

with g ∈ and ς0 ∈ H1 , there exists a unique solution u ∈ 2 wAC ([0, T ); H0 )∩C([0, T ); H1weak ). A corresponding higher-order regularity theory for nearly symmetric problems, parallel to that of Theorem 2.7, can then be developed, but this requires considerable effort, similar to the extensive analysis of section 2.4, and will not be included here. On the other hand this improvement in regularity can often be obtained in particular problems rather easily by applying the results in Theorems 2.11 and 2.12 below to the strong forms (2.18) with j > 0. See Theorem 6.14 for an

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44

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

application in which this idea is applied to a large class of quasilinear systems in two and three space dimensions. It is instructive to first consider the example in section 1.1 of an initialboundary value problem for a scalar unknown u : [0, T ) → H01 (Ω), where Ω is an open set in Rn and the differential equation includes a lower order term:   ut = (aij (x, t)uxi )xj + bi (x, t)uxi + g(x, t), (x, t) ∈ Ω × (0, T ), u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ),  u(x, 0) = ς0 (x), x ∈ Ω. The following purely formal argument parallels that of Remark 4 following Theorem 2.1: if we multiply the differential equation by ut and integrate over Ω × [0, t], taking ut to be zero on ∂Ω, we obtain Z tZ Z tZ
u2t dxdt = − aij uxi utxj + bi uxi ut + gut dxdt. 0

Ω

0

Ω

If the symmetry condition aij = aji holds for all i and j, then
aij uxi utxj = 12 aij uxi uxj t − 21 (aij )t uxi uxj so that Z tZ 0

Ω

Z

t aij uxi uxj dx 0 Ω Z tZ   1 = − 2 (aij )t uxi uxj + bi uxi ut + gut dxdt.

u2t dxdt+ 12

0

Ω

It follows that under suitable hypotheses on the coefficients aij and bi , Z t¯Z u2t dxdt + sup |u(t)|2H 1 0

Ω

0

0≤t≤t¯

h ≤ C(t¯) |ς0 |2H 1 + 0

Z t¯Z 0

i (|∇x u|2 + |g|2 )dxdt ,

Ω

which is stronger than the minimal regularity result of Theorem 2.1 but weaker than Z the m = 1 case of Theorem 2.7. Observe that the bilinear form A(u, v) = (aij uxi vxj + bi uxi v)dx in this example is not symmetric, but Ω

rather is the sum A = Asym + B of a symmetric part Asym and a lower-order term B satisfying |B(u, v)| ≤ C|u|H01 |v|L2 for bounded bi . The following two theorems give both a rigorous justification for the above formal argument as well as a generalization to the abstract problem (2.47). In Theorem 2.11 we assume that ς0 is in H1 as in the example, and in Theorem 2.12 we consider the case that ς0 ∈ H0 and an initial-layer regularization occurs. Several remarks follow the statements of the theorems and the proofs are given at the end of the section.

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2. SYSTEMS WITH SYMMETRY

45

Theorem 2.11. Let H0 , H1 and T be as in Theorem 2.1 and assume the following: (a) Asym , B and A = Asym + B are weakly measurable mappings from [0, T ) to B(H1 × H1 ). (b) Asym satisfies (i ) Asym ∈ wAC 2 ([0, T ); B(H1 × H1 )); (ii ) for every t¯ ∈ [0, T ) there are positive numbers θ(t¯) and CA (t¯) and a nonnegative number λ(t¯) such that for all t ∈ [0, t¯] and z, w ∈ H1 , |Asym (t)(z, w)| ≤ CA (t¯)|z|H1 |w|H1 , Z t¯ kA0sym k2B(H1 ×H1 ) dt ≤ CA (t¯) 0

and Asym (t)(z, z) ≥ θ(t¯)|z|2H1 − λ(t¯)|z|2H0 ; (iii ) for all t ∈ [0, T ) and w, z ∈ H1 , Asym (t)(z, w) = Asym (t)(w, z). (c) There is a function β˜ ∈ L2loc ([0, T )) such that for almost all t ∈ [0, T ), ˜ |B(t)(z, w)| ≤ β(t)|z| H1 |w|H0 for all z, w ∈ H1 . Then given ς0 ∈ H1 and g ∈ L2loc ([0, T ); H0 ) there is a unique element u ∈ wAC 2 ([0, T ); H0 ) ∩ C([0, T ); H1weak )

(2.48)

which satisfies (2.47) for all v ∈ H1 . In addition, given specific representatives of g and the derivative u0 of u, there is a null set E ⊂ [0, T ) such that for all t ∈ / E and all v ∈ H1 , hu0 (t), viH0 = −A(u(t), v) + hg(t), viH0 .

(2.49)

The energy equality (2.50)

2 1 2 |u(t)|H0

Z +

t

A(s)(u(s), u(s))ds = 0

2 1 2 |ς0 |H0

Z

t

hg(s), u(s)iH0 ds

+ 0

holds for t ∈ [0, T ), and given t¯ ∈ [0, T ) there is a constant C(t¯) depending only on upper bounds for the norm of the inclusion H1 ⊆ H0 and on ˜ L2 ([0,t¯]) such that t¯, CA (t¯), θ−1 (t¯), λ(t¯) and |β| Z t¯ Z t¯ h i 2 0 2 2 ¯ (2.51) sup |u(t)|H1 + |u (t)|H0 dt ≤ C(t) |ς0 |H1 + |g(t)|2H0 dt . 0≤t≤t¯

0

0

If ς0 is in H0 but not in H1 , the solution u attains the regularity in (2.48) instantaneously in positive time, this effect being similar to the behavior described in Theorem 2.10. The precise result is as follows:

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46

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

Theorem 2.12. Let H0 , H1 , T, A, Asym and B be as in Theorem 2.11. Then given ς0 ∈ H0 and g satisfying (2.52)

tδ/2 g(t) ∈ L2loc ([0, T ); H0 ) and g ∗ ∈ L2loc ([0, T ); H1∗ )

for some δ ≥ 1, there is a unique u in the regularity class (2.53)

C([0, T ); H0 ) ∩ wAC 2 ((0, T ); H0 ) ∩ C((0, T ); H1weak )

with equivalence class on [0, T ) in L2loc ([0, T ); H1 ) which satisfies (2.47) for all v ∈ H1 . The strong form (2.49) holds for almost all t ∈ [0, T ) and the energy equality (2.50) holds for all t ∈ [0, T ), just as in Theorem 2.11; and given t¯ ∈ [0, T )) there is a constant C(t¯) as described in that theorem such that Z t¯ δ 2 sup t |u(t)|H1 + tδ |u0 (t)|2H0 dt ¯ 0 0 0. Denote by v ∈ H0 the integral with respect to s on the left above and z(s) ∈ H0 its integrand. Then omitting the domain (0, L] and the indicator dE(µ) in integrals with

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2. SPECTRAL REPRESENTATIONS

53

respect to E and applying Definition 1.5 and (2.62), we compute that Z Z t Z  χε (µ)µ−σ viH0 χε (µ)µ−σ dE(v,v) = h z(s)ds, (0,L]

0

Z

t

Z

 χε (µ)µ−σ ψ(s, µ) viH0 ds 0 1/2  Z t h Z Z t i 1/2 2 χε (µ)2 µ−2σ ψ(s, µ)2 dE(v,v) ds |g|H0 ≤ hg(s),

=

0

0

=

Z 0

≤

γσ1/2

t

|g|2H0 Z 0

(0,L]

1/2  Z

χε (µ)2 µ−σ

(0,L]

t

|g|2H0

hZ

t

i 1/2 µ−σ ψ(s, µ)2 ds dE(v,v)

0

1/2  Z

χε (µ)µ−σ dE(v,v)

1/2

(0,L]

where γσ is the sup in (2.68). Thus Z t Z χε (µ)µ−σ dE(v,v) ≤ γσ |g|2H0 (0,L]

0

and the result follows by letting ε → 0 on the left and applying (2.62) for σ = 0 and (2.63) for σ = 1.  PROOF of Theorem 2.13. We first project the solution u(t) away from µ = 0 as follows. Let χε be as in the proof of Lemma 2.15 and Pε = Z χε (µ)dE(µ). Then Pε/2 Pε = Pε and if v ∈ H0 then Pε v ∈ H1 . Setting ˜ ∈ L2 ([0, T ); H0 ) we then have that for v ∈ H0 , g˜ = g + Bu loc hPε u(t), viH0 = hu(t), Pε viH0 Z t (2.69)   = hPε ς0 , viH0 + − A0 (u, Pε v) + λhPε u, viH0 + hPε g˜, viH0 . 0 Z However, µ dE(µ) = S where S is as in (2.59), so that Z Z

−1 χ A0 (u, Pε v) = A0 (u, S µ µ−1 χε/2 Pε u, viH0 . ε v) = h Substituting into (2.69) and applying the fact that H1 is dense in H0 we conclude that Pε u ∈ wAC([0, T ); H0 ) with derivative Z
0 (Pε u) = − µ−1 χε/2 Pε u + λPε u + Pε g˜. We may regard this as a linear differential equation for the unknown Pε u for which the operator Z −λt Mε (t) ≡ e et/µ χε (µ)dE(µ)

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54

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

functions as an integrating factor; that is, Mε Pε u is absolutely continuous with derivative Mε Pε g˜ (the proof is straightforward and is given in Theorem A.11). Thus for t ∈ [0, T ), Z t 2 Mε (t)Pε u(t) = Pε ς0 + Mε (s)Pε g˜(s)ds. 0

Applying the operator Qε (t) ≡ eλt

Z

e−t/µ dE to both sides we then obtain

Pε2 u(t) = Qε (t)Pε2 ς0 + Qε (t)

(2.70)

Z

t

Mε (s)Pε g˜(s)ds. 0

We now let ε → 0 as follows. The square of the H0 norm of the difference between the ς0 terms in (2.70) and (2.66) is Z Z 2λt −2t/µ 2 2 2λt e e (1 − χε (µ)) dE(ς0 ,ς0 ) ≤ e (1 − χ2ε (µ))2 dE(ς0 ,ς0 ) (0,L]

(0,L]

which goes to zero uniformly on [0, t¯] for every t¯ ∈ [0, T ). Also, by (2.63) the square of the A0 norm of this same difference is bounded by Z 2λt e µ−1 e−2t/µ (1 − χ2ε (µ))2 dE(ς0 ,ς0 ) (0,L] Z 2λt −1 −2s ≤ (e t sup se ) (1 − χ2ε (µ))2 dE(ς0 ,ς0 ) , s≥0

(0,L]

which goes to zero uniformly on [τ, t¯] for 0 < τ < t¯ < T . Similar considerations apply to the term on the left side of (2.70). Finally, the difference between the integral on the right and the corresponding term in (2.66) can be written Z thZ i eλt e(s−t)/µ (χε (µ)2 − 1)dE e−λs g˜(s)ds, 0

so that the square of its H0 norm is bounded by Z t h iZ t 2λt 2(s−t)/µ χ 2 2 e sup e ( ε (µ) − 1) ds e−2λs |˜ g (s)|2H0 ds µ

0

0

by Lemma 2.15. The sup here may be taken over µ ∈ [0, 2ε] and therefore goes to zero as ε → 0 by direct computation. This proves (2.66) as well as the statements concerning the convergence of the two series in (2.67), and completes the proof of Theorem 2.13.  PROOF of Corollary 2.14. First, since the continuity of eλt is not in doubt, we may assume that λ = 0 (or see Exercise 2.1). We can also assume that ς0 ∈ H1 , otherwise the initial time can be taken to be τ > 0. Let S be as in (2.59), E its spectral measure and L = kSkL(H0 ) .

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2. SPECTRAL REPRESENTATIONS

55

We let 0 ≤ t1 ≤ t2 ≤ t¯ < T and prove that u(t2 ) → u(t1 ) in H1 as ˜ so that |˜ t2 → t1 . Again let g˜ = f + Bu g |L2 ([0,t¯];H0 ) and supt∈[0,t¯] |u(t)|H1 are ¯ bounded by a positive constant C(t). If z(t) is the first term on the right side of (2.66) then by (2.63), Z 2 θ|z(t2 ) − z(t1 )|H1 ≤ µ−1 (e−t2 /µ − e−t1 /µ )2 dE(ς0 ,ς0 ) , (0,L]

which goes to zero as |t2 − t1 | → 0 because the integrand goes to zero for every µ and is bounded by µ−1 , which is integrable with respect to the measure E(ς0 ,ς0 ) by (2.63). Next let w(t) be the second term on the right side of (2.66) and let χε and Pε be as in the proof of Theorem 2.13. Then Z t2  Z  w(t2 ) − w(t1 ) = e(s−t2 )/µ g˜(s)ds t1

(2.71)

Z +

t1  Z

  (s−t2 )/µ  e − e(s−t1 )/µ χε (µ) g˜(s)ds

0

Z +

t1  Z

 (s−t2 )/µ  e − e(s−t1 )/µ (Pε g˜(s) − g˜(s))ds.

0

The square of the H1 norm of the first term on the right is bounded by Z t2   Z t2 −1 −1 2(s−t2 )/µ θ sup µ e ds |˜ g (s)|2H0 ds µ∈(0,L] t1

t1

by Lemma 2.15. The sup here is bounded by C(t¯) by direct computation and the second factor is arbitrarily small if |t2 − t1 | is sufficiently small. Next, the square of the H1 norm of the second term on the right in (2.71) is bounded by Z t1  2 C(t¯) sup µ−1 e(s−t2 )/µ − e(s−t1 )/µ ds ε≤µ≤L 0

which by the mean value theorem applied to the integrand is bounded by C(t¯)|t2 − t1 |2 /ε3 . Finally by Lemma 2.15 the square of the H1 norm of the third term on the right in (2.71) is bounded by Z t ¯ C(t) |˜ g (s) − Pε g˜(s)|2H0 . 0

The integrand here goes to zero as ε → 0 for almost all s and is bounded by |˜ g (s)|2H0 , which is integrable; this integral therefore goes to zero as ε → 0. Combining these bounds, we conclude that the right side of (2.71) can be made arbitrarily small by choosing first ε small then requiring that t2 and t1 be sufficiently close depending on ε. This proves that u : [0, t¯] → H1 is (uniformly) continuous. 

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2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

Exercises 2.1 Show that if u is the solution of (2.3) described in Theorem 2.1 and w(t) = e−λ0 t u(t) for a constant λ0 , then w satisfies the corresponding problem with F (t) replaced by e−λ0 t F (t) and A replaced by A+λ0 h·, ·iH0 . 2.2 Assume that the hypotheses in (2.1) and (2.2) hold with A and F constant in t and λ ≤ 0. Prove that there is an element u ¯ ∈ H1 such that the ¯ as t → ∞. solution u of Theorem 2.1 converges strongly in H0 to u Suggestion: look for a time-independent solution of (2.3). 2.3 The construction of the basis used in the proof of Theorem 2.1 depends on the theory of spectral measures and spectral integrals (see Theorem A.10 in the Appendix). An easier and more accessible construction is available for the case that H0 and H1 are separable Hilbert spaces with H1 dense and compactly contained in H0 , as follows: Let S ∈ L(H0 , H1 ) be the operator satisfying hSf, viH1 = hf, viH0 for all f ∈ H0 and v ∈ H1 . Show that S is compact and self-adjoint as an element of L(H0 ). There is therefore an H0 -orthonormal basis {ϕk }k for H0 whose elements are eigenvectors of S; that is, Sϕk = µk ϕk where the µk are positive 1/2 and decrease to zero. Show that the set {ψk ≡ µk ϕk }k is an H1 orthonormal basis for H1 and that the H0 –projection onto the span of ϕ1 , . . . , ϕK restricted to H1 coincides with the H1 -projection onto the span of ψ1 , . . . , ψK . Then check the conclusions of Theorem A.10. 2.4 Let u be the solution of (2.3) described in Theorem 2.1(b) and let HK be a finite dimensional subspace of H1 . Define the so-called Galerkin approximation uK : [0, T ) → HK by the requirement that (2.6) holds for all v K ∈ HK . Show that given t¯ ∈ [0, T ) there is a constant C(t¯) depending on t¯ and on A but independent of K, HK , ς0 and F such that Z t¯ K 2 sup |u(t) − u (t)|H0 + |u(t) − uK (t)|2H1 dt 0≤t≤t¯

0

h

≤C(t¯) sup |u(t) − ΠK u(t)|2H0 + 0≤t≤t¯

Z 0

t¯

 |u(t) − ΠK u(t)|2H1 dt

where ΠK : H0 → HK is the H0 -orthogonal projection onto HK . Thus modulo the constant C(t¯) and in the norm implied above, the Galerkin approximation uK is as accurate as the H0 -orthogonal projection ΠK u. Suggestion: first show that uK − ΠK u is orthogonal to HK in the inner Z t

product h·, ·iH0 +

A(·, ·). 0

2.5 While it is clear from the proof of Theorem 2.1 that each of the two summands on the left side of (2.4) is bounded by the expression on the right, it is less clear that their sum is so bounded. Prove this.

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2. SPECTRAL REPRESENTATIONS

57

2.6 Let H be a separable Hilbert space and I a nontrivial interval in R. Show that if u, v ∈ wAC(I; H) then hu, viH is absolutely continuous on I with derivative hu0 , viH + hu, v 0 iH . Conclude that |u|2H is absolutely continuous with derivative 2hu, u0 iH . 2.7 Let F ∈ wAC(I; X ∗ ) where I is a nontrivial interval and X is a separable Banach space. (a) Fix a specific representative F 0 and show that for almost all t ∈ I the difference quotients (F (t + h) − F (t))/h are weak-∗ convergent to F 0 (t) as h → 0. Suggestion: exploit the fact that almost every t is a Lebesgue point of |F 0 (·)|X ∗ . (b) Assume in addition that F (t) ∈ V for all t ∈ I where V is a weak-∗ sequentially closed subspace of X ∗ . Prove that F 0 (t) ∈ V for almost all t ∈ I. 2.8 Let H be a separable Hilbert space, I a nontrivial interval in R and u X∈ wAC(I; H). Show that given ε > 0 there is a δ > 0 such that |u(bi ) − u(ai )|H < ε for every finite collection of nonoverlapping i X intervals {[ai , bi ]} contained in I with |bi − ai | < δ. i

2.9 Prove the assertion (2.20) made in the first paragraph of the proof of Theorem 2.5. 2.10 Prove the assertion in Remark 3 following the statement of Theorem 2.7 in section 2.3. Suggestion: given F ∈ Vj construct the solution z to the equation Aε (t)(z, ·) + λhz, ·iH0 = F by iteration in Hj+2 . 2.11 Suppose that the results of Theorem 2.7 hold for a particular solution u of (2.3) with T = ∞ and with the left side of (2.28) bounded independently of t¯. Show that for j = 0, . . . , m − 1, u(j) → 0 strongly in H2m−2j−1 as t → ∞. Suggestion: bound the total variation of |u(j) |2H2m−2j−1 on [0, ∞). 2.12 Show that the rate of regularization |u(t)|H1 ≤ Ct−1/2 |ς0 |H0 proved in Theorem 2.12 for the case that g = 0 and ς0 ∈ H0 is optimal in the sense that the bound |u(t)|H1 ≤ Ct−p |ς0 |H0 cannot hold in general in the context of the theorem for any p < 1/2. Suggestion: consider the example in Remark 4 in section 2.6 with initial data ςˆ0 (ξ) = |ξ|−q for |ξ| ≥ 1 and for various q > 0; the constant C is of importance here. 2.13 Let H0 = L2 (Rn ), H1 = H 1 (Rn ), S ∈ L(H0 , H1 ) the operator defined by hSf, viH1 = hf, viH0 for f ∈ H0 and v ∈ H1 , as in (2.59), and E the spectral measure of S regarded as a symmetric operator in L(H0 ). Identify E(V ) for Borel sets V in terms of the Fourier transform, apply the result to compute the spectral integral of a bounded measurable function, and then compute the first integral on the right side of (2.66)

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58

2. DIFFERENTIAL EQUATIONS IN HILBERT SPACE

in terms of the standard heat kernel. Suggestion: Let ϕ(ξ) = (1 + |ξ|2 )−1 and show that for Borel sets V , E(V ) = F −1 MχV ◦ϕ F where F is Fourier transform, χV is the characteristic function of V and Mf v = f v for f ∈ L∞ and v ∈ L2 . 2.14 Let H0 and H1 be separable Hilbert spaces with H1 dense and continuously contained in H0 and let S ∈ L(H0 , H1 ) be the operator defined by hSf, viH1 = hf, viH0 for all f ∈ H0 and v ∈ H1 . Let E be the spectral measure of S as a symmetric operator in L(H0 ) and for s ≥ 0 define vector spaces Z ˜ Hs = {v ∈ H0 : µ−s dE(v,v) < ∞}. ˜ 0 = H0 and H ˜ 1 = H1 , by (2.63).) Show that H ˜ s is a Hilbert (Thus H Z space with inner product hu, viHs ≡ µ−s dE(u,v) . Now let u(t) be as ˜ all zero. Show that if ς0 ∈ H ˜ s with s ∈ [0, 1] in (2.66) with λ, g and B (s−1)/2 then |u(t)|H1 ≤ Ct |ς0 |Hs , and that if ς0 ∈ H0 then for all s ≥ 0 −s/2 |u(t)|H˜ s ≤ Ct |ς0 |H0 . 2.15 Suppose that u satisfies the spectral representation (2.66) with g and B both zero and with the hypotheses and notations of Theorem 2.13 in force. Assume that there is an interval (µ1 , µ2 ] with 0 < µ1 < µ2 < L such that E((µ1 , µ2 ])ς0 = 0. Let P1 = E((0, µ1 ]) and P2 = E((µ2 , L]). Show that if P2 ς0 6= 0 then the ratio |P1 u(t)|H0 /|u(t)|H0 → 0 as t → ∞. This shows that u is well-approximated by P2 u for large time. The requirement that E((µ1 , µ2 ])ς0 = 0 may be a special property of ς0 but also holds for all ς0 if there is a “spectral gap,” that is, an interval (µ1 , µ2 ] as above on which E is zero. For example, if H1 is compactly contained in H0 , then E is zero outside a countable set whose only limit point is zero; in this case there are infinitely many such spectral gaps.

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10.1090/surv/251/03

CHAPTER 3

Linear Parabolic Systems: Basic Theory In this chapter we apply the abstract theory in sections 2.1, 2.6 and 2.7 to parabolic systems of partial differential equations. Specifically we let Ω be an open set in Rn and we take H0 and H1 to be closed subspaces of [L2 (Ω)]N and [H 1 (Ω)]N respectively where H1 is dense in H0 and N ≥ 1. Elements of these spaces will be (column) vector functions of x ∈ Ω with differentiation carried out component-wise. We will consider the following specific case of the abstract problem (2.3): find u(t) for t ∈ [0, T ) ⊆ [0, ∞) such that for all v ∈ H1 and t ∈ [0, T ), Z t   (3.1) hu(t), viL2 (Ω) = hς0 , viL2 (Ω) + − A(s)(u(s), v) + F (s) · v ds 0

where ς0 ∈ H0 , Z (3.2) A(u, v) = Ω



 (aη,ω uxη ) · vxω + (bη uxη ) · v + (cω u)·vxω + (du) · v dx Z + (f u) · v dσ ∂Ω

(summation over repeated indices is understood) and Z Z
(3.3) F ·v = g · v + G : ∇x v + Γ · v dσ. Ω

∂Ω

Here aη,ω , bη , cω and d are N × N matrix–valued functions of (x, t) ∈ Ω × [0, T ), f is an N × N matrix–valued function of (x, t) ∈ ∂Ω × [0, T ), and the dot denotes the usual inner product in RN ; g and Γ are RN -valued, G is N × n matrix-valued, ∇x v is the N × n x-derivative matrix, and G : ∇x v is the scalar product Gij vxi j . Finally, σ denotes surface measure on ∂Ω (σ is constructed in section 3.2) and the boundary integrals involving f and Γ are omitted in the case that Ω = Rn . The basic existence result is given in Theorem 3.6 below and may be paraphrased as follows: if Ω is a uniformly Lipschitz domain (or satisfies the weaker cone condition if the boundary integrals are omitted), if the components of the coefficients of A and F are in Lploc ([0, T ); Ls (Ω)) or Lploc ([0, T ); Ls (∂Ω)) for certain p and s, and if the ensemble of matrices aη,ω satisfies a certain positive definiteness condition, then given ς0 ∈ H0 there is a unique solution u ∈ C([0, T ); H0 ) of (3.1) whose Lebesgue equivalence class on [0, T ) is in L2loc ([0, T ); H1 ) and which satisfies the energy equality (2.5) and the bound 59

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3. LINEAR PARABOLIC SYSTEMS: BASIC THEORY

(2.4) in term of ς0 and F . The precise statement and its proof are given in Theorem 3.6 in section 3.3 below and a parallel existence theory is given in section 3.4 for systems which are symmetric to leading order. Solutions of symmetric systems are somewhat more regular and can be represented in terms of an associated spectral measure if the aη,ω are independent of t. In section 3.1 we consider specific examples of such systems, paying particular attention to boundary conditions and the corresponding choices of subspaces H0 and H1 . In section 3.2 we review basic material concerning boundary measures and normal vector fields, including the divergence and trace theorems for Sobolev functions (complete proofs of which are given in section A.4 in the Appendix) and we discuss an important question of analysis concerning measurability and differentiability on Ω × [0, T ) of elements of L2loc ([0, T ); H 1 (Ω)).

§3.1

Linear Parabolic Systems and Their Weak Forms

In this section we consider specific examples of initial-boundary value problems for systems of partial differential equations whose weak forms are included in (3.1)–(3.3). Our perspective will be that the weak form is the problem of interest and that the differential equation is a symbolic equivalent which may be useful in some situations. Differential equations will therefore be treated in a purely formal way in this section; recovery of differential equations from their weak forms is considered in Theorem 5.5 in Chapter 5. Assume that Ω is a proper open subset of Rn with unit outer normal vector field ν and surface measure σ on ∂Ω and that the divergence theorem holds on Ω. We can then integrate by parts in (3.2) to obtain that Z Z A(u, v) = − (Lu) · v dx + (L∂ u) · v dσ Ω

∂Ω

where Lu = (aη,ω uxη )xω − bη uxη + (cω u)xω − du, L∂ u = ν ω (aη,ω uxη + cω u) + f u, Z F ·v =

Z (g − Div G) · v dx +

Ω

(Gν + Γ) · v dσ ∂Ω

and where Div G is the RN -valued function (Div G)i = Gij xj . The strong form (2.13) of (3.1), that is, hut , viH0 = −A(u, v) + F · v for each t, may then be written Z Z

(3.4) ut − Lu − g + Div G · v dx = − L∂ u + Gν + Γ · v dσ. Ω

∂Ω

Now consider specific examples in which u is a formal solution of (3.4) with initial value u(·, 0) = ς0 :

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3. LINEAR PARABOLIC SYSTEMS AND THEIR WEAK FORMS

61

Example 1. If H0 = [L2 (Ω)]N and H1 = [H01 (Ω)]N , then the boundary integral in (3.4) is zero hence so is the integral on the left, and we conclude formally that u is a solution of the system   ut = Lu + g − Div G, (x, t) ∈ Ω × (0, T ), u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ),  u(x, 0) = ς0 (x), x ∈ Ω. The “essential” Dirichlet boundary condition that u = 0 on ∂Ω is a consequence of the requirement that u(·, t) ∈ H1 . Note that a nonhomogeneous boundary condition u = ϕ on ∂Ω can be accommodated in this formalism by extending ϕ to a map on Ω and replacing the unknown u by u − ϕ. Example 2. If H0 = [L2 (Ω)]N and H1 = [H 1 (Ω)]N then we can still apply (3.4) with test functions v which are zero on ∂Ω to obtain the same differential equation as in Example 1. The left side of (3.4) is therefore zero for arbitrary v, so that the term in parentheses on the right side is zero on ∂Ω (provided that the restrictions to ∂Ω of elements v ∈ H1 are suitably dense). That is,   ut = Lu + g − Div G, (x, t) ∈ Ω × (0, T ), L u = Gν + Γ, (x, t) ∈ ∂Ω × [0, T ),  ∂ u(x, 0) = ς0 (x), x ∈ Ω. The “natural” boundary condition that L∂ u = Gν + Γ is thus derived as a consequence of the weak form (3.1) rather than as a requirement that u ∈ H1 . Notice that in the special case that N = 1 and L is the Laplace operator, aη,ω is the Kronecker delta and cω = 0 so that L∂ u = ν ω uxω = ∇x u · ν. The natural boundary condition for the Laplace operator is therefore the familiar ∂u Neumann condition ≡ ∇x u · ν = Gν + Γ on ∂Ω. ∂n Example 3. Examples 1 and 2 can be generalized to cases in which different components of u satisfy different boundary conditions and these may be coupled and may differ on different components of ∂Ω. Thus let {Cl } be the connected components of ∂Ω and for each l let Ql be given mapping Cl into the set of Nl × N matrices, where either 1 ≤ Nl ≤ N and Ql (x) has full rank at each x ∈ Cl or Nl = 0 and Ql is the N × N zero matrix on Cl . Let H0 = [L2 (Ω)]N and H1 = {v ∈ [H 1 (Ω)]N such that Ql v = 0 on Cl for all l}. As in Examples 1 and 2 we find that the quantity in the parentheses on the left side of (3.4) is zero, hence the integral on the right side is zero for arbitrary v ∈ H1 . Thus at each x ∈ Cl , L∂ u − Gν − Γ ⊥ v for all v satisfying Ql v = 0. Since the solution u itself is to be in H1 , we conclude formally that  ut = Lu + g − Div G, (x, t) ∈ Ω × (0, T ),    Ql u = 0, (x, t) ∈ Cl × [0, T ), L u − Gν − Γ ∈ (nullspace Ql )⊥ , (x, t) ∈ Cl × [0, T ),    ∂ u(x, 0) = ς0 (x), x ∈ Ω

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62

3. LINEAR PARABOLIC SYSTEMS: BASIC THEORY

where ⊥ denotes orthogonal complement in Rn . Observe that the second and third conditions, which are to hold for all l, comprise respectively Nl and N − Nl scalar conditions, as expected. Example 4. In this example we let n = N = 3 and define H1 = {v ∈ [H01 (Ω)]3 such that div v = 0 in Ω} where div is the divergence with respect to x, and take H0 to be the L2 closure of H1 , whose elements are thus divergence–free in the sense of weak derivatives. Proceeding as in the previous examples, we find that the integral on the left in (3.4) is zero for vector fields v which are divergence–free. Therefore if Π denotes the L2 projection onto H0 , then u formally satisfies 
u = Π Lu + g − Div G , (x, t) ∈ Ω × (0, T ),  t   div u = 0, (x, t) ∈ Ω × (0, T ), u = 0, (x, t) ∈ ∂Ω × [0, T ),    u(x, 0) = ς0 (x), x ∈ Ω. Example 5. In this example we reverse the logic and begin with an initialboundary value problem, now with a nonstandard boundary condition, and illustrate its weak formulation in the format above in (3.1)–(3.3). Again let n = N = 3 and consider the initial-boundary value problem

(3.5)

  ut = (µuxη )xη + ∇x (λ div u) − ∇x P + g, (x, t) ∈ Ω × (0, T ), u = −κ[(∇x u + ∇tr x u)ν]tan , (x, t) ∈ ∂Ω × [0, T ),  u(x, 0) = ς0 (x), x ∈ Ω

where tr denotes transpose and tan denotes projection orthogonal to the normal vector on ∂Ω. The differential equation here is a simplified, linearized model of Newton’s law applied to a compressible fluid in which u, P and g represent the fluid velocity, pressure and external force, and µ and λ are positive functions describing certain material properties of the fluid (see Batchelor [4], for example, for a detailed discussion of the rational mechanics of fluids). The boundary condition, which is the point of interest here, expresses a proposal of Navier [32] that on the boundary of the region containing the fluid the velocity u is proportional to the tangential component of the “stress.” The so-called stress tensor is the linear map taking the unit normal to a hypothetical hypersurface inside Ω, or to the boundary surface ∂Ω, into the force exerted by the fluid on one side of the surface on the fluid on the other, or on the boundary. In standard compressible fluid mechanics the stress tensor is proportional to the matrix 12 [(∇x u + ∇tr x u)]; this accounts for the second equation in (3.5), which implies in particular that u is to be tangential to the boundary. We therefore define H1 = {v ∈ [H 1 (Ω)]3 such that v · ν = 0 on ∂Ω}

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3. PRELIMINARIES

63

and H0 = [L2 (Ω)]3 . To derive the weak form we multiply the differential equation in (3.5) by v ∈ H1 and apply the divergence theorem; the result is that hut , viH0 = −A(u, v) + F · v where Z Z
A(u, v) = µ∇x u : ∇x v + λ divu div v dx − (µ∇x u ν) · v dσ. Ω

∂Ω

This appears to be inconsistent with (3.2), however, because the integrand in the boundary integral involves ∇x u, whose restriction to ∂Ω is not in general well-defined for u(·, t) ∈ [H 1 (Ω)]3 (see Theorem 3.2 below). We can reduce the order of u appearing here, however, by making use of the boundary condition in (3.5) as follows. If v ∈ H1 then v is tangential on the boundary, so that at points of ∂Ω, tr (∇x u ν) · v = [(∇x u + ∇tr x u)ν]tan · v − [∇x u ν]tan · v

= −κ−1 u · v − [∇tr x u ν]tan · v. The order of the second term on the right here can be reduced by ordinary calculus: Let ∂Ω be given locally by {x : h(x) = 0} where h : R3 → R and ν = ∇x h/|∇x h|, and let s → x(s) be a curve taking values locally on ∂Ω. Then differentiating the relation u(x(s), t) · ∇x h(x(s)) = 0 with respect to s 00 for fixed t we find that the tangential component of the vector ∇tr x u ∇x h+h u tr −1 00 is zero on ∂Ω and therefore that [∇x u ν]tan = −|∇x h| [h u]tan . Thus the boundary integral in the expression for
A above may be written exactly as in (3.2) with f = µ κ−1 I − |∇x h|−1 h00 . Systems of partial differential equations such as those considered in the above examples are loosely described as “parabolic” if the associated bilinear forms A satisfy the fourth condition in (2.1). More precisely, the pair of operators (L, L∂ ) is said to be locally uniformly parabolic on [0, T ) with respect to the spaces H0 and H1 if the four conditions in (2.1) are satisfied.

§3.2

Preliminaries

In this section we review various facts and notations concerning boundary measures, normal vectors and measurability on product spaces, preliminary to the application of Theorem 2.1 to the examples of section 3.1. Complete statements and proofs are given in sections A.4 and A.5 in the Appendix. Boundary Measures and Trace and Divergence Theorems The following theorem shows that if Ω is a Lipschitz domain then there is a Radon measure σ (see section A.1) on the Borel subsets of ∂Ω corresponding to surface area and there is a unit outer normal vector field ν defined σ-almost everywhere: Theorem 3.1. Let Ω be a Lipschitz domain in Rn as described in Definition 1.7 and let Ui , ψi and Ti be as in that definition.

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64

3. LINEAR PARABOLIC SYSTEMS: BASIC THEORY

(a) Define a cover {Wi }i of ∂Ω by W1 = U1 and Wi = Ui − ∪i−1 j=1 Wi for i ≥ 2; and for f ∈ Cc (∂Ω) define XZ p I(f ) = (f ◦ Ti )(y) 1 + |∇ψi (y)|2 dy. i

Ti−1 (∂Ω∩Wi )

Then there is a unique Radon measure σ on the Borel sets in ∂Ω such that Z (3.6) f dσ = I(f ) ∂Ω

for all f ∈ Cc (∂Ω). (b) Lp (∂Ω, σ) is separable for p ∈ [1, ∞). (c) If f : ∂Ω → R is Borel measurable and nonnegative and E ⊆ ∂Ω ∩ Ui is a Borel set, then Z Z p (3.7) f dσ = (f ◦ Ti )(y) 1 + |∇ψi (y)|2 dy E

Ti−1 (E)

(whose value could be +∞). (d) If f : ∂Ω → R is Borel measurable then f ∈ L1 (∂Ω) if and only if I(|f |) < ∞ (which is meaningful by (c)), in which case Z f dσ = I(f + ) − I(f − ) ∂Ω

where f + = f ∨ 0 and f − = f + − f . (e) There is a set E ⊂ ∂Ω of σ-measure zero and a measurable mapping ν from ∂Ω − E into the set of unit vectors in Rn such that for x0 ∈ ∂Ω − E, x0 + rν(x0 ) is in Ω for small negative r and in (Ω)c for small positive r; and if s → x(s) is a continuous mapping from a nontrivial interval into ∂Ω with x(0) = x0 and x differentiable at s = 0, then x(0) ˙ · ν(x0 ) = 0. The trace of an element f ∈ Cc (Rn ) with respect to a proper subset Ω of Rn is its restriction f ∂Ω to ∂Ω. The trace operator is thus a linear map from Cc (Rn ) to Cc (∂Ω). The following theorem shows that the trace operator extends to an operator T on W 1,p (Ω) with values in Lp (∂Ω): Theorem 3.2. Let Ω be a uniformly Lipschitz domain in Rn with parameters ε0 and L as described in Definition 1.7 and let p ∈ [1, ∞). Then there is an operator T ∈ L(W 1,p (Ω), Lp (∂Ω)) such that for f ∈ Cc (Rn ), T (f Ω ) = f ∂Ω . The norm of T is determined by n, L, ε0 and p. Finally, the divergence theorem holds for Sobolev functions on uniformly Lipschitz domains:

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3. PRELIMINARIES

65

Theorem 3.3. Let Ω be a uniformly Lipschitz domain in Rn , ν the unit normal vector field on ∂Ω defined in Theorem 3.1(e) and f ∈ W 1,1 (Ω). Then for j = 1, . . . , n, Z Z (3.8) fxj dx = (T f )ν j dσ. Ω

∂Ω

The Spaces Lp,q Next we consider an important issue concerning measurability on product spaces. The application of Theorem 2.1 to the problems discussed in section 3.1 will show that, under appropriate conditions, solutions are in the space C([0, T ); [L2 (Ω)]N ) with equivalence classes in a subspace of L2loc ([0, T ); [H 1 (Ω)]N ). It is natural to ask whether for such a solution u we can choose for each t a representative w(·, t) of u(t), which is therefore a measurable function on Ω, in such a way that w(·, ·) is measurable on the product space Ω × (0, T ). If so, then w would be amenable to analyses involving the Fubini and Tonelli theorems, and its weak differentiability with respect to x on the (n + 1)-dimensional domain Ω × (0, T ) could be considered. This question is somewhat subtle. An example due to Sierpinski (see [13] pg. 69 or [36] pg. 152) shows that for each t ∈ [0, 1] there is (or may be—see the final sentence in this paragraph) a countable set St ⊆ [0, 1] such that S ≡ {(x, t) ∈ [0, 1]2 : t ∈ [0, 1] and x ∈ St } is not measurable with respect to Lebesque measure on R2 . Its characteristic function is therefore not measurable on [0, 1]2 and no redefinition on a set of measure zero can make it so. On the other hand, if u(t) is the Lebesgue equivalence class on [0, 1] of the characteristic function of St , then u(t) = 0 in L2 ([0, 1]) for all t, and therefore u ∈ C([0, 1]; L2 ([0, 1]). The construction in the paragraph above, however, in which w is derived from u, could lead to a function w which is not in fact measurable on the product space, namely, the characteristic function of S. While rather striking and disconcerting, this example depends on the continuum hypothesis, and the continuum hypothesis is known to be undecidable. Of course, we could take w ≡ 0 in the above example, thus obtaining a representative of u which is measurable on the product space. We will show that such a choice is always possible for the functions we will encounter. Here we give a minimal discussion, sufficient for the statement of the basic existence result in the next section; a complete and self-contained exposition is given in section A.5. Fix a nontrivial interval I ⊂ R, an open set Ω in Rn and p ∈ [0, ∞] with H¨older conjugate p0 , andZlet fZ ∈ Lp (I; L2 (Ω)). Then given v ∈ L2 (Ω) and   0 ψ ∈ Lp (I), the integral f (t)v dx ψ(t)dt is well-defined, that is, is I

Ω

independent of representatives and is finite. Thus f defines a real-valued

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66

3. LINEAR PARABOLIC SYSTEMS: BASIC THEORY 0

bilinear function on the vector space L2 (Ω) × Lp (I) in a natural way. We will see that this same action is induced by an equivalence class fm whose members are measurable on the product space Ω × I. We begin with the following definition: Definition 3.4. Let I, Ω and p be as above. Then Lp,2 (Ω × I) is the set of equivalence classes [ϕ]Ω×I of functions ϕ on Ω × I which are measurable with respect to the (Lebesgue) product measure on Ω × I and for which the Lp norm on I of |ϕ(·, t)|L2 (Ω) , denoted by |ϕ|Lp,2 (Ω×I) , is finite. The vector space Lp,2 (t)loc (Ω × I) is the set of equivalence classes of functions which are measurable on Ω × I and whose restrictions to Ω × I 0 are in Lp,2 (Ω × I 0 ) for every compact interval I 0 ⊆ I. The cross product spaces [Lp,2 (Ω × I)]N and N are then defined in the obvious ways. [Lp,2 (t)loc (Ω × I)] Notice that Tonelli’s theorem underlies the definition: if ϕ is measurable on the product space, then |ϕ(·, t)|L2 (Ω) defines a measurable class on I. Moreover, the Fubini and Tonelli theorems show Zthat,Z if ϕ ∈ Lp,2 (Ω×I), then   0 given v ∈ L2 (Ω) and ψ ∈ Lp (I), the integral ϕ(x, t)v(x)dx ψ(t)dt I

Ω

is well-defined and is equal to the integral over Ω × I with respect to the product measure. The following results are extracted from the more extensive discussion in section A.5. We note first that if f ∈ Lp (I; H 1 (Ω)) and if ϕ is any representative of f , then for almost all t each weak derivative ϕxi (t) exists in the weak sense on Ω, thereby generating a weakly measurable map from I into L2 (Ω). Its equivalence class over I, which we denote by fxj , is then an element of Lp (I; L2 (Ω)). Theorem 3.5. Let Ω be an open set in Rn and I a nontrivial interval in R and let p ∈ [0, ∞] with H¨ older conjugate p0 . (a) There is a one-to-one correspondence f ←→ fm between elements f ∈ Lp (I; L2 (Ω)) and fm ∈ Lp,2 (Ω × I) such that for all v ∈ L2 (Ω) and 0 ψ ∈ Lp (I), Z Z Z Z   f (t)v ψ(t)dt = fm (x, t)v(x)dx ψ(t)dt. I

Ω

I

Ω

(b) If f ∈ Lp (I; L2 (Ω)) and ψ ∈ Cc∞ (Ω × I ◦ ) (I ◦ is the interior of I) then Z Z Z Z   f (t)ψ(·, t) dt = fm (x, t)ψ(x, t)dx dt. I

Ω

I

Ω

(c) If f ∈ Lp (I; H 1 (Ω)) then fm has partial derivatives Dxj fm in the weak sense on Ω × I ◦ and Dxj fm = (fxj )m . (d) If f ∈ L∞ (I; L∞ (Ω)) and v ∈ L∞ (I; L2 (Ω)) then f v ∈ L∞ (I; L2 (Ω)) and (f v)m = fm vm .

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3. THE BASIC EXISTENCE THEOREM

67

The first three of these results are special cases of Corollaries A.16 and A.17 and Theorem A.19 in the Appendix and the last is a special case of Lemma 5.6. We remark that if f ∈ C(I; L2 (Ω)) then its equivalence class [f ]I is 2 in L∞ (t)loc (I; L (Ω)) and therefore its measurable counterpart is included in Definition 3.4. There will be no ambiguity in writing fm in place of the more complicated ([f ]I )m .

§3.3

The Basic Existence Theorem

In this section we apply Theorem 2.1(b) to obtain a basic result on the existence and uniqueness of solutions to the problems described in (3.1)–(3.3). We show that under fairly mild conditons on Ω, A, F and ς0 , solutions exist, are unique and satisfy the energy equality (2.5) and the bound (2.4). (The corresponding application of Theorem 2.1(a) is straightforward but more technical and will play no role in the subsequent development.) Theorem 3.6. Let T ∈ (0, ∞] and assume the following: (a) Ω is an open set in Rn , n ≥ 2, not necessarily bounded, satisfying one of the following: either (I) Ω is a uniformly Lipschitz domain strictly contained in Rn , or (II) Ω satisfies the cone condition (this includes the case Ω = Rn ). (b) H0 and H1 are closed subspaces of [L2 (Ω)]N and [H 1 (Ω)]N , N ≥ 1, respectively, with the inherited norms and with H1 dense in H0 ; Z (c) A is as defined in (3.2) (the term (f u) · v dσ is omitted in case (II)) ∂Ω

with coefficient matrices satisfying the following: ∞ • the components of aη,ω are elements of L∞ loc ([0, T ); L (Ω)) • the components of bη and cω are finite sums in which each summand s is an element of L∞ loc ([0, T ); L (Ω)) for some s ∈ (n, ∞] • the components of d are finite sums in which each summand is in r L∞ loc ([0, T ); L (Ω)) for some r ∈ (n/2, ∞] • in case (I) the components of f are finite sums in which each sums mand is in L∞ loc ([0, T ); L (∂Ω)) for some s ∈ (n, ∞] (in each of the three items above the values of s and r may depend on the component and the summand in question) • given t¯ ∈ [0, T ) there is a positive number θ(t¯) such that for almost all (x, t) ∈ Ω × (0, t¯) ωη (M tr aηω ≥ 2θ(t¯)|M |2 m (x, t)M ) for all N × n matrices M ; here tr denotes transpose, summation over η and ω is understood, any choice of matrix norm is allowed and aηω m is any member of the equivalence class of matrix-valued

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68

3. LINEAR PARABOLIC SYSTEMS: BASIC THEORY

functions on Ω × [0, T ) whose elements correspond to those of aη,ω as in Theorem 3.5. Then the following hold: (d) The conditions in (2.1) are satisfied with A ∈ L∞ loc ([0, T ); B(H1 × H1 )). In particular, for t¯ ∈ [0, T ) and almost all t ∈ [0, t¯], kA(t)kB(H1 ×H1 ) ≤ C(t¯) and A(t)(v, v) ≥ θ(t¯)|v|2H 1 − λ(t¯)|v|L2

(3.9)

¯ for all v ∈ H1 , where λ ∈ L∞ loc ([0, T )) and |λ|L∞ ([0,t¯]) and C(t) depend on the cone parameters δ and r in Definition 1.6(b) and on upper bounds for θ(t¯)−1 , for the norms of the coefficients of A on [0, t¯] and the numbers of summands described in (c), and for (s − n)−1 and (r − n/2)−1 for all values of s and r occurring in (c). (e) Given • g as in (3.3) with components which are finite sums of elements in L2loc ([0, T ); Lq (Ω)) for some q ∈ [2n/(n + 2), 2] for n ≥ 3 and in (1, 2] for n = 2 • G as in (3.3) with components in L2loc ([0, T ); L2 (Ω)) • (in case (I)) Γ as in (3.3) with components in L2loc ([0, T ); L2 (∂Ω)) • ς0 ∈ H0 , there is a unique u ∈ C([0, T ); H0 ) whose [0, T )-equivalence class is in L2loc ([0, T ); H1 ) and which satisfies the conclusions of Theorem 2.1: • u satisfies (3.1) for all v ∈ H1 • the energy equality Z t 2 1 A(s)(u(s), u(s))ds 2 |u(t)|[L2 (Ω)]N + 0 (3.10) Z tZ Z tZ   = 21 |ς0 |2[L2 (Ω)]N + g · u + G : ∇x u dx + 0

Ω

0

Γ · u dσ

∂Ω

holds for all t ∈ [0, T ) (integrals over ∂Ω are omitted in case (II)) • the bound Z t¯ R t¯ sup |u(t)|2[L2 (Ω)]N + θ(t¯) e t 2λ(s)ds |u(t)|2[H 1 (Ω)]N dt (3.11)

0≤t≤t¯

0

Z t¯ R t¯ R t¯ 2λ(t)dt 2 −1 ≤ e 0 |ς0 |[L2 (Ω)]N + θ(t¯) e t 2λ(s)ds CF (t)dt 0

holds for t ∈ [0, T ); here CF (t) = C1 Cg (t) + CG (t) + C2 CΓ (t)

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10.1090/surv/251/04

3. THE BASIC EXISTENCE THEOREM

69

where Cg (t) is the sum of the squares of the Lq (Ω) norms in the first item in (e), C1 depends on n and in the case that n = 2 on an upper bound for (q − 2)−1 for all such q, CG (t) and CΓ (t) are the sums of the squares of the L2 (Ω) and L2 (∂Ω) norms of the components of G(t) and Γ(t), and C2 depends on an upper bound for the norm of the trace operator which maps H 1 (Ω) to L2 (∂Ω) (the term C2 CΓ is omitted in case (II))  N • um ∈ L∞,2 and the weak derivatives Dxη um exist on (t)loc (Ω × [0, T ))  2,2 N Ω × (0, T ) and are in L(t)loc (Ω × [0, T )) ; and if [Cc∞ (Ω)]N ⊂ H1 then the differential equation

∂ ∂ ∂ η um = aη,ω cωm um m Dxη um − bm Dxη um + ∂t ∂xω ∂xω −dm um + gm − Div Gm

(3.12)

holds in the sense of distributions on Ω × (0, T ), where each term here is either locally integrable on Ω × (0, T ) or is a distribution ∂ ∂ derivative , or Div of a locally integrable function. ∂t ∂xω Remarks: (1) The conclusion in (d) that λ ∈ L∞ loc ([0, T )) is stronger than the minimal 1 requirement in (2.1) that λ ∈ Lloc ([0, T )). This results from integrability hypotheses in (c) which are slightly stronger than is strictly necessary. Small improvements can be made at considerable technical expense involving more complicated index arithmetic, as will be evident from the proof. (2) For N = 1 the coefficients aηω are scalar-valued and the last condition in (c) reduces to the requirement that, for almost all (x, t) ∈ Ω × (0, t¯), 2 n η ω aη,ω m (x, t)ξ ξ ≥ 2θ(t¯)|ξ| for all ξ ∈ R . (3) See also Remarks (1), (2) and (4) following the statement of Theorem 2.1 for further comments related to the material of this section. PROOF of Theorem 3.6. To prove (d) we need to check that the following hold for z, w ∈ H1 : (i) the mapping t → A(t)(z, w) is measurable; (ii) |A(t)(z, w)| ≤ C(t¯)|z|H1 |w|H1 for t¯ ∈ (0, T ); and (iii) for t ∈ [0, t¯], A(t)(z, z) ≥ θ(t¯)|z|2H1 − λ(t¯)|z|2H0 where C(t¯) and λ(t¯) are as described in (d). Consider first the contribution to A(z, w) from (aη,ω zxη ) · wxω . (We will suppress here the distinction between A and any specific weakly measurable representative on [0, T ).) Since |zxη ||wxω | ∈ L1 (Ω) and |aη,ω | ∈

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70

3. LINEAR PARABOLIC SYSTEMS: BASIC THEORY

∞ L∞ loc ([0, T ); L (Ω)), the mapping t →

Z


aη,ω (·, t)zxη · wxω dx is a measur-

Ω

able function of t which is bounded in absolute value as required for (ii). Concerning (iii) we record that by the last hypothesis in (c), Z Z
 ω,η η,ω a (·, t)zxη · zxω dx = (∇x z)tr aη,ω (·, t)∇x z Ω Ω
2 ≥ 2θ(t¯)|∇x z|L2 = 2θ(t¯) |z|2H1 − |z|2H0 . Next, abusing notation slightly, we denote by bzx w any one of the summands in the expansion of the term (bω zxη ) · w in the expression (3.2) defining A(z, w). Then for almost all t, b(·, t) ∈ Ls (Ω) for a particular s ∈ (n, ∞] so that the H¨older conjugate s0 of s is in [1, n/(n − 1)). Then γ ≡ 2s0 /(2 − s0 ) satisfies 1/γ + 1/2 = 1/s0 , and γ is in [2, p∗ ) where p∗ = 2n/(n − 2) for n ≥ 3 and p∗ ∈ [2, ∞) is sufficiently large depending on s for n = 2. It follows that H 1 (Ω) is continuously contained in Lp (Ω) for p ∈ [2, p∗ ] (see Theorem A.3; the cone condition is applied here) and therefore |zx w|Ls0 ≤ C|zx |L2 |w|Lγ ≤ C|z|H 1 |w|H 1 . Z s (Ω)), the mapping t → Since b ∈ L∞ ([0, T ); L bzx w dx is measurable in t loc (3.13)

Ω

with image bounded in absolute value by C|b(·, t)|Ls (Ω) |w|H 1 |z|H 1 for almost all t, and is therefore in L∞ loc ([0, T )) as required for (i) and (ii). For (iii) we ∗ recall that γ ∈ [2, p ), so that, applying (3.13) with w = z and interpolating the Lγ norm between and 2 and p∗ we obtain that for some τ ∈ (0, 1] and arbitrary δ > 0, Z bz z dx ≤ C|b|Ls |zx |L2 |z|τL2 |z|1−τ x Lp∗ Ω

≤ C|b|Ls |z|2−τ |z|τL2 H1 ≤ δ|z|2H 1 + C(δ)|z|2L2 . Bounds for the terms (cω z) · wxω and (dz) · w in (3.2) are obtained in a similar way. For the boundary integral in case (I) we denote by f wz any one of s the summands in the expression for (f w) · z. Then f ∈ L∞ loc ([0, T ); L (∂Ω)) 0 where s and its conjugate s are in the same ranges as in (3.13), and therefore by Theorem 3.2, |zw|Ls0 (∂Ω) ≤ C|zw|W 1,s0 (Ω) ≤ C(|zw|Ls0 (Ω) + |zwx |Ls0 (Ω) + |zx w|Ls0 (Ω) ). The argument is now the same as for the term (bzxη ) · w, proceeding from (3.13) above. These observations prove that A ∈ L∞ loc ([0, T ); B(H1 × H1 )) with the ¯ required bound and that, given t ∈ (0, T ) and δ > 0, there is a constant C depending on t¯ and δ and on the same quantities listed in (d) for λ such that

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3. THE BASIC EXISTENCE THEOREM

71

for almost all t ∈ [0, t¯] and for all z ∈ [H 1 (Ω)]N ⊇ H1 , A(t)(z, z) ≥ 2θ(t¯)|z|2H 1 (Ω) − δ|z|2H 1 (Ω) − C(δ)|z|2L2 (Ω) . Thus (iii) holds for appropriate choices of constants δ and λ(t¯). This completes the proof of (d). To prove (e) we have to check that the mapping F defined in (3.3) is in L2loc ([0, T ); H1∗ ). The required measurability and boundedness are obvious for the contributions from the terms involving G and Γ (in case (I)). For the term involving a particular summand of a particular component of g, still denoted g ∈ L2loc ([0, T ); Lq (Ω)), we have that, by the hypothesis on q, the H¨older conjugate q 0 of q is in the range [2, p∗ ], where p∗ is as above preceding Z (3.13). Therefore for v ∈ H 1 , g(·, t) · v dx ≤ C|g(·, t)|Lq (Ω) |v|H 1 (Ω) . The Ω

contribution of the term involving g to |F (t)|2H∗ is therefore bounded by 1 C1 Cg (t), which is in L1loc ([0, T )), as required. Theorem 2.1(b) now applies to show that (3.1)–(3.3) has a unique solution u as described in (e) and which satisfies (3.10) and (3.11). The statements  N  N that um ∈ L∞,2 and Dxj um ∈ L2,2 then follow from Theorem 3.5(c). (t)loc (t)loc ∞ To prove (3.12) we fix ϕ ∈ Cc (Ω × (0, T )) so that by Theorem 3.5(b), Z

−1

Z TZ u(t)[ϕ(·, t) − ϕ(·, t − h)]dx dt.

um ϕt = lim h h→0

Ω×(0,T )

0

Ω

Moving the difference quotient onto u and applying (3.2) and the fact that ϕ(·, t) ∈ H1 , we then obtain Z um ϕt = lim h Ω×(0,T )

−1

h→0

Z TZ

t+h h

A(s)(u(s), ϕ(·, t))

0

t

Z −


i g(s) · ϕ(·, t) + G : ∇ϕ(·, t) dx ds dt.

Ω

For fixed h we can reverse the s and t integrals above (to justify this approximate ϕ by a function which is piecewise constant in t and which when substituted into the brackets on the right yields a measurable function of (s, t)) then take the limit as h → 0 to obtain Z TZ

Z um ϕt = Ω×(0,T ) Z =

0



 (aη,ω uxη ) · ϕxω + . . . − g · ϕ − G : ∇ϕ dxds

Ω

 η,ω
 (am umxη ) · ϕxω + . . . + − gm + Div Gm · ϕ ,

Ω×(0,T )

the last step here being justified by Theorem 3.5(d). 

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72

§3.4

3. LINEAR PARABOLIC SYSTEMS: BASIC THEORY

Parabolic Systems with Symmetry

In this section we consider parabolic systems satisfying the symmetry conditions of Theorems 2.11 and 2.12. Specifically, we examine solutions of the problem (3.1)–(3.3) with the terms involving cj , G and Γ absent, with d = d1 + d2 where d1 and f are symmetric, and under the assumption that (aη,ω )tr = a(ω,η) for all η and ω. The problem is therefore to find u ∈ C([0, T ); H0weak ) with equivalence class in L2 ([0, T ); H1 ) such that hu(t), viL2 (Ω) = hς0 , viL2 (Ω) Z t (3.14)   + − Asym (s)(u(s), v) − B(s)(u(s), v) + hg(s), viL2 (Ω) ds 0

for all v ∈ H1 , where Z (3.15)

Asym (z, w) =



η,ω

(a



Z

zxη ) · wxω + (d1 z) · w dx +

Ω

(f z) · w dσ ∂Ω

and Z (3.16)

B(z, w) =

 η  (b zxη ) · w + (d2 z) · w dx

Ω

aη,ω ,

bη ,

for z, w ∈ H1 . Again, d1 and d2 are N × N matrix-valued and g is an RN -valued function of (x, t) ∈ Ω×[0, T ); and f is an N ×N matrix-valued function of (x, t) ∈ ∂Ω × [0, T ) (the term involving f is omitted if Ω = Rn ). The following theorem gives the application of Theorems 2.11 and 2.12 to the above problem, part (a) for initial data ς0 ∈ H1 and part (b) for ς0 ∈ H0 : Theorem 3.7. Let T, Ω, H0 and H1 satisfy hypotheses (a) and (b) of Theorem 3.6 and let A = Asym + B be as above in (3.15)–(3.16) with coefficients satisfying the following: • The components of each aη,ω are in wAC 2 ([0, T ); L∞ (Ω)), those of d1 are finite sums of elements of wAC 2 ([0, T ); Lr1 (Ω)) for values of r1 in (n/2, ∞], and in case (I) of Theorem 3.6 those of f are finite sums of elements of wAC 2 ([0, T ); Ls (∂Ω)) for values of s in (n, ∞]. • The components of each bη are in L2loc ([0, T ); L∞ (Ω)) and those of d2 are finite sums in which each summand is in L2loc ([0, T ); Lr2 (Ω)) for some r2 ∈ [n, ∞] if n ≥ 3 and in (2, ∞] if n = 2. • For each t and for all η and ω, (aη,ω )tr = a(ω,η) a.e. in Ω, and at each t, d1 is symmetric a.e. in Ω and in case (I) of Theorem 3.6 f is symmetric a.e. in ∂Ω. • The matrix-valued functions aη,ω m satisfy the coerciveness condition in the last item in Theorem 3.6(c).

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3. PARABOLIC SYSTEMS WITH SYMMETRY

73

(Different values of r1 , r2 and s are allowed for different components and summands of d1 , d2 and f .) (a) Then given ς0 ∈ H1 and g ∈ L2loc ([0, T ); H0 ), 2 ([0, T ); H ) ∩ C([0, T ); Hweak ) which • there is a unique u ∈ wACloc 0 1 satisfies (3.14) for all v ∈ H1 and t ∈ [0, T );

• and given specific representatives t 7→ u0 (t), A(t) and (in case (I)) f (t) into H0 , B(H1 × H1 ) and L2 (∂Ω) respectively, there is a set E ⊂ [0, T ) of measure zero such that for t ∈ / E and for all v ∈ H1 , Z Z  η,ω u0 (t) · v dx = − (a uxη ) · vxω +(bη uxη ) · v Ω Ω  + ((d1 + d2 )u) · v dx (3.17) Z Z − (f u) · v dσ + g · v dx; ∂Ω

Ω

• the energy equality Z tZ  η,ω 2 1 (a uxη ) · uxω +(bη uxη ) · u 2 |u(t)|L2 (Ω) + 0

(3.18)

Ω

 +((d1 + d2 )u) · u dx + Z tZ 2 1 = 2 |ς0 |L2 (Ω) + g · u dx 0

Z tZ (f u) · u dσ 0

∂Ω

Ω

holds for all t ∈ [0, T ); • and if t¯ ∈ (0, T ) then there is a constant C(t¯) depending on t¯, n, H0 , H1 , on the values of r1 , r2 and s and the numbers of summands referred to in the hypotheses and on the norms on [0, t¯] of the coefficients aη,ω , bη , d1 , d2 and f in the hypotheses and on θ(t¯), such that Z t¯ 2 sup |u(t)|H1 + |u0 (t)|2H0 dt (3.19)

0≤t≤t¯

0

Z t¯ h i ≤ C(t¯) |ς0 |2H1 + |g(t)|2L2 (Ω) dt . 0

(b) In place of the hypotheses of (a) assume that g is as in (3.14) with components which are finite sums of elements in L2loc ([0, T ); Lq (Ω)) for values of q ∈ [2n/(n + 2), 2] for n ≥ 3 and in (1, 2] for n = 2 and with tδ/2 g(t) in L2loc ([0, T ); L2loc (Ω)) for some δ ≥ 1, and that ς0 ∈ H0 . Then • there is a unique u ∈ C([0, T );H0 ) ∩ wAC 2 ((0, T ); H0 ) ∩ C((0, T ); H1weak )

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74

3. LINEAR PARABOLIC SYSTEMS: BASIC THEORY

with equivalence class in L2loc ([0, T ); H1 ) which satisfies (3.14) for all v ∈ H1 and t ∈ [0, T ); • the strong form (3.17) of (3.14) holds for almost all t and the energy equality (3.18) holds for all t ∈ [0, T ); • given t¯ ∈ [0, T ) there is a constant C(t¯) depending on the same quantities as in (3.19) and on δ and the values of q occurring in the hypotheses above such that Z t¯ δ 2 sup t |u(t)|H1 + tδ |u0 (t)|2H0 dt 0 0, and if the components of g are in L2 ([0, ∞); L2 (Ω)), then |u(·, t) − u(t)|L2 (Ω) → 0 as t → ∞. Thus the solution is asymptotially spatially homogeneous with its average coalescing with an orbit of an associated ordinary differential equation.

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10.1090/surv/251/04

CHAPTER 4

Elliptic Systems: Higher Order Regularity In this chapter we prove two regularity results for weak solutions of elliptic systems of second order. The first is independent of boundary conditions and shows that solutions have weak derivatives which are locally bounded in L2 of the spatial domain (interior regularity). The second applies to a broad class of boundary-value problems in domains with smooth boundaries and shows that these derivatives are in L2 of the entire domain (global regularity). In both cases the underlying partial differential equations hold in the sense of weak derivatives, and if the order of weak differentiability is sufficiently high, then in the sense of continuous derivatives as well. These local and global results are stated in Theorems 4.1 and 4.2 respectively and are proved in subsequent sections. Both are fundamental in the theory of partial differential equations and Theorem 4.2 will be an essential ingredient in the regularity theory for parabolic systems in Chapters 5 and 6. Specifically, we again let Ω be an open set in Rn , n ≥ 2, not necessarily bounded, and H1 a closed subspace of H 1 (Ω)N , N ≥ 1, and consider the bilinear form A ∈ B(H1 ×H1 ) of Chapter 3, now with coefficients independent of t: Z  η,ω A(u, v) = (a uxη ) · vxω + (bη uxη ) · v Ω Z (4.1)  ω + (c u) · vxω + (du) · v dx + (f u) · v dσ ∂Ω

where aη,ω , bη , cω and d are N × N –matrix valued functions of x ∈ Ω, f is an N × N –matrix valued function of x ∈ ∂Ω and the dot denotes the usual inner product in RN . We impose fairly general conditions on these coefficient functions and consider the regularity of an element u ∈ H1 which is assumed to satisfy A(u, v) = F · v

(4.2)

for all v ∈ H1 , where F ∈ H1∗ is given by Z Z
(4.3) F ·v = g · v + G : ∇x v + Ω

Γ · v dσ.

∂Ω

Here g and Γ are RN -valued, G is N × n–matrix-valued, ∇x v is the N × n x–derivative matrix and G : ∇x v is the scalar product Gij vxi j . (Boundary integrals are omitted in both (4.1) and (4.3) if Ω = Rn .) 77

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78

4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

In Theorem 4.1 we consider the case that (4.2) holds for all v ∈ [H01 (Ω)]N , so that the boundary integrals are absent and no regularity conditions on ∂Ω are imposed. The components of u will then be in H k+2 (Ω0 ) for all open subsets Ω0 of Ω a positive distance from ∂Ω provided that those of g are in H k (Ω) and those of G are in H k+1 (Ω) (in the special case that Ω = Rn , u will be in H k+2 (Rn )). In Theorem 4.2 we strengthen the hypotheses by requiring that (4.2) holds for all v ∈ H1 , which will be the subspace of [H 1 (Ω)]N discussed in Example 3 of section 3.1 corresponding to a variety of possible essential, natural and mixed boundary conditions, and requiring further that Ω and Γ satisfy certain C k+2 and H k+1 regularity conditions respectively. The existence and uniqueness of a “solution” u of (4.2) can be proved when A is coercive in the sense that A(u, u) ≥ C|u|2H1 for a positive constant C and for all u ∈ H1 . However, by allowing fairly general lower order terms in (4.1) we preclude coerciveness in general and therefore can make no statement about existence or uniqueness. In fact, solutions may fail to exist for some g, G and Γ, and there may be multiple solutions for others. See the discussion in [12], pg. 321, for example. In particular there may be nonzero u for which (4.2) holds with g, G and Γ all zero. Consequently, in the generality considered here, solutions cannot be bounded in a given Sobolev norm, say, solely by a sum of bounds for g, G, and Γ.

§4.1

Statements of Results

Throughout this chapter Ω will be an open set in Rn , n ≥ 2. For k+2 the global regularity result of Theorem 4.2, Ω will be a Cbdd domain, not necessarily bounded (Definition 1.7(c)) and the integer k ≥ 0 will be fixed throughout. Hypotheses on Coefficients First we make measurability and positivity conditions similar to those in Theorem 3.6: (4.4) The components of the N × N matrices aη,ω , bη , cω and d are Lebesgue equivalence classes of measurable functions on Ω, those of f are σ-equivalence classes of measurable functions on ∂Ω (σ is the boundary measure described in Theorem 3.1; this condition is omitted if Ω = Rn ), and there is a positive number θ such that for any given representatives aη,ω and for a.a. x ∈ Ω, (M tr aη,ω (x)M )ω,η ≥ 2θ|M |2 for all N × n matrices M . Here tr denotes transpose, summation over η and ω is understood and any matrix norm is allowed.

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4. STATEMENTS OF RESULTS

79

Before proceeding we need to introduce a measure of differentiability for functions defined on the boundary of Ω. Let Bi and Ti be as in Definition 1.7 and for ϕ : ∂Ω → R, q ∈ [1, ∞] and a multi-index α with |α| ≤ k + 2 define  1/q  P |Dyα˜ (ϕ ◦ Ti )|qLq (Bi ) , q ∈ [1, ∞), α i (4.5) hϕiLq , ∂Ω ≡ sup |Dα (ϕ ◦ T )| ∞ , q = ∞, i L (Bi ) i y˜ provided that the derivatives Dyα˜ (ϕ ◦ Ti ) exist in the weak sense on each Bi . The following conditions on the coefficients of A in (4.1) will suffice for most applications but are somewhat stronger than necessary:  the components of each aη,ω and each cω are in W k+1,∞ (Ω)    the components of each bη and d are in W k,∞ (Ω) (4.6) α  hf iL∞ , ∂Ω < ∞ for all multi-indices α with |α| ≤ k + 1   (for Theorem 4.2 only). Weaker but more technical conditions can be given in terms of index sets Il ⊆ [2, ∞] defined as follows for nonnegative integers l:   [n/l, ∞], 0 ≤ l < n/2, (4.7) Il = (2, ∞], l = n/2,   [2, ∞], l > n/2. Observe that Il increases with l, that ∞ ∈ Il for every l and that I0 = {∞}. (The dependence of Il on n is suppressed.) We can now describe weaker sufficient conditions on coefficients and in doing so we will abuse notation somewhat and make the simple statement “Dα d ∈ LI|α|+1 ”, for example, in place of the precise but more complicated statement “each component dµν of the matrix-valued function d has a weak derivative Dα dµν which is a finite P µν µν sµν sum i dα,i where each dα,i is in L α,i (Ω) for some sµν α,i ∈ I|α|+1 .” The most general hypotheses on the coefficients of A that the theory accommodates are then as follows:

(4.8)

 α D a ∈ LI(|α|−1)∨0 and Dα c ∈ LI|α| for 0 ≤ |α| ≤ k + 1    Dα b ∈ LI|α| and Dα d ∈ LI|α|+1 for 0 ≤ |α| ≤ k  hf iαLsα , ∂Ω < ∞ for 0 ≤ |α| ≤ k + 1 and some sα ∈ I|α|    (for Theorem 4.2 only).

Observe that the conditions in (4.6) imply those in (4.8) because ∞ ∈ Il for every l. We denote by CA an upper bound for θ−1 and for the norms, seminorms and numbers of summands in the descriptions above in (4.8). If n is even and if any summand of any derivative of any component of any of the coefficients in (4.8) is required to be in Ls (Ω) or Ls (Bi ) for some s in In/2 , then CA must also be an upper bound for (s − 2)−1 .

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4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

Observe that (4.4) and (4.8) are stronger than the hypotheses on the coefficients of A in Theorem 3.6. Consequently A ∈ B(H1 × H1 ) and as in (3.9), A(v, v) ≥ θ|v|2[H 1 (Ω)]N − C|v|2[L2 (Ω)]N

(4.9)

for all v ∈ [H 1 (Ω)]N , where the positive constant C depends on upper bounds for θ−1 and CA . We can now state a local regularity result in which boundary considerations enter neither the hypotheses nor the conclusions and solutions are shown to be in [H k+2 ]N on interior sets: Theorem 4.1. Let k be a nonnegative integer and assume the following: • Ω is an open set in Rn , n ≥ 2, not necessarily bounded; • A is the bilinear form defined in (4.1) with the boundary integral omitted and with coefficients aη,ω , bη , cω , and d satisfying the positive definiteness condition in (4.4) and either of the sets of regularity conditions (4.6) or (4.8) with references to f omitted and with the constant CA as defined above. Then the following hold: (a) If Ω = Rn : there is a constant C depending on n, N and on upper bounds for k and CA such that if u ∈ [H 1 (Rn )]N satisfies Z
A(u, v) = g · v + G : ∇x v dx Rn

for all v ∈ [Cc∞ (Rn )]N where the components of g are in H k (Rn ) and those of G are in H k+1 (Rn ), then u ∈ H k+2 (Rn ) and
|u|[H k+2 (Rn )]N ≤ C |u|[L2 (Rn )]N + |g|[H k (Rn )]N + |G|[H k+1 (Rn )]N ×n . (b) If Ω is a proper open subset of Rn : given a proper open subset Ω0 of Ω with γ ≡ dist(Ω0 , ∂Ω) > 0, there is a constant C depending on n, N and on upper bounds for k, CA and γ −1 such that if u ∈ [H 1 (Ω)]N satisfies Z
A(u, v) = g · v + G : ∇x v dx Ω

for all v ∈ of g are in and

[Cc∞ (Ω)]N , H k (Ω) and

hence for all v ∈ [H01 (Ω)]N , where the components those of G are in H k+1 (Ω), then u ∈ [H k+2 (Ω0 )]N


|u|[H k+2 (Ω0 )]N ≤ C |u|[L2 (Ω)]N + |g|[H k (Ω)]N + |G|[H k+1 (Ω)]N ×n .

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4. STATEMENTS OF RESULTS

81

(c) If Ω and Ω0 are as in (b), then the components of aη,ω uxη and cω u are in H k+1 (Ω0 ) ⊆ H 1 (Ω0 ) and those of bη uxη and du are in H k (Ω0 ) ⊆ L2 (Ω0 ), and the equation (4.10)

−(aη,ω uxη )xω + bη uxη − (cω u)xω + du = g − Div G,

in which subscripts denote weak derivatives in Ω and Div G is the vector N k n field whose ith component is Gij xj , holds in [H (Ω0 )] . If Ω = R as in n (a) then these same statements hold with Ω0 replaced by R . (d ) Let k0 = k0 (n) be the positive integer defined by (4.11)

k0 − 1 ≤ n/2 < k0

and let λ = k0 − n/2 if k0 − n/2 < 1 and choose λ arbitrarily in (0, 1) if k0 − n/2 = 1. Assume that k = k0 + l where l is a nonnegative integer. Then in either of the cases (a) or (b) there are specific representatives aη,ω , bη , cω , d, g, G and u such that, for every ball B whose closure l,λ is contained in Ω the components of bη , d and g are in Cbdd (B) (see l+1,λ η,ω ω section 1.2), those of a , c and G are in Cbdd (B), and those of u are l+2,λ in Cbdd (B); and (4.10) holds pointwise in Ω with subscripts denoting continuous derivatives in the sense of calculus. Theorem 4.1 is proved in section 4.2. Boundary Conditions and the space H1 The second result of this chapter concerns global regularity of solutions of (4.1)–(4.3) in which Ω is a proper subset of Rn and both solutions and test functions satisfy certain essential boundary conditions. Briefly, for each connected component Cl of ∂Ω we allow Nl linearly independent zeroconditions on u|Cl , where Nl ∈ {0, . . . , N }. Thus if Nl = 0 then no condition is imposed and if Nl = N then elements v of H1 will be required to satisfy v = 0 on Cl . For intermediate values of Nl the boundary condition will take the form Ql v = 0 on Cl , where Ql maps Cl into the set of Nl × N matrices and has full rank at each x. The detailed hypotheses are as follows: k+2 Ω is a Cbdd domain in Rn , not necessarily bounded, where n ≥ 2 and k ≥ 0 (Definition 1.7(c)), and there is a positive constant CBC , and for each connected component Cl of ∂Ω there is an integer Nl ∈ {0, . . . , N } such that for each l for which Nl ≥ 1 (if any) the following hold: there are continuous mappings Ql and Sl from Cl into the set of Nl × N matrices and N × N matrices respectively such that   (4.12) Ql Sl−1 = INl ×Nl 0 on Cl

and such that, in the notation of Definition 1.7, for all i for which Cl ∩ Ui 6= φ, the functions Ql ◦ Ti and Sl±1 ◦ Ti have continuous, bounded derivatives up to order k + 2 on Ti−1 (Cl ∩ Ui ) ⊂ Bi , and for y˜ ∈ Ti−1 (Cl ∩ Ui ) and |α| ≤ k + 2, (4.13)

|Dyα˜ (Ql ◦ Ti )(˜ y )|, |Dyα˜ (Sl±1 ◦ Ti )(˜ y )| ≤ CBC .

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82

4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

(Observe that if Cl ∩ Ui is nonempty, then it coincides with ∂Ω ∩ Ui , whose inverse image under Ti is an open subset of Bi .) We take CBC = 1 if Nl = 0 for all l. The space H1 will then be the closed subspace of elements v ∈ [H 1 (Ω)]N such that Ql v = 0 on Cl in the sense of the trace theorem (Theorem 3.2) for all l for which Nl ≥ 1. We can now state the global regularity result in which solutions are shown to be in H k+2 (Ω): Theorem 4.2. Let k be a nonnegative integer and assume the following: k+2 • Ω is a Cbdd domain in Rn where n ≥ 2 (Ω need not be bounded)

• the hypotheses and notations regarding H1 and the Nl , Ql and Sl in the paragraph above (in which (4.13) occurs) are in effect • the coefficients aη,ω , bη , cω , d and f of the bilinear form A in (4.1) satisfy the positive definiteness condition (4.4) and either of the sets of regularity conditions (4.6) or (4.8). Then there is a positive constant C determined by n, N and by upper bounds for k, MΩk+2 (defined in Definition 1.7), CA and CBC (defined in the subsections above) such that if u ∈ H1 satisfies Z Z
(4.14) A(u, v) = g · v + G : ∇x v + Γ · v dσ Ω

∂Ω

for all v ∈ H1 and for some g ∈ G ∈ [H k+1 (Ω)]N ×n and Γ ∈ 2 N [L (∂Ω)] satisfying X 1/2 (4.15) CΓ ≡ |Γ ◦ Ti |2[H k+1 (Bi )]N n/2 (see Theorem A.3(a)). The norm of the inclusion is evidently independent of r (by interpolation) unless m = n/2, in which case it depends also on a

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86

4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

finite upper bound for r. By checking cases we find that if r−1 + s−1 = 12 0 . then s ∈ Im (see (4.7)) if and only if r ∈ Im To prove (a) we let v be an element of H l (W), α a multi-index with |α| ≤ i and α = β + γ. Without loss of generality we can take Dzβ ζ ∈ Ls for 0 some s ∈ Il−i+|β| . Define r by r−1 + s−1 = 12 so that r ∈ Il−i+|β| . Then since l − i + |β| = l − i + |α| − |γ| ≤ l − |γ| we have that Dγ v ∈ H l−|γ| ⊂ H l−i+|β| , which is continuously included in Lr . Thus

Dβ ζ Dγ v ∈ Ls · Lr ⊂ L2 .

(4.25)

This proves that Dα (ζv) ∈ L2 for all |α| ≤ i, as required. Observe that the inclusion constant in (4.25) is independent of r, hence of s, unless l − i + |β| = n/2, in which case it depends on an upper bound for r, hence on an upper bound for (s − 2)−1 . To prove (b) we derive a bound for the L2 norm of the term on the left in (4.25) assuming now that s ∈ I˜l−i+|β| . In this case there is a δ ∈ (0, 1) such that s˜ ≡ (1 − δ)s ∈ Il−i+|β| , so that if r˜ is defined by r˜−1 + s˜−1 = 12 , 0 then r˜ ∈ Il−i+|β| . Consequently since l − γ = l − i + |β|,

|Dγ v|Lr˜ ≤ C|Dγ v|H l−i+|β| ≤ C|v|H l .

Applying this together with the fact that s˜/2(1 − δ), r˜/2(1 − δ) and 1/δ are conjugate indices, we obtain for the case s˜ < ∞ that

β

|D ζ D

γ

v|2L2

Z

|Dβ ζ|2 |Dγ v|2(1−δ) |Dγ v|2δ Z 2(1−δ)/˜s  Z 2(1−δ)/˜r  Z δ β s˜/(1−δ) γ r˜ ≤C |D ζ| |D v| |Dγ v|2

=

2(1−δ)

= C|Dβ ζ|2Ls |v|H l

|v|2δ Hi ;

a similar bound holds for the case that s˜ = ∞. To complete the proof of (b) we take the minimum over all such δ occurring in the above argument noting that if 0 < a < b, then aδ b1−δ is a decreasing function of δ.



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4. INTERIOR REGULARITY: PROOF OF THEOREM 4.1

87

Difference Quotients Now fix µ ∈ {1, . . . , n − 1}, let eµ ∈ Rn be the corresponding standard basis vector, and let Ω0 and Ω0σ be as in (4.19) for a particular σ ∈ (0, ε ∧ r). Then if h > 0 is sufficiently small depending on σ, z ± heµ will be in Ω0 if z ∈ Ω0σ , and if ζ : Ω → R we can therefore define ( (τh± ζ)(z) = ζ(z ± heµ )   (4.26) (δh± ζ)(z) = ±h−1 ζ(z ± heµ ) − ζ(z) . It is easy to check that these definitions extend to Lebesgue equivalence classes, that the product rule δh± (ζη) = ζ(δh± η) + (δh± ζ) (τh± η)

(4.27)

holds where defined, and that if ζ, η ∈ L2 (Ω) with at least one having support in Ω0σ , then Z Z ± (4.28) (δh ζ) η = − ζ (δh∓ η) Ω

Ω

for h sufficiently small depending on σ. Further properties of difference quotients are given in the following lemma: Lemma 4.6. Let Ω0 be as in the statement of Proposition 4.3 and let σ ∈ (0, ε ∧ r) and q ∈ (1, ∞] with H¨ older conjugate q 0 . Then the following hold for h sufficiently small depending on σ: (a) For integers l ≥ 0 the maps ϕ ∈ Cc∞ (Ω0 ) → ϕ± h defined by Z 1 ϕ(z ± hseµ )ds, z ∈ Ω0σ , ϕ± h (z) = 0

0

0

extend to bounded linear transformations from W l,q (Ω0 ) to W l,q (Ω0σ ) with norm less than or equal to one. (b) If ζ and its weak derivative ζzµ are in Lq (Ω0 ), then δh± ζ ∈ Lq (Ω0σ ), 0 |δh± ζ|Lq (Ω0σ ) ≤ |ζzµ ||Lq (Ω0 ) , and for all v ∈ Lq (Ω0 ) with support in Ω0σ , Z Z ± (4.29) (δh ζ) v dz = (ζzµ ) v ∓ h dz. Ω0

Ω0

L1loc (Ω0 )

(c) If ζ ∈ and if its weak derivative ζzµ is in W l,q (Ω0 ) for some nonnegative integer l, then δh± ζ ∈ W l,q (Ω0σ ) and |δh± ζ|W l,q (Ω0σ ) ≤ C|ζzµ |W l,q (Ω0 ) where C depends on n, l, i and on upper bounds for ε−1 , r−1 (defined in Proposition 4.3), σ −1 and the Lipschitz constant for ψ in Br . (d) If ζ ∈ L1loc (Ω0 ) and if its weak derivative ζzµ is in Ml,i (Ω0 ) for some nonnegative integers i ≤ l, then δh± ζ ∈ Ml,i (Ω0σ ) and XX |δh± ζ|Ml,i (Ω0σ ) ≤ C |(ζzµ )α,m |Lsα,m (Ω0 ) |α|≤i m

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88

4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

where (ζzµ )α,m and sα,m are as in Definition 4.4 (with ζ replaced by ζzµ ) and C depends on n, l, i and on upper bounds for ε−1 , r−1 (defined in Proposition 4.3), the Lipschitz constant for ψ in Br and in the case that n is even and n/2 ∈ {l − i, . . . , l}, on an upper bound for (sα,m − 2)−1 for all α satisfying l − i + |α| = n/2. PROOF.

Part (a) follows immediately from the bound |ϕ± h |W l,q0 (Ω0 ) ≤ σ

0

|ϕ|W l,q0 (Ω0 ) and the fact that Cc∞ is dense in W l,q (Ω0 ). To prove (b) we first take v = ϕ ∈ Cc∞ (Ω0σ ) so that by (4.28) and the assumption on the support of ϕ, Z Z ± (δh ζ) ϕ dz = − ζ (δh∓ ϕ)dz Ω0 Ω0 Z Z Z 1  =− ζ(z)Dzµ ϕ(z ∓ hseµ ds dz = ζzµ (z)ϕ∓ h (z)dz. Ω0

Ω0

0

The bound stated in (b) now follows and (c) and (d) are immediate from (b) and the fact that the operators δh± commute with weak differentiation.  PROOF of Proposition 4.3. We will prove the following statement by induction on i = −1, . . . , k: P(i) : If u0 ∈ H10 has support in Ω0σ for some σ ∈ (0, ε ∧ r) and satisfies Z
(4.30) A0 (u0 , v) = g 0 · v + G0 : ∇z v dz Ω0

for all v ∈ H10 , for some bilinear form A0 satisfying the hypotheses of Proposition 4.3 and for some g 0 and G0 whose components are in [H i∨0 (Ω0 )]N and [H i+1 (Ω0 )]N ×n respectively, then u0 ∈ [H i+2 (Ω0 )]N and (4.31)

|u0 |[H i+2 (Ω0 )]N ≤ C |u0 |[L2 (Ω0 )]N + |g 0 |[H i∨0 (Ω0 )]N + |G0 |[H i+1 (Ω0 )]N ×n




where C is as described in Proposition 4.3. The statement P(−1) is true by (4.9), so we may assume that i ≥ 0 and prove P(i) assuming P(i−1) . There is no loss of generality in taking i = k. We therefore assume that P(k−1) has been proved and fix A0 and u0 as in the 0 hypotheses in P(k) . In particular u0 ∈ H10 has support in Ωσ and satisfies (4.30) with the given A0 and where the components of g 0 and G0 are in H k (Ω0 ) and H k+1 (Ω0 ) respectively. Abbreviating [H k (Ω0 )]N by H k (Ω0 ), etc., we therefore have from P(k−1) that
(4.32) |u0 |H k+1 (Ω0 ) ≤ C |u0 |L2 (Ω0 ) + |g 0 |H k (Ω0 ) + |G0 |H k+1 (Ω0 ) ≡ CCG 0 .

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4. INTERIOR REGULARITY: PROOF OF THEOREM 4.1

89

A convenient reduction is now immediate. Since (b0 )η ∈ Mk,k , (c0 )ω ∈ Mk+1,k+1 , d ∈ Mk+1,k

(4.33)

by (4.8) and Lemma 4.5, we have that (b0 )η u0xη , d0 u0 ∈ H k (Ω0 ) and c0 u0 ∈ H k+1 (Ω0 ) with respective norms bounded by CCG 0 . These terms can therefore be absorbed into g 0 and G0 , affecting only a change in the constant C in (4.32). We therefore proceed to show that u0 ∈ H k+2 (Ω0 ) and |u0 |H k+2 (Ω0 ) ≤ CCG 0 assuming that A0 is fixed as in the statement of Proposition 4.3 but with the terms (b0 )η u0xη , c0 u0 and d0 u0 absent. The argument will be divided into three steps, as follows. Step 1: An integration by parts. Let τh± and δh± be the operators defined in (4.26) for a given µ ∈ {1, . . . , n − 1} (µ will be fixed through 0 Step 2). Choose a function χ ∈ Cc∞ (Rn ) satisfying χ ≡ 1 on Ωσ/2 and 0

0

(supp χ ∩ Ω ) ⊂ Ωσ/4 and whose derivatives up to order k + 2 are bounded by C. We claim that, for h sufficiently small depending on σ, there are functions gh00 and G00h with components in H (k−1)∨0 and H k respectively satisfying |g 00 |H (k−1)∨0 (Ω0 ) , |G00h |H k (Ω0 ) ≤ CCG 0

(4.34) and such that (4.35)

0

A

(δh+ u0 , v)

0

+ A (u

0

, δh− (χv))

Z = Ω0

(gh00 · v + G00h : ∇z v)dz

for all v ∈ H10 (Ω0 ). 0 To prove this we first note that since u
0 has support in Ωσ , the integrals on the left may be taken over Ω0σ + Bh ∩ Ω0 ⊂ Ω0σ/2 , on which χ = 1. Applying (4.27) and (4.28) to each summand in the obvious way, denoting weak derivatives with subscripts and suppressing the prime on a, we obtain Z  η,ω + 0  (a δh uzη ) · vzω + (aη,ω u0zη ) · (δh− (χv)zω ) dz Ω0 Z  η,ω + 0  = (a δh uzη ) · vzω − δh+ (aη,ω u0zη ) · vzω dz Ω0 Z  + η,ω + 0  =− ((δh a )(τh uzη )) · vzω dz. Ω0

This has the form of the terms on the right in (4.35) with gh00 = 0 and the nonzero components of G00h being components of (δh+ aη,ω )(τh+ u0 )zη . The H k norm of the latter is bounded by |(δh+ aη,ω )(τh+ u0 )zη )|H k (Ω0

σ/2

)

≤ |δh+ aη,ω |Mk,k (Ω0

σ/2

0 ) |u |H k+1 (Ω0 )

≤ CCG 0

by (4.32), Lemma 4.5, Lemma 4.6(d) and by the first hypothesis in (4.8), I|α| which includes the requirement that Dα aη,ω (in the shorthand of zµ ∈ L (4.8)) for |α| ≤ k. This proves the claim (4.34)–(4.35).

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90

4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

Step 2: H k+1 bound for transverse derivatives. We first claim that, for h sufficiently small depending on σ, there are new functions gh00 and G00h , still satisfying (4.34), such that Z (4.36) A0 (δh+ u0 , v) = (gh00 · v + G00h : ∇z v) dz Ω0

H10 .

for all v ∈ To prove this we first choose a function χ ˜ ∈ Cc∞ which is 0 0 identically one on supp χ + Bh and such that (supp χ) ˜ ∩ Ω ⊂ Ωσ/8 . Applying (4.28)–(4.30), we compute the second term on the left side of (4.35): Z   0 − χg ˜ · δh (χv) + G0 : ∇z δh− (χv) dz Ω0 Z   + 0 = (χg ˜ 0 )+ h · (χv)zµ − (δh G ) : ∇z (χv) dz 0 ZΩ h i

+ 0 0 )+ etr − δ + G0 : ∇ v dz. G )∇ χ · v + χ ( − (δ = χzµ (χg ˜ 0 )+ χg ˜ z z µ h h h h Ω0

This has the form of the integral on the right side of (4.36) in which the H (k−1)∨0 (Ω0 ) norm of the contribution to gh00 is bounded by
+ 0 0 0 |χzµ (χg ˜ 0 )+ h ) − (δh G )∇z χ|H (k−1)∨0 (Ω0 ) ≤ C |g |H (k−1)∨0 + |G |H k ≤ CCG 0 σ/8

H k (Ω0 )

by Lemma 4.6; and the norm of the contribution to G00h is bounded by

+ 0 0 0 tr |χ (g 0 )+ h eµ − δh G |H k (Ω0 ) ≤ C |g |H k + |G |H k+1 ≤ CCG 0 . σ/8

This proves the claim (4.36). Observe that δu0 ∈ H10 (the specific form of the essential boundary condition in the definition of H10 is important here). Therefore P(k−1) applies to (4.36) to show that, for h small depending on σ, δh+ u0 ∈ [H k+1 (Ω0 )]N with [H k+1 (Ω0 )]N -norm bounded by CCG 0 . Since this bound is independent of h, there is a sequence h → 0 such that δh+ u0 converges weakly in [H k+1 (Ω0 )]N , say to w. Then for ϕ ∈ [Cc∞ (Ω0 )]N , Z Z 0 u · ϕzµ dz = lim u0 · δh− ϕ dz h→0 0 0 Ω Ω Z Z + 0 = − lim δh u · ϕ dz = − w · ϕ dz. h→0 Ω0

Ω0

This proves that w is the weak Dzµ -derivative of u0 . Since µ ∈ {1, . . . , n − 1} was arbitrary, we conclude that for all multi-indices α of length less than or equal to k + 2, with the possible exception of (0, . . . , k + 2), the weak derivative Dzα u0 exists, is an element of [L2 (Ω0 )]N and satisfies the bound |Dzα u0 |L2 (Ω0 ) ≤ CCG 0 , where C and CG 0 are as described in Proposition 4.3 and (4.32) respectively. Step 3: H k+1 bound for the normal derivative. The required bound for the remaining derivative of order k + 2 can now be deduced from (4.30)

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10.1090/surv/251/05

4. INTERIOR REGULARITY: PROOF OF THEOREM 4.1

91

and the bounds obtained in Step 2. In this argument we mix Dα notation with subscripts to denote weak derivatives in Ω0 and we take all integrals over Ω0 . Let ϕ ∈ [Cc∞ (Ω0 )]N and let α be a multi-index with |α| ≤ k. Applying (4.30) with v = Dα ϕ ∈ H10 and integrating by parts we obtain Z Z 
 η,ω 0 α Dα g 0 − Dα Div G0 · ϕ dz (a uzη ) · D ϕzω dz = (4.37) ≤ CCG 0 |ϕ|L2 by the assumptions on g 0 and G0 . Next we show that, with the possible exception of the the term for which η = ω = n, the integral of each of the summands on the left in (4.37) is bounded in absolute value by CCG 0 |ϕ|L2 . First note that, by (4.8) and Lemma 4.5, aη,ω ∈ Mk+1,k+1 ⊂ Mk+1,k and k,k . Thus if η 6= n is fixed then the term in question is aη,ω , aη,ω zω ∈ M Z Dα (aη,ω u0z )zω · ϕ dz ≤ sup |aη,ω u0z |H k+1 |ϕ|L2 ≤ CCG 0 |ϕ|L2 , η η ω

and if ω 6= n is fixed the term is Z
0 0 u ) · ϕ dz Dα (aη,ω u0zη zω + aη,ω ≤ (|aη,ω u0zη zω |H k + |aη,ω zω zη zω uzη |H k |ϕ|L2     0 ≤ sup |aη,ω |Mk,k |u0zω |H k+1 + |aη,ω zω |Mk,k |uzη |H k ϕ|L2 . η

(sum over η in the second term on the right). We therefore conclude from (4.37) that Z (4.38) (an,n u0zn ) · Dα ϕzn dz ≤ CCG 0 |ϕ|L2 . Now replace the RN -valued function ϕ by ϕξ where ϕ ∈ Cc∞ (Ω0 ) is scalarvalued and ξ ∈ RN is a (constant) unit vector, and let ζ(x) ≡ (an,n (x))tr ξ and α = (0, . . . , k). If k ≥ 1 the integral on the left in (4.38) is then Z Z 0 k k (ζ · uzn )Dzn ϕzn dz = (−1) Dzkn (ζ · u0zn )ϕzn dz Z k = (−1) Dzk−1 (ζ · u0zn zn + ζzn · u0zn )ϕzn dz. n The second term on the right here is bounded by |ζzn · u0zn |H k |ϕ|L2 ≤ |azn |Mk,k |u0 |H k+1 |ϕ|L2 ≤ CCG 0 |ϕ|L2 and if k ≥ 2 the first term can be written Z k (ζ · u0zn zn zn + ζzn u0zn zn )ϕzn dz. (−1) Dzk−2 n Now repeat the argument: the second term here is bounded by CCG 0 |ϕ|L2 , exactly as in the previous step. Continuing in this way we thus find that Z   n,n tr k+1 0 ((a ) ξ) · Dzn u ϕzn dz ≤ CCG 0 |ϕ|L2 .

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4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY


 This proves that the weak derivative Dzn ξ · an,n Dzk+1 u0 exists and is n bounded in L2 by CCG 0 . Letting ξ range over standard basis vectors we conclude that the same is true for the components of the vector an,n Dzk+1 u0 . n To complete the argument we observe that the positivity hypothesis in (4.4) shows that the moduli of the (possibly complex) eigenvalues of an,n are bounded below away from zero on Ω0 , hence so is its determinant. Then since 1,∞ by (4.8), so too are those of (an,n )−1 the components of an,n zn are in W and its transpose. Combining this with the previous observation we conclude that Dzn Dzk+1 u0 exists and is bounded in L2 (Ω0 ) by CCG 0 . This completes n the proof of Step 3 and together with the conclusion above in Step 2 shows that u0 is in H k+2 (Ω0 ) and satisfies the bound in (4.21). This in turn proves P(k) and completes the proof of Proposition 4.3.  PROOF of Theorem 4.1. The proof of Theorem 4.1(a) is essentially contained in that of Proposition 4.3 but is substantially simpler. First for the case that Ω is a proper subset of Rn we proceed by induction, as described at the beginning of the previous subsection, assuming that u ∈ H k+1 (Ω1 ) where Ω1 is open, Ω0 ⊂ Ω1 ⊂ Ω, dist(Ω0 , ∂Ω1 ) ≥ γ/2, dist(Ω1 , ∂Ω) ≥ C −1 and Ω1 satisfies the cone condition with parameters bounded below by C −1 (so that Lemma 4.5 applies). Then for h > 0 sufficiently small depending on γ and now for arbitrary µ ∈ {1, . . . , n}, x ± heµ will be in Ω1 if x ∈ Ω0 . We can then define difference operators δh± and a cutoff function χ ∈ Cc∞ (Ω1 ) with χ ≡ 1 on Ω0 (in place of the cutoff function χ in Step 1 of the proof of Proposition 4.3) and then proceed as in Steps 1 and 2 to prove the claim in (4.36). This leads immediately to the conclusion that uxµ ∈ H k+1 (Ω0 ) with the required bound. The argument in Step 3 is avoided because µ is arbitrary. The proof is even simpler for part (b) in which Ω = Rn : the cutoff function χ is avoided altogether and the conclusion of the induction is that uxµ ∈ H k+1 (Rn ). To prove (c) we check from (4.8) that a and c are in Mk+1,k+1 (Ω) and that b and d are in Mk+1,k (Ω). The components of aη,ω uxη and cω u are therefore in H k+1 (Ω0 ) ⊂ H 1 (Ω0 ) and those of bη uxη and du are in H k (Ω0 ) ⊂ L2 (Ω0 ). We can therefore take v ∈ H01 (Ω0 ) in (4.2) and apply the divergence theorem to obtain that Z   (−aη,ω uxη )xω + bη uxη − (cω u)xω + du − g + Div G · v dx = 0. Ω0

It follows that the term in brackets is zero in L2 (Ω0 ) hence in H k (Ω0 ). To prove (d) we note that Il ⊆ [2, ∞] for every l (see (4.7)) so that by (4.8) the components of aη,ω and cω are in H k+1 (B) and those of bη and d are in H k (B) for every ball B whose closure is contained in Ω. The conclusions of (d) therefore follow directly from Theorem A.6 together with the inclusions u ∈ H k+2 (B), G ∈ H k+1 (B) and g ∈ H k (B).

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4. GLOBAL REGULARITY: PROOF OF THEOREM 4.2

93



§4.3

Global Regularity: Proof of Theorem 4.2

In this section we prove Theorem 4.2 by applying Theorem 4.1 away from the boundary of Ω and a reduction to the model problem of Proposition 4.3 near the boundary. PROOF of Theorem 4.2. First, Theorem A.26 shows that the integrals over ∂Ω in (4.1)–(4.3) involving f and Γ can be expressed as integrals over Ω of terms having the same forms as those involving b, c, d, g and G. It is easily checked from (4.8) and (4.15) that if the former are absorbed into the latter, then the augmented coefficients b, c and d and the augmented inhomogeneous terms g and G continue to satisfy the hypotheses of Theorem 4.2. We may therefore assume that the boundary integrals in (4.1)–(4.3) are absent. We prove the theorem by induction on k ≥ −1 (for all A satisfying the given hypotheses), as in the proof of Proposition 4.3. The case k = −1 again follows from (4.9) and for the induction step we assume that the theorem holds with k replaced by k − 1, that the hypotheses of the theorem hold exactly as stated and therefore that
(4.39) |u|H k+1 (Ω) ≤ CCG ≡ C |u|L2 (Ω) + |g|H k (Ω) + |G|H k+1 (Ω) (we again abbreviate [H k (Ω)]N by H k (Ω), etc.) where here and throughout the proof C will denote a generic positive constant as described in the statement of the theorem. We have to show that (4.40)

|u|H k+2 (Ω) ≤ CCG .

We can eliminate the terms involving b, c and d by observing that, by (4.8) and Lemma 4.5, (4.41)

bη ∈ Mk,k , cω ∈ Mk+1,k+1 and d ∈ Mk+1,k ,

so that bη uxη , d ∈ H k (Ω) and cu ∈ H k+1 (Ω) with respective norms bounded by CCG . These terms can therefore be absorbed into g and G, and we may proceed assuming that the terms involving b, c and d in the definition of A are absent. We begin by applying Theorem 1.8 with X = ∂Ω and δ > 0 determined by r, s, and ε0 (see Definition 1.7(c)) to obtain a partition of unity {ϕi } of ∂Ω subordinate to the cover {Ui } and satisfying the conclusions of Theorem 1.8. We will obtain the required H k+2 bound for u restricted to ∂Ω ∩ Ui , where i is now fixed, as follows. Let ∂Ω∩Ui be contained in the connected component Cl of ∂Ω and let S = Sl be as in (4.13) if 1 ≤ Nl ≤ N −1 and S ≡ I otherwise. Then define the following in terms of the notations in Definition 1.7: • Ω0 ≡ {z = (˜ z , zn ) ∈ Rn : z˜ ∈ Bi and − ψi (˜ z ) < zn < 0} • R(˜ z , zn ) ≡ Ri (˜ z , zn + ψi (˜ z )).

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4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

Thus R is a C k+2 diffeomorphism of Ω0 onto Ω ∩ Ui with derivatives up to order k + 2 bounded pointwise by C. Continuing, • let H10 be the set of elements v 0 ∈ [H 1 (Ω0 )]N such that v 0 (z) = 0 z ), and if Nl ≥ 1 then (v 0 )j (˜ z , 0) = 0 for if z˜ ∈ ∂Bi or if zn = −ψi (˜ z˜ ∈ Bi and j = 1, . . . , Nl • let u0 ≡ (ϕi ◦ R)(S ◦ R ◦ Π)(u ◦ R) where Π(˜ z , zn ) ≡ (˜ z , 0) (the parentheses here indicate the multiplication of a scalar times a matrix times a vector) • define T mapping functions on Ω0 to functions on Ω by (  (S −1 ◦ R ◦ Π ◦ R−1 )(x) (v 0 ◦ R−1 )(x), x ∈ Ω ∩ Ui , 0 (T v )(x) = 0, x ∈ (Ω ∩ Uic ) (S −1 denotes a matrix inverse and R−1 a functional inverse) • define A0 (w0 , v 0 ) ≡ A(T w0 , T v 0 ). The following are then easily checked: • u0 ∈ H10 ∩ H k+1 (Ω0 ) with |u0 |H k+1 (Ω0 ) ≤ C|ϕi u|H k+1 (Ω) ( (ϕi u)(x), x ∈ Ω ∩ Ui 0 • (T u )(x) = 0, x ∈ (Ω ∩ Uic ) • T is a bounded linear transformation from H10 to H1 with norm bounded by C (H1 is defined in the paragraph in which (4.13) occurs); and if v 0 ∈ H10 ∩ H m (Ω0 ) for some m ∈ {1, . . . , k + 2} then T v 0 ∈ H m (Ω ∩ Ui ) and |T v 0 |H m (Ω∩Ui ) ≤ C|v 0 |H m (Ω0 ) • A0 ∈ B(H10 × H10 ) with norm bounded by C, and for all v 0 ∈ H10 , A0 (v 0 , v 0 ) ≥ C −1 |v 0 |2H 1 (Ω0 ) − C|v 0 |2L2 (Ω0 ) . We also claim the following: • A0 has the same form as A in (4.1) but with the boundary integral omitted, Ω replaced by Ω0 , and aη,ω , bη , cω and d replaced by (a0 )η,ω , (b0 )η , (c0 )ω and d0 , these coefficients satisfying the positivity and regularity conditions in (4.4) and (4.8) with Ω replaced by Ω0 and CA replaced by CCA . In particular, A0 satisfies the third itemized hypothesis of Proposition 4.3. To prove this we let w0 , v 0 ∈ H10 and w = T w0 and v = T v 0 . Then by the chain rule applied in the definition of T we find that to highest order, ˜ z0 + . . . (sum over q) where S˜ = S −1 ◦ R ◦ Π is C k+2 and vxη ◦ R = X qη Sv q

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95

X qη = (R−1 )qxη ◦ R is C k+1 with derivatives bounded in Ω0 . Computing the highest–order terms in A0 we then have that Z 0 0 0 A (w , v ) = A(w, v) = (aηω wxη ) · vxω dx i ZΩ∩U    = (aηω ◦ R)(wxη ◦ R) · (vxω ◦ R) dz 0 ZΩ     ˜ z0 ) · (Sv ˜ z0 ) + . . . dz. = X pη X qω (aηω ◦ R)(Sw p q Ω0

This shows that (a0 )pq = X pη X qω S˜tr (aη,ω ◦ R)S˜ (sum over η and ω; note that aηω and (a0 )pq are matrices and X pη and X qω are scalars). Thus for |α| ≤ k + 1, each component of Dzα (a0 )pq is a finite sum of derivatives Dxβ aηω of order |β| ≤ |α|, modulo multiplication by bounded functions and composition with R. Since the aηω satisfy (4.8) so too do the (a0 )pq because these conditions are stronger for lower order derivatives and the sets Il in (4.7) are increasing in l. Similar arguments apply to terms involving (b0 )η , (c0 )ω and d0 . This proves that the coefficients of A0 satisfy the required regularity conditions (4.8). To check the positivity condition in (4.4) we let M be an N × n matrix and compute
qp
qp ˜ M tr (a0 )pq M =X pη X qω M tr S˜tr (aη,ω ◦ R)SM
˜ X)tr (aη,ω ◦ R)(SM ˜ X) ωη = (SM ˜ X|2 ≥ C −1 |M |2 ≥C −1 |SM by the positivity condition satisfied by the aηω . This proves the above claim. The last hypothesis of Proposition 4.3 to check is the following: • there are functions g 0 and G0 in H k (Ω0 ) and H k+1 (Ω0 ) respectively such that Z (4.42) A0 (u0 , v 0 ) = (g 0 · v 0 + G0 : ∇z v 0 ) dz Ω0

for all v 0 ∈ Ω0 and with respective norms bounded by CCG,i , where (4.43)

CG,i = |u|H k+1 (Ω∩Ui ) + |g|H k (Ω∩Ui ) + |G|H k+1 (Ω∩Ui ) .

To prove this we let v 0 ∈ H10 be given, v = T v 0 and X and S˜ be as above. Then   A0 (u0 , v 0 ) = A(u, ϕi v) + A(ϕi u, v) − A(u, ϕi v) Z   = g · (ϕi v) + G : ∇(ϕi v) dx Ω∩Ui Z  ηω  + (a (ϕi u)xη ) · vxω − (aη,ω uxη ) · (ϕi v)xω dx. Ω∩Ui

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4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

Our hypotheses on g and G show that the contributions to g 0 and G0 from the first integral on the right are bounded in H k and H k+1 respectively by CCG,i . The second integral may be written Z h
˜ 0 ) ϕi,xη ◦ R) aηω ◦ R)(u ◦ R) · (X qω Sv zq 0 Ω i
˜ 0 ) dz. − (ϕi,xω ◦ R) aη,ω ◦ R))(uxη ◦ R · (Sv Recalling that X and S˜ have bounded derivatives on Ω0 up to orders k + 1 and k + 2 respectively, we thus have that the H k (Ω0 ) norm of the contribution of the above integrand to the g 0 term in (4.42) is bounded by C sup |aηω uxη |H k (Ω∩Ui ) ≤ C sup |aη,ω |Mk,k |u|H k+1 (Ω∩Ui ) ≤ CCG,i , ω

ω

H k+1 (Ω0 )

and the bounded by

norm of the contribution to the G0 term in (4.42) is

C sup |aη,ω u|H k+1 (Ω∩Ui ) ≤ C sup |aη,ω |Mk+1,k+1 |u|H k+1 (Ω∩Ui ) ≤ CCG,i , η,ω

η,ω

as required. We have now checked that the hypotheses of Proposition 4.3 apply to (4.42) and therefore that u0 ∈ H k+2 (Ω0 ) with |u0 |H k+2 (Ω0 ) ≤ CCG,i . It follows that ϕi u ∈ H k+2 (Ω) with |ϕi u|H k+2 (Ω) ≤ CCG,i . P Now recall from Theorem 1.8 that i ϕi ≡ 1 in a neighborhood V of ∂Ω for which dist(∂V, ∂Ω) ≥ C −1 . We can therefore choose ϕ0 ∈ Cc∞ (Rn ) with ϕ0 identically one on Ω0 ≡ Ω − V, with support in Ω a positive distance from ∂Ω bounded below by C −1 and with derivatives of order up to k + 2 bounded by C. Thus X (4.45) u = ϕ0 u + (1 − ϕ0 ) ϕi u. (4.44)

i

Theorem 4.1 applies to show that the first term on the right here is in H k+2 (Ω) with norm bounded by CCG (CG is defined in (4.39)). Concerning the second term, we have that for finite sums over i, X X X Z 2 ϕi u H k+2 (Ω) = |Dα (ϕi u)|2 i

i

=

X i

|α|≤k+2 Ω

|ϕi u|2H k+2 (Ω∩Ui ) ≤ C

X

2 CG,i

i

by (4.44). However, the last condition in Definition 1.7 applied to each of the integrals in the definition (4.43) of CG,i together with the induction hypothesis (4.39) shows that X
2 CG,i ≤ C |u|2H k+1 (Ω) + |g|2H k (Ω) + |G|2H k+1 (Ω) = CCG2 . i

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4. GLOBAL REGULARITY: PROOF OF THEOREM 4.2

97

The sum in (4.45) therefore converges strongly in H k+2 (Ω) with limit bounded by CCG , so that u ∈ H k+2 (Ω) and |u|H k+2 (Ω) ≤ CCG . This completes the proof of Theorem 4.2(a). Next, the assertion in (b) that (4.10) holds has already been proved in Theorem 4.1(c), and to prove that the boundary conditions hold as stated in (b) we fix a connected component Cl of ∂Ω and let Nl , Ql and Sl be as in (4.12) and (4.13), taking Sl to be the identity matrix on Cl if Nl = 0. The essential boundary condition Ql u = 0 on Cl holds (if Nl > 0) because u ∈ H1 . Concerning the natural boundary condition we assume that Nl ∈ {0, . . . , N − 1} and let x0 ∈ ∂Ω ∩ Ui ⊆ Cl be a σ-Lebesgue point of representatives of the functions appearing in (4.18). Let ξ0 be an arbitrary N -vector in the null space of Ql (x0 ) if Nl > 0 and an arbitrary vector in Rn if Nl = 0. We will construct corresponding C k+2 test functions vh ∈ H1 as follows. Define ξ(x) = Sl−1 (x)Sl (x0 )ξ0 on Cl , so that by (4.12) and (4.13), k+2 ξ ◦ Ti ∈ Cbdd (Bi ) and Ql (x)ξ(x) = 0 on Cl . Next extend the domain of ξ by choosing a nonnegative function χh ∈ Cc∞ (Rn ) having support in the ball of radius h centered at Ri−1 (x0 ) and whose integral over ∂Ω equals one; then define vh = (χh ◦ Ri−1 )(ξ ◦ Ri ◦ Π ◦ Ri−1 ) on Ω ∩ Ui , where Π(˜ y , yn ) = (˜ y , ψi (˜ y )), and vh = 0 elsewhere in Ω. Then vh ∈ H1 if h is sufficiently small, and we can therefore apply (4.1), integrate by parts, and apply (4.10) to obtain Z  ω η,ω  ν (a uxη + cω u) + f u − Gν − Γ · vh dσ = 0. ∂Ω

Letting h → 0, we conclude that, at x0 ,  ω η,ω  ν (a uxη + cω u) + f u − Gν − Γ · ξ0 = 0. Thus if Nl = 0, ξ is arbitrary and the term in brackets is zero; otherwise the term in brackets is orthogonal to every ξ0 in the null space of Ql , as asserted in (b). The proofs of the regularity statements in (c) are identical to those of Theorem 4.1(d) with B now an arbitrary ball in Rn , and the remaining statements then follow immediately from these and the conclusions in (b). 

Exercises 4.1 Let τh± and δh± be as in (4.26) for a given µ ∈ {1, . . . , n}, let p ∈ [1, ∞) and abbreviate Lp (Rn ) = Lp . • Show that if u ∈ Lp then τh+ u → u in Lp as h → 0 but τh+ does not converge to the identity in the norm of L(Lp ). • Let u ∈ Lp and show that u ∈ W 1,p (Rn ) if and only if there is a positive number h0 and a constant C such that |δh+ u|Lp ≤ C for all h with |h| ≤ h0 and all µ.

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4. ELLIPTIC SYSTEMS: HIGHER ORDER REGULARITY

These results hold as well with τh+ and δh+ replaced by τh− and δh− . 4.2 Suppose that ζ ∈ L1loc (Rn ) and that there is a constant C such that for all v ∈ H 1 (Rn ) the product ζv is in H 1 (Rn ) with norm bounded by C|v|H 1 (Rn ) . Show that ζ satisfies the condition in (4.22) for α = 0 but need not satisfy this condition for |α| = 1. Suggestion: for the latter let Ωj be disjoint balls in R2 and choose pj > qj > 2 with pj , qj → 2 as j → ∞. Construct ζj ∈ Cc1 (Ωj ) such that ∇x ζj ∈ (Lqj − Lpj )(Ωj ). 4.3 Suppose that the hypotheses and notations in Theorem 3.7 are in force and that u is the solution of (3.14)–(3.16) described in that theorem. Assume also that the coefficients aη,ω , bη , d1 , d2 and f satisfy the conditions in (4.4) and (4.8) with k = 0, uniformly on compact intervals in [0, T ), and that g ∈ L2loc ([0, T ); L2 (Ω)). Show that u ∈ L2loc ([0, T ); H 2 (Ω)). 4.4 Complete the details of the following example in which the hypotheses of Theorem 4.2 hold with k = 0 with the exception that Ω is a uniformly 2 Lipschitz domain but not a Cbdd domain. Let x = (r cos θ, r sin θ) be the usual polar coordinate representation in R2 − {x : x2 ≤ 0} in which r(x) > 0 and |θ(x)| < π, and let Ω be the set of points x for which r(x) ∈ (0, 1) and |θ(x)| < 2π/3. The function w(x) = r(x)3/4 cos (3θ(x)/4) is then the real part of the analytic complex-valued function (x + iy)3/4 and is therefore harmonic in Ω. Define u(x) = ϕ(r(x))w(x) where ϕ is a nonincreasing C ∞ function on R with ϕ(r) = 1 or 0 according as r ≤ 1/3 or r ≥ 2/3. Then u is in H01 (Ω) and satisfies the weak form of the equation ∆u = −g for a particular g ∈ L2 (Ω); that is, Z Z ∇x u · ∇x v dx = g v dx Ω

Ω

H01 (Ω).

for all v ∈ Check that u is not in H 2 (Ω). (The fact that Ω is not convex also plays a role in this example; see Grisvard [20] for a thorough discussion.) 4.5 Assume that the hypotheses and notations of Theorem 4.2 are in force for a given k ≥ 0 and in addition that A is coercive on H1 × H1 ; that is, there is a positive number λ such that A(u, u) ≥ λ|u|2H1 for all u ∈ H1 . There is then an operator S ∈ L([L2 (Ω)]N , H1 ) such that A(Sf, v) = hf, viL2 for all v ∈ H1 (Ω). Show that any eigenfunction of S must be in [H k+2 (Ω)]N ; consequently if these hypotheses hold for all k ≥ 0 then any such eigenfunction is in C ∞ (Ω) and all its derivatives have continuous extensions to Ω.

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10.1090/surv/251/05

CHAPTER 5

Parabolic Systems: Higher Order Regularity In this chapter we derive higher order regularity results for solutions of the parabolic systems considered in Chapter 3, now corresponding to a wide range of essential, natural and mixed boundary conditions. The general problem is restated below in section 5.1 and the main result is given in Theorem 5.5: under appropriate regularity conditions on coefficients, inhomogeneities and spatial domains and with compatible initial data, solutions are m times differentiable with respect to t ∈ (0, T ) and 2m + 1 times differentiable with respect to x ∈ Ω, where m ≥ 1 will be fixed throughout. Differentiability is interpreted in three distinct ways in three parts of the theorem: (a) weak absolute continuity in time of the Sobolev space–valued solution u and Bochner space differentiability of its Lebesgue equivalence class on [0, T ) (Definition A.18), (b) differentiability in the weak sense on Ω × (0, T ) of the measurable counterpart um , and (c) continuous differentiability of a representative uc of um in the usual calculus sense when m is sufficiently large. Initial layer regularization and higher order regularity away from t = 0 for solutions of problems with incompatible initial data are described in Theorem 5.8.

§5.1

Boundary Conditions, Coefficients and Inhomogeneities

The formal problem is as in Chapter 3: let Ω be an open set in Rn , n ≥ 2, T ∈ (0, ∞] and N ≥ 1, and let H1 be a closed subspace of [H 1 (Ω)]N which is dense in [L2 (Ω)]N (specific choices for H1 are described below). Then given ς0 ∈ [L2 (Ω)]N the problem is to find u : [0, T ) → [L2 (Ω)]N whose (0, T )-equivalence class is a weakly measurable map into H1 and which satisfies Z t   (5.1) hu(t), vi[L2 (Ω)]N = hς0 , vi[L2 (Ω)]N + − A(s)(u(s), v) + F (s) · v ds 0

for all v ∈ H1 where A : [0, T ) → B(H1 × H1 ) and F : [0, T ) → H1∗ are given by Z  η,ω  A(u, v) = (a uxη ) · vxω + (bη uxη ) · v + (cω u)·vxω + (du) · v dx Ω Z (5.2) + (f u) · v dσ ∂Ω 99

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100

5. PARABOLIC SYSTEMS: HIGHER ORDER REGULARITY

(summation over repeated indices η and ω is understood) and Z Z
Γ · v dσ. g · v + G : ∇x v + (5.3) F ·v = Ω

∂Ω

Here aη,ω , bη , cω and d are N ×N matrix-valued functions of (x, t) ∈ Ω×[0, T ), f is an N × N matrix–valued function of (x, t) ∈ ∂Ω × [0, T ) and the dot denotes the usual inner product in RN ; g and Γ are RN -valued, G is N × n matrix-valued, ∇x v is the N × n x-derivative matrix, G : ∇x v is the scalar product Gij vxi j and σ is the boundary measure described in Theorem 3.1 (the boundary integrals involving f and Γ are omitted in the case that Ω = Rn ). We saw in Theorem 3.6 that under fairly mild regularity hypotheses on A and F this problem has a unique solution u ∈ C([0, T ); [L2 (Ω)]N ) with [0, T )-equivalence class in L2loc ([0, T ); H1 ). Higher order regularity properties of u will be derived from the application of Theorems 2.7 and 2.10. Three preliminary discussions will be required: description of the Hilbert subspaces Hj of [H 1 (Ω)]N and subspaces Vj of H1∗ and verification of the hypotheses preceding Theorem 2.7(a); formulation of the assumptions on the coefficients of A, construction of the bilinear forms A(j) and verification of the hypotheses in Theorem 2.7(b) and (c); and formulation of the assumptions on the inhomogeneities g, G and Γ and verification of the hypotheses (2.26)–(2.27) in Theorem 2.7. Spatial Domain and Boundary Conditions. A specific realization of the general Hilbert space structure of Theorem 2.7 will now be described and will be fixed for this and the following chapter. This structure corresponds to the essential boundary conditions in Example 3 of section 3.1 and (4.13) in Chapter 4 with k now replaced by 2m − 1 ≥ 1. Definitions, notations and hypotheses are as follows: Definition 5.1. Fix an ensemble (5.4)

2m−1 N, n, m, Ω, {Nl , Ql , Sl }, CBC , {Hj }2m+1 j=0 , {Vj }j=−1

in which N, n, m are integers with N ≥ 1, n ≥ 2 and m ≥ 1; Ω is either Rn , in which case Nl , Ql , Sl and CBC are not defined and play no role, or Ω is a 2m+1 Cbdd domain in Rn (Definition 1.7), not necessarily bounded, and Nl , Ql and Sl are defined for each connected component Cl of ∂Ω as follows (as in the italicized paragraph preceding the statement of Theorem 4.2 but with k replaced by 2m − 1): Nl ∈ {0, . . . , N } and Ql and Sl map Cl into the set of Nl × N matrices and N × N matrices respectively; if 0 < Nl < N then   Ql Sl−1 = INl ×Nl 0 on Cl , where INl ×Nl is the Nl × Nl identity; if Nl = 0 then Ql = 0 and if Nl = N then Ql = IN ×N ; and in either of the latter two cases Sl = IN ×N . Also, in the notation of Definition 1.7, for all i for which Ui intersects Cl , 2m+1 Ql ◦ Ti and Sl± ◦ Ti are in Cbdd (Bi ) with derivatives bounded pointwise in Bi by CBC .

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In all cases H0 will be [L2 (Ω)]N and if Ω = Rn then Hj will be [H j (Rn )]N for j ≥ 1. If Ω is a proper subset of Rn then H1 will be the set of elements v ∈ [H 1 (Ω)]N such that Ql T v = 0 (T is the trace operator of Theorem 3.3) on Cl for each l (this condition is vacuous if Nl = 0) and Hj = H1 ∩[H j (Ω)]N for j ≥ 1. Finally, V−1 ≡ H1∗ and for j = 0, . . . , 2m − 1, Vj is the set of elements F ∈ H1∗ for which there is a representation (5.3) in which the components of g are in H j (Ω), those of G are in H j+1 (Ω), and either Ω = Rn and the boundary integral involving Γ is absent, or in the notation of Definition 1.7, X |Γ ◦ Ti |2H j+1 (Bi ) < ∞. i

The above structure is consistent with the requirements of Theorem 2.7: Lemma 5.2. The ensemble (5.4) in Definition 5.1 satisfies the hypotheses in the statement preceding part (a) of Theorem 2.7. PROOF. We need only identify a Hilbert space structure on Vj . Observe that if w ∈ Hj+2 then the mapping v ∈ H1 7→ hw, vi[H 1 (Ω)]N is in Vj , and conversely given F ∈ Vj there is a unique wF ∈ Hj+2 for which F · v = hwF , vi[H 1 (Ω)]N for all v ∈ H1 , by Theorem 4.2. Thus Vj can be identified with the Hilbert space Hj+2 and so is a Hilbert space itself, as required.  As the proof suggests, we could have taken Vj = {hz, ·iH1 such that z ∈ Hj+2 } and wrtten F = hz, ·i[H 1 (Ω)]N in place of the more complicated expression in (5.3). Applications typically present g, G and Γ, however, but not z, and this has determined our choice. Hypotheses on Coefficients. These can be given both in terms of W k,q (Ω)– valued mappings on [0, T ) as well as in terms of weak derivatives of equivalence classes of functions which are measurable on Ω × (0, T ) (the same is true for the conclusions of the major theorems of this chapter). Both formulations are important, and to clarify the connection we extract several results from section A.5 in the Appendix where complete, detailed statements and proofs are given. These generalize Definition 3.4 and Theorem 3.5 and are summarized as follows, in which Ω is an arbitrary open set in Rn , I a nontrivial interval in R, p ∈ [1, ∞], q ∈ (0, ∞] and prime denotes H¨older conjugate: • (Definition A.13) The space Lp,q (Ω × I) is the Lebesgue equivalence class of real-valued functions w on Ω × I which are measurto the product measure on Ω × I and for which able with respect |w(·, t)|Lq (Ω) p (which is meaningful by Tonelli’s theorem) is L (I) p,q finite; and L(t)loc (Ω × I) is the set of such equivalence classes which are in Lp,q (Ω × I 0 ) for every compact subinterval I 0 of I.

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• (Corollary A.16) There is a one-to-one correspondence w ↔ wm between elements w ∈ Lp (I; Lq (Ω)) and elements wm ∈ Lp,q (Ω × I) 0 0 in which w and wm define the same action on Lp (I) × Lq (Ω) by iterated integration. • (Theorem A.23) If w ∈ Lp (I; Lq (Ω)) then w ∈ Lp (I; W k,q (Ω)) (whose definition requires a brief discussion–see Definition A.22) if and only if for every multi-index α with |α| ≤ k, the derivative Dxα wm with respect to x ∈ Ω exists in the weak sense on Ω × I ◦ and is in Lp,q (Ω × I). • (Theorem A.24) If w ∈ L1loc (I; Lq (Ω)) then the equivalence class w contains an element wAC which is in wAC(I; W k,q (Ω)) if and only if for every multi-index α with |α| ≤ k, the derivatives Dxα wm and Dt Dxα wm with respect to x ∈ Ω and t ∈ I exist in the weak sense on Ω × I ◦ and are in L1,q (t)loc (Ω × I). The correspondence w ↔ wm extends componentwise to vector–valued and matrix–valued functions in the obvious way. Also, if w ∈ C(I; Lq (Ω)), then q its equivalence class [w]I is in L∞ loc (I; L (Ω)), and no ambiguity will result from our writing wm in place of ([w]I )m . We now describe hypotheses on the coefficients of A required for the higher order regularity result of the next section. The following positive definiteness assumption is equivalent to the last hypothesis in the basic existence result Theorem 3.6(c): For each η and ω in {1, . . . , n}, aη,ω is a [0, T )–equivalence class of mappings from [0, T ) into the set of N × N matrices, each component ∞ ¯ of which is in L∞ loc ([0, T ); L (Ω)); and given t ∈ [0, T ) there is a positive number θ(t¯) such that for almost all (x, t) ∈ Ω × (0, t¯), (5.5)

ω,η (M tr aη,ω ≥ 2θ(t¯)|M |2 m (x, t)M )

for all N × n matrices M ; (here tr denotes transpose, summation over η and ω is understood and any choice of matrix norm is allowed). Next, in order to formulate regularity conditions for functions defined on ∂Ω we recall that if ψ : ∂Ω → R and α is a multi-index, then in the notation of Definition 1.7,  1/q  P α (ψ ◦ T )|q |D , q ∈ (1, ∞), i q y˜ i L (Bi ) (5.6) hψiαLq , ∂Ω ≡ sup |Dα (ψ ◦ T )| ∞ , q = ∞ i

i L (Bi )

y˜

provided the above derivatives exist in the weak sense on each Bi . Also, if I ⊂ R is a nontrivial interval, j ≥ 0 and ψ ∈ L1loc (I; Lq (∂Ω)), we define  1/q  P j α q |D D [(ψ ◦ T ) ]| , q ∈ (1, ∞), (α,j) i m Lq (Bi ) t y˜ i (5.7) hψiLq , ∂Ω ≡ j sup |D Dα [(ψ ◦ T ) ]| ∞ , q = ∞, i

t

y˜

i m L (Bi )

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103

provided the above derivatives exist in the weak sense on each Bi × I ◦ . If so, (α,j) then hψiLq , ∂Ω is a measurable function of t. We now describe three sets of regularity conditions for the coefficients of A, each more general but also more complicated than the previous. The first set is unnecessarily strong but is easy to apply and suffices for many applications: (5.8) • Each component of each of aη,ω , bη , cω and d is an element h in ∞ L∞ loc ([0, T ); L (Ω)) satisfying the following: for j = 0, . . . , m − 1, Dtj Dxα hm ∈ L∞,∞ (t)loc (Ω × [0, T )) Dtj+1 Dxα hm ∈ L2,∞ (t)loc (Ω × [0, T )) for |α| ≤ 2m − 2j if h represents a component of aη,ω or cω and for |α| ≤ 2m − 2j − 1 if h represents a component of bη or d. ∞ • (If Ω 6= Rn ) each component h of f is in L∞ loc ([0, T ); L (∂Ω)) and for j = 0, . . . , m − 1 and |α| ≤ 2m − 2j, (α, j)

hhiL∞ , ∂Ω ∈ L∞ loc ([0, T )) (α, j+1)

hhiL∞ , ∂Ω ∈ L2loc ([0, T )). (Readers for whose purposes the above conditions suffice may choose to proceed directly to Lemma 5.3 below.) More general regularity hypotheses can be formulated in which less stringent conditions are imposed on higher order derivatives (which is to be expected, since a term such as Dtj Dxα (aη,ω uxη ), for example, involves products of higher derivatives of one factor and lower derivatives of the other). An intermediate set of hypotheses is available offering a broader range of allowable Lebesgue spaces for higher order t–derivatives of the coefficients, although not for higher order x–derivatives. These are stated in terms of index sets I˜k , which are the sets Ik defined in (A.1) with the left-hand endpoints removed, with the exception that I˜0 = {∞}. Thus

(5.9)

  {∞}, k = 0, ˜ Ik = (n/k, ∞], 0 < k ≤ n/2,   (2, ∞], k > n/2.

The following intermediate regularity conditions will suffice for most applications:

(5.10)

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5. PARABOLIC SYSTEMS: HIGHER ORDER REGULARITY

1,∞ (Ω)); • Each component of each aη,ω is an element h in L∞ loc ([0, T ); W and given j ∈ {0, . . . , m − 1} and α with |α| ≤ 2m − 2j, there is an s ∈ I˜2j , which may depend on η, ω, the particular component of aη,ω , j and α, such that ∞,s Dtj Dxα hm ∈ L(t)loc (Ω × [0, T ))

Dtj+1 Dxα hm ∈ L2,s (t)loc (Ω × [0, T )). • The above conditions hold as well for the components of bη , cω and d with values of s in I˜2j+1 , I˜2j+1 and I˜2j+2 respectively and with the upper bound for |α| replaced by 2m − 2j − 1, 2m − 2j and 2m − 2j − 1 respectively. ∞ • (If Ω 6= Rn ) each component h of f is in L∞ loc ([0, T ); L (∂Ω)) and for each j = 0, . . . , m−1 there is an s ∈ I˜2j+1 , which may depend on the particular component of f and on j, such that for |α| ≤ 2m − 2j, (α, j)

hhiLs , ∂Ω ∈ L∞ loc ([0, T )) (α, j+1)

hhiLs , ∂Ω ∈ L2loc ([0, T )). Observe that the conditions in (5.8) imply those in (5.10) with s = ∞ in each case. (Again, readers for whose purposes the above conditions suffice may choose to proceed directly to Lemma 5.3 below.) Finally we describe the most general regularity hypotheses on coefficients which the theory can accommodate. These are stated in terms of index sets Ik and Kk , defined in (A.1) and (A.2), and I˜k defined above in (5.9). First, the following conditions guarantee that the bilinear form A in (5.2) satisfies the absolute continuity hypotheses (ci –ciii ) of Theorem 2.7: (5.11) q • Each component of each aη,ω is an element h in L∞ loc ([0, T ); L (Ω)) with q = ∞; and given j ∈ {0, . . . , m − 1} there is an s ∈ I2j+1 , which may depend on η, ω, the particular component h of aη,ω and j, such that

Dtj hm ∈ L∞,s (t)loc (Ω × [0, T )) Dtj+1 hm ∈ L2,s (t)loc (Ω × [0, T )). • The above conditions hold as well for the components of bη , cω and d with values of q in {∞}, {∞} and I1 respectively and I2j+1 replaced by K2j+1 , I2j+2 and K2j+2 respectively. The following hypotheses will be applied to verify the ellipticity condition (b) of Theorem 2.7:

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5. BOUNDARY CONDITIONS, COEFFICIENTS AND INHOMOGENEITIES

105

(5.12) • For all η and ω, for each component h of aη,ω and for |α| ≤ 2m, there is an s ∈ I(|α|−1)∨0 such that Dxα hm ∈ L∞,s (t)loc (Ω × [0, T )). • The above condition holds as well for the components of bη , cω and d with values of s in I|α| , I|α| and I|α|+1 respectively and with the upper bound for |α| replaced by 2m − 1, 2m and 2m − 1 respectively. The following hypotheses will be applied to verify condition (civ ) of Theorem 2.7: (5.13) • If m ≥ 2 then for all η and ω, for each component h of aη,ω , and for j = 1, . . . , m − 1 and |α| ≤ 2m − 2j, there is an s ∈ I˜2j+|α| such that Dtj Dxα hm ∈ L∞,s (t)loc (Ω × [0, T )). • The above condition holds as well for the components of bη , cω and d with I˜2j+|α| replaced by I˜2j+|α|+1 , I˜2j+|α|+1 and I˜2j+|α|+2 respectively and the upper bound for |α| replaced by 2m − 2j − 1, 2m − 2j and 2m − 2j − 1 respectively. ∞ Finally, for the case that Ω 6= Rn , we require that f ∈ L∞ loc ([0, T ); L (∂Ω)) and that each component h of f satisfies the following:

(5.14) • for each α with |α| ≤ 2m there is an s ∈ I|α| such that (α,0)

hhiLs , ∂Ω ∈ L∞ loc ([0, T )) • for all j = 0, . . . , m − 1 there is an s ∈ I2j+2 such that (0, j)

hhiLs , ∂Ω ∈ L∞ loc ([0, T )) (0, j+1)

hhiLs , ∂Ω ∈ L2loc ([0, T )) • and if m ≥ 2 then for j = 1, . . . , m − 1 and |α| ≤ 2m − 2j there is an s ∈ I˜2j+|α|+1 such that (α, j)

hhiLs , ∂Ω ∈ L∞ loc ([0, T )).

Observe that all the above conditions are satisfied automatically in the important case that the coefficient matrices are independent of x and t. We now construct bilinear forms A(j) satisfying the hypotheses of Theorem 2.7:

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Lemma 5.3. Assume that the hypotheses and notations in Definition 5.1 are in force, let T ∈ (0, ∞] and let Ik and Kk be the index sets defined in (A.1) and (A.2). Assume that N × N matrix-valued [0, T )–equivalence classes aη,ω , bη , cω , d and f are given satisfying the descriptions and regularity conditions in one of (5.8), (5.10) or (5.11)–(5.14) and that the positive definiteness condition (5.5) holds. Then: (a) For j = 0, . . . , m there are mappings (a(j) )η,ω such that for j < m 2 ([0, T ); Ls (Ω)) for some each component of each (a(j) )η,ω is in wACloc s ∈ I2j+1 depending on j, η, ω and the particular component; the derivative of each such component is the equivalence class in L2loc ([0, T ); Ls (Ω)) of the corresponding component of (a(j+1) )η,ω if j < m − 1 and the corresponding component of (a(m) )η,ω ∈ L2loc ([0, T ); Ls (Ω)) if j = m; (a(0) )η,ω is an element of the equivalence class aη,ω as a mapping into the set of measurable mappings on Ω, and more generally, [(a(j) )η,ω ]m = Dtj aη,ω m

(5.15) for j = 0, . . . , m.

The same statements hold as well for (b(j) )η , (c(j) )ω and d(j) with I2j+1 replaced by K2j+1 , I2j+2 , and K2j+2 respectively and in the case that Ω 6= Rn , for f (j) with Ω replaced by ∂Ω and I2j+1 by I2j+2 . (b) For j = 0, . . . , m − 1, |α| ≤ 2m − 2j and any particular component h of any aη,ω , the Bochner space derivative Dxα h(j) exists and is in α C([0, T ); Lsweak (Ω)) for some sα ∈ I2j+|α| which may depend j, α, η, ω and the particular component. The same statement holds as well for (b(j) )η , (c(j) )ω , and d(j) with the upper bound for |α| replaced by 2m − 2j − 1, 2m − 2j, and 2m − 2j − 1 respectively and I2j+|α| replaced by I2j+|α|+1 , I2j+|α|+1 and I2j+|α|+2 respectively. (c) The bilinear forms A(0) , . . . , A(m) given by Z  (j) η,ω (j) A (z, w) = (a ) zxη ) · wxω + ((b(j) )η zxη ) · w Ω

 + ((c(j) )ω z) · wxω + (d(j) z) · w dx Z +

(f (j) z) · w dσ

∂Ω

satisfy the conditions in (a)–(c) of Theorem 2.7 with constant CA (t¯) determined by n, m, N, CBC (see Definition 5.1), t¯, θ(t¯) and the norms of the coefficients on [0, t¯] occurring in the hypotheses (5.8), (5.10) or (5.11)–(5.14). PROOF. We first show that the boundary integral in the definition of A, if it occurs, can be absorbed into terms in A involving integrals over Ω. To do

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107

this we first let z, w ∈ Cc∞ (Ω) and apply (A.26) in Theorem A.26 to obtain Z Z  1 kl k l  kl k l (5.16) f z w dσ = S (f )z w +S 2 (f kl )·(wl ∇x z k +z k ∇x wl ) dx. ∂Ω

Ω

The middle term on the right, for example, has the same form as the term involving bη in the definition of A. The reader can easily check via Theorem A.26 that the three hypotheses on the components of f in (5.14) have been correctly tailored to insure that the components of S 2 (f ) satisfy the three hypotheses in (5.11)–(5.13) imposed on bη . The same holds for the other two terms in (5.16), which then extends to elements z, w ∈ H 1 (Ω). The term on the left in (5.16) may therefore be absorbed into the terms in A involving bη , cω and d, thereby affecting only the constant CA . We may therefore proceed assuming that the boundary integral in the definition of A is omitted. ∞ To prove part (a) we let h ∈ L∞ loc ([0, T ); L (Ω)) denote a particular η,ω component of a particular a . The hypotheses in (5.11) that Dtj hm ∈ L∞,s (t)loc and Dtj+1 hm ∈ L2,s (t)loc for some s ∈ I2j+1 together with Theorem A.24 then show that there is an h(j) ∈ wAC 2 ([0, T ); Ls (Ω)) whose measurable counterpart is Dtj hm and that [h(j) ]m = Dtj hm ,

(5.17)

which proves (5.15), and whose derivative satisfies [(h(j) )0 ]m = Dt [h(j) ]m . Applying (5.17) twice, we conclude that [(h(j) )0 ]m = Dtj+1 hm = [h(j+1) ]m . Thus the derivative of h(j) is the [0, T ) equivalence class of h(j+1) (or of h(m) if j = m − 1). This proves the statements in (a) for aη,ω , and the corresponding statements for the other coefficients are proved in a similar way. To prove (b) we let j ≤ m − 1 and |α| ≤ 2m − 2j and again denote by h a component of a particular aη,ω . Then by the hypotheses (5.12) and α ∞ (5.13), Dtj Dxα hm ∈ L∞,s (t)loc for some sα ∈ I2j+|α| . Therefore if ϕ ∈ Cc (Ω) and t0 ∈ [0, t¯), Z t0 +∆tZ Z h(j) (t)Dxα ϕ dxdt h(j) (t0 )Dxα ϕ(x)dx = lim |∆t−1 ∆t→0

Ω

Z = lim ∆t−1 ∆t→0

t0

t0 +∆tZ

t0

Ω

Ω

Dxα Dtj hm ϕ dxdt ≤ C(t¯)|ϕ|Ls0α (Ω)

s0α

where is the H¨older conjugate of sα . Since s0α ∈ [1, ∞), Cc∞ (Ω) is dense 0 s in L α and the above therefore shows that the weak derivative Dxα h(j) (t0 ) exists and is bounded in Lsα by C(t¯). The weak continuity of the map t → Dxα h(j) (t) then follows from the same density together with the fact that h(j) is a continuous mapping into Lsα (Ω). This proves the statements in (b) for aη,ω and the corresponding statements for the other coefficients are proved in a similar way.

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To prove (c) we need to check that the hypotheses of Theorem 2.7 are satisfied. The first of these, Theorem 2.7(a), follows from (5.11) and (5.12): these are stronger than the conditions in Theorem 3.6, part (d) of which guarantees the requirement in Theorem 2.7(a). To prove the hypothesis in Theorem 2.7(b) we can (and leave to the reader to) check that, in the notation of Definition 2.6, the bilinear form Aε is the same as in the definition (5.2) of A but with mollified coefficients: aη,ω is replaced by aη,ω where ε Z η,ω aε (·, t) = hε (t − s)aη,ω (·, s + ε)ds, |s−t| 0 determined by r, s, and ε0 (see Definition 1.7(c)) subordinate to the cover {Ui }. Define Γ(j) (x, t) =

X

(0)

˜ (T −1 (x), t) ϕi (x)Γ i,c i

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where the sum is over i for which ϕi (x) 6= 0. The regularity statements in the second item in (ciii) for Γc are now easily checked, although we note that important use is made of the final item in Definition 1.7(c). The corresponding statements for fc are proved in a similar way; we note again, however, that for each space Ls (Bi ) occurring in the hypotheses (5.14), the index s is greater than or equal to two, so that Ls (Bi ∩ Ti−1 (B)) ⊆ L2 (Bi ∩ Ti−1 (B)) for every ball B ⊂ Rn . The final item in Theorem 5.5(ciii ) follows from the final item (av ), just as in the proof of the first item in (ci ), because every function appearing in (5.37) is H¨older continuous. This completes the proof of (c) and of Theorem 5.5. 

§5.3

Incompatible Data and Initial Layer Regularization

In this section we consider higher order regularity of solutions of the problem (5.1)–(5.3) when the regularity conditions on the coefficients of A in Theorem 5.5 are satisfied but the compatibility conditions (5.23) are not. Specifically, given initial data ς0 ∈ [L2 (Ω)]N and given inhomogeneous terms g, G and Γ satisfying the hypotheses of the basic existence result Theorem 3.6 on [0, T ) but the higher order regularity conditions of Theorem 5.5 only away from t = 0, the corresponding solution u of Theorem 3.6 attains the higher order regularity described in Theorem 5.5 instantaneously for t > 0 with certain specific rates. The precise results are given in the following theorem, which is a special case of Theorem 2.10: Theorem 5.8. Assume that the hypotheses and notations in Definition 5.1 are in force, let T ∈ (0, ∞] and let N × N matrix-valued [0, T )–equivalence classes aη,ω , bη , cω , d and (if Ω 6= Rn ) f be given satisfying the positive definiteness condition (5.5) and the regularity conditions in one of (5.8), (5.10) or (5.11)–(5.14). Let ς0 ∈ [L2 (Ω)]N and let g, G and Γ be given satisfying the following (references to Γ and integrals over ∂Ω are omitted if Ω = Rn and we again abbreviate [H k (Ω)]N by H k , etc.): • The conditions in Theorem 3.6(e) are satisfied. • There is a δ ∈ {0} ∪ ( 12 , ∞) such that for j = 0, . . . , m − 1 and k = j + 1, . . . , m and on every set Ω × [0, t¯] ⊂ Ω × [0, T ) tk+δ Dtj Dxα gm ∈ L∞,2 , 0 ≤ 2j + |α| ≤ 2k − 2, (5.38)

tk+δ Dtj Dxα Gm ∈ L∞,2 , 0 ≤ 2j + |α| ≤ 2k − 1, X (α,j) tk+δ hΓiL2 , ∂Ω ∈ L∞ loc ([0, T )). |α|≤2k−2j−1

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5. INCOMPATIBLE DATA AND INITIAL LAYER REGULARIZATION

123

• For j = 0, . . . , m and k = j, . . . , m and on every set Ω × [0, t¯], tk+δ Dtj Dxα gm ∈ L2,2 , 0 ≤ |α| ≤ max{2k − 2j − 1), 0} tk+δ Dtj Dxα Gm ∈ L2,2 , 0 ≤ 2j + |α| ≤ 2k, X (α,j) tk+δ hΓiL2 , ∂Ω ∈ L2loc ([0, T )).

(5.39)

|α|≤2k−2j

• For j = 0, . . . , m − 1 there is an element γj+1 ∈ L2 (Ω) such that for all v ∈ H1 , Z tZ h
−1 lim t sj+1+δ Dtj gm + S 1 (Γ)m · v + t→0 0 Ω i
(5.40) + Dtj Gm + S 2 (Γ)m : ∇x v dx ds Z = γj+1 · v dx Ω

S1

S2

where and are the operators defined in Theorem A.26 and j Dt denotes a weak derivative on Ω × (0, T ) (see Exercise 5.5 for an alternative formulation of this hypothesis). Then the solution u of (5.1)–(5.3) described in Theorem 3.6 satisfies the following: (a)(Bochner-Sobolev space description) The mappings F (j) defined in (5.22) satisfy the hypotheses of Theorem 2.10(a)–(c); the statements in Theorem 5.5(a) concerning u(j) therefore hold but with [0, T ) replaced by (0, T ) in every instance and with the following modifications and additions: • The bound in Theorem 5.5(aii ) is replaced by m X k  X k=0 j=0

sup t2(k+δ) |u(j) (t)|2[H 2k−2j (Ω)]N

0 0, {u(0) (τ ), . . . , u(m) (τ )} satisfies the compatibility condition (5.23) required for the application of Theorem 5.5 to the initial value problem for (5.1)-(5.3) posed at initial time τ . The conclusions of Theorem 5.5 therefore hold with [0, T ) replaced by [τ, T ) in each case. 

Exercises (See also exercises 2.11 and 2.12.) 5.1 Rewrite the statements of Lemmas 5.3 and 5.4 and Theorem 5.5 for the case N = 1, n = 2 or 3 and m = 1. 5.2 Assume that A(j) and F (j) are as in Lemmas 5.3 and 5.4 with Ω = Rn and let m ≥ 1. Check that if ς0 ∈ H 2m (Rn ) then there exist ςj ∈ H 2m−2j (Rn ), j = 1, . . . , m, satisfying the compatibility conditions (5.23). 5.3 In this problem we formulate the compatibility conditions (5.23) in Theorem 5.5 solely in terms of ς0 for two additional cases. Assume that the other hypotheses and notations in Theorem 5.5 hold, that the coefficients of A are independent of t and that ς0 ∈ H2m is given. Let L (0) and L∂Ω be the operators L(0) and L∂Ω defined in (5.29) and (5.31) and let h(j) = g (j) (0) − Div G(j) (0). Prove that there exist ςj ∈ H2m−2j for which the compatibility conditions (5.23) hold in each of following two cases:

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5. PARABOLIC SYSTEMS: HIGHER ORDER REGULARITY

(a) H1 = [H01 (Ω)]N , and either m = 1 or m ≥ 2 and Lj [ς0 ] +

j−1 X

Lj−1−i [h(i) ] ∈ [H01 (Ω)]N , j = 1, . . . , m − 1.

i=0

(b) H1 = [H 1 (Ω)]N and L∂Ω Lj [ς0 ] +

j−1 X


Lj−1−i [h(i) ] = G(j) (0) + Γ(j) (0) on ∂Ω

i=0

for j = 0, . . . , m − 1 (the sum is absent if j = 0). 5.4 Let u be the solution described in Theorem 5.5. Assume that the coefficients of A are constants and show that u(j) ∈ C([0, T ); H2m−2j ) for j = 0, . . . , m provided that g (j) ∈ C([0, T ); [H 2m−2j−2 ]N ) and G(j) ∈ C([0, T ); [H 2m−2j−2 ]N ×n ) for j = 0, . . . , m − 1. 5.5 Reformulate the hypotheses on Γ in (5.40) of Theorem 5.8 by replacing S 1 (Γ) and S 2 (Γ) by appropriate sums of norms on the coordinate domains Bi of Definition 1.7 in such a way that the argument in the proof of (ciii) of that theorem remains valid. 5.6 Let u be the solution described in Theorem 5.8. Show that if j < k ≤ m λ,λ0 and 2k − 2j − |α| ≥ k0 , then tk+δ Dtj Dxα uc ∈ Cbdd 0 (Ω × [0, t¯]) for every t¯ < T and that its value at t = 0 is zero.

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10.1090/surv/251/06

CHAPTER 6

Applications to Quasilinear Systems Nonlinearity by itself cannot serve as the foundation for a useful mathematical structure. Ivar Stakgold [40]

In this chapter we apply the linear theory of Chapters 3 and 5 to study problems in which coefficients and inhomogeneities depend on the unknown. A basic theory is given in sections 6.1–6.3 for solutions with minimal regularity, including the theorems of Leray and Hopf for the Navier-Stokes equations as well as results for more general problems in which nonlinearities have specific, controlled growth at infinity. More highly regular solutions, derived by iteration from the corresponding regularity theory for linear problems, are considered in sections 6.4–6.7. This discussion includes local-in-time existence with large compatible data, global existence for data near an attracting rest point, invariant regions for parabolic systems, and a large data, global existence theorem in two and three space dimensions for certain systems with symmetry.

§6.1

A Quasilinear System in Hilbert Space

In this section we prove a basic existence theorem for an abstract problem posed in Hilbert spaces, similar to that in (2.1) but with the inhomogeneous term F depending on the unknown function. Thus (2.3) is replaced by Z t   (6.1) hu(t), viH0 = hς0 , viH0 + − A(s)(u(s), v) + F (s, u(s)) · v ds 0

where A is the bilinear map considered in Theorem 2.1 and F maps [0, T )×H1 into H1∗ . The basic result, stated in Theorem 6.2 below, is proved by an approximation and limiting procedure similar to that in the proof of Theorem 2.1, modified to accommodate the nonlinearity. The two key ingredients in the proof are an estimate, which leads to compactness of approximate solutions, and a commutation property, which allows for the interchange of weak convergence with F . These are stated in Theorem 6.2 as assumptions (hypotheses (d) and (e)) rather than as consequences of assumptions, because their verification typically requires ad hoc arguments specific to the particular application, as will be apparent in the examples discussed in sections 6.2 and 6.3. 127

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128

6. APPLICATIONS TO QUASILINEAR SYSTEMS

Before stating the main theorem we address the question of the measurability of the composition t → F (t, u(t)) in (6.1): Lemma 6.1. Let H be a separable Hilbert space, I a nontrivial interval in R and F a mapping from I × H to H∗ . Assume that F (·, w) · v is measurable on I for every fixed w and v in H and that for almost all t ∈ I, F (t, w) · v is a continuous function of w ∈ H for every fixed v ∈ H. If w : I → H is weakly measurable on I and v ∈ H, then the map → F (t, w(t)) · v is measurable on I. PROOF. Let w : I → H be weakly measurable and let wK (t) be its projection onto the first K elements of an orthonormal basis {ψk }. Then for almost all t and for all v ∈ H, F (t, wK (t)) · v → F (t, w(t)) · v as K → ∞ by the continuity assumption. It therefore suffices to assume that w maps to the span of {ψk }K 1 , say w(t) = ak (t)ψk (sum over k) where a = (a1 , . . . , aK ) is measurable in t. Let {Rih }i be cubes of side h > 0 covering RK and fix ai,h ∈ Rih for each i. Then for almost all t, X χRh (a(t))F (t, ai,h F (t, w(t)) · v = lim k ψk ) · v h→0

i

i

where χRi is the characteristic function of Rih . Each summand on the right here is measurable in t; hence so is the limit.  The following is the main result of this section: Theorem 6.2. Let T, H0 , H1 and A be as in (2.1) and assume also that A ∈ L∞ loc ([0, T ); B(H1 × H1 )), and let F be a mapping from [0, T ) × H1 into H1∗ . Assume the following: (a) For fixed w and v in H1 the mapping t 7→ F (t, w)·v is Lebesgue measurable on [0, T ). (b) There is a set E ⊂ [0, T ) of measure zero such that if t ∈ [0, T ) − E then for each fixed v ∈ H1 the mapping w 7→ F (t, w) · v is continuous from H1 into R. (c) Given t¯ ∈ [0, T ) and M > 0 there is a real-valued, locally integrable function ψ on (0, t¯) and a nonnegative number C(M, t¯) such that for w ∈ H1 satisfying |w|H0 ≤ M and for t ∈ (0, t¯) − E, X r |F (t, w)|H1∗ ≤ C(M, t¯) |w|Hi 1 + ψ(t) i

where the sum is finite and each ri is in [0, 2). (d) There is a positive nonincreasing function θ0 and locally integrable, nonnegative functions λ0 , δi and γ on (0, T ) such that for t¯ ∈ [0, T ), t ∈ (0, t¯)−E

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6. A QUASILINEAR SYSTEM IN HILBERT SPACE

129

and w ∈ H1 ,

(6.2)

−A(t)(w, w) + F (t, w) · w ≤ −θ0 (t¯)|w|2H1 + λ0 (t)|w|2H0 X + δi (t)|w|pHi0 + γ(t) i

where the sum is finite and pi > 2 for each i (compare the final item in (2.1)). (e) For all t¯ ∈ (0, T ) and v ∈ H1 , Z t¯ Z t¯ F (t, wk (t)) · v dt → F (t, w(t)) · v dt 0

0

for any sequence of weakly measurable functions from [0, t¯] into H1 whose Lebesgue equivalence classes are bounded independently of k in L∞ ((0, t¯); H0 ) and in L2 ((0, t¯); H1 ) and which converge to a weakly measurable function w in the following sense: hwk (t), ziH0 → hw(t), ziH0 uniformly on [0, t¯] for every z ∈ H0 and the (0, t¯)-equivalence class of wk converges to the equivalence class of w weakly in L2 ((0, t¯); H1 ). {wk }k

Then there is a local–in–time solution of (6.1) for given initial data as follows: (f) Given ς0 ∈ H0 , T0 ∈ (0, T ) and η > 0 there is a time τ ∈ (0, T0 ] and an element u ∈ C([0, τ ], H0weak ) whose equivalence class is in L2 ((0, τ ); H1 ), for which the equivalence class on [0, τ ] of the mapping t → F (t, u(t)) defines an element of L1loc ([0, τ ]; H1∗ ) and which satisfies (6.1) for every v ∈ H1 and t ∈ [0, τ ] as well as the bound Z τ 2 0 1 |u(t)|2H1 dt sup 2 |u(t)|H0 + θ (T0 ) (6.3)

0≤t≤τ

0

≤(1 + η)e The time τ is chosen so that

R T0

0 0 2λ (t) dt

XZ i



2 1 2 |ς0 |H0

Z +

T0

 γ(t) dt .

0

τ

δi is sufficiently small depending on

0

η and on the exponential and the ς0 and γ terms on the right side of (6.3). If δi ≡ 0 for every i then there is an element u ∈ C([0, T ), H0weak ) whose equivalence class is in L2loc ([0, T ); H1 ) and which satisfies (6.1) for every v ∈ H1 and t ∈ [0, T ) as well as the bound in (6.3) with η = 0, τ = T0 and T0 arbitrary in [0, T ); as a consequence, |u(t)−ς0 |H0 → 0 as t → 0+ . (g) If the hypothesis in (c) holds with all ri ≤ 1 and with ψ ∈ L2loc ([0, T )) then the solution u in (f) is in C([0, τ ]; H0 ) (that is, H0 with the norm topology) and the energy equality Z t Z t (6.4) 12 |u(t)|2H0 + A(s)(u(s), u(s)) ds = 21 |ς0 |2H0 + F (s, u(s)) · u(s) ds 0

0

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130

6. APPLICATIONS TO QUASILINEAR SYSTEMS

holds for all t ∈ [0, τ ]. If additionally each δi ≡ 0, then the solution in (f) is in C([0, T ); H0 ) and the above energy equality holds for all t ∈ [0, T ). (h) If the hypothesis in (c) holds with all ri ≤ 1 and with ψ ∈ L2loc ([0, T )), as in (g), and if in addition for every M > 0 there exist CM > 0, s ∈ [0, 2) and ζ square integrable on [0, τ ] such that for almost all t,
| F (t, w0 ) − F (t, w) · (w0 − w)| (6.5) 
2−s 0 ≤ CM ζ(t) + |w0 |H1 + |w|H1 |w0 − w|H0 |w − w|sH1 for w, w0 ∈ H1 with H0 –norms bounded by M , then solutions of (6.1) are unique in the class of elements of C([0, τ ]; H0w ) with equivalence classes in L2 ((0, τ ); H1 ). And if in addition δi ≡ 0 for all i, then there is a unique solution of (6.1) on all of [0, T ). PROOF. We first treat the case that the δi are not all zero. Choose T0 ∈ (0, T ) and define Z T0 R T0 2 0 λ0 (t) dt 2 1 Cλ0 (T0 ) = e , Cdata (T0 ) = 2 |ς0 | + γ(t) dt 0

and Z ∆i (t) =

t

δi (s) ds. 0

We then leave to the reader to check that, given pi > 2 and η > 0 as in the hypotheses, there is a τ ∈ (0, T0 ] as described in (a) and which will now be fixed, such that the following holds: If H is a continuous nonnegative function on [0, τ ] satisfying   X H(t) ≤ Cλ0 (T0 ) Cdata (T0 ) + ∆i (t)(2H(t))pi /2 i

and H(0) ≤ 12 |ς0 |2H0 , then H(t) ≤ (1 + η)Cλ0 (T0 )Cdata (T0 ) for all t ∈ [0, τ ]. The proof is an elementary calculus P argument comparing the relative positions of the curves y = x and y = 1 + i ∆i (t)(2x)pi /2 in the (x, y) plane for small, fixed t, together with a scaling. To construct a solution we let {ϕk }k be the orthonormal basis for H0 described in Theorem A.10(b). Thus if v ∈ H1 then the H0 –projections K X K v ≡ hv, ϕk iH0 ϕk converge to v in H1 as K → ∞ as well as in H0 , and 1

|v K |H1 ≤ 2|v|H1 for all K. Approximate solutions uK mapping a subinterval

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6. A QUASILINEAR SYSTEM IN HILBERT SPACE

131

of [0, T ) into HK ≡ span{ϕ1 , . . . , ϕK } are then to be found satisfying Z t A(s)(uK (s), v K ) ds huK (t), v K iH0 = hς0 , v K iH0 − 0 (6.6) Z t F (s, uK (s)) · v K ds + 0

for all v K ∈ HK . If uK (t) =

K X

ak (t)ϕk then the RK -valued function

1

a = (a1 , . . . , aK ) is to satisfy Z (6.7)

a(t) = a0 +

t

f (s, a(s)) ds 0

where (a0 )k = hς0 , ϕk iH0 , f ≡ (f1 , . . . , fK ) is defined by fk (t, a) = −A(t)(wa , ϕk ) + F (t, wa ) · ϕk and wa ≡

K X

ak ϕk . We will check that Theorem A.7 applies to guarantee a

1

local-in-time solution a(t). The continuity hypothesis of Theorem A.7 is obvious from assumption (b), and the measurability condition is checked exactly as in the proof of Lemma 6.1. The last requirement is that |f (t, ·)|L∞ (Vj ) is integrable on [0, T0 ] for each Vj in some increasing sequence Vj ⊂ RK of open sets covering RK . For this we can take Vj to be the ball of radius j centered at the origin and then apply hypothesis (c) and the fact that |wa | is bounded by C|a| for a constant C depending on K. The conclusions of Theorem A.7 and Corollary A.8 therefore hold and there is a solution a of (6.7), hence a solution uK of (6.6), on an interval [0, tmax ) ∈ (0, T0 ] which is maximal in the sense of Corollary A.8 and which at this point may depend on K. We will show that tmax is at least as large as the time τ defined above, which is independent of K. To do this we note that since the solution a of (6.7) is absolutely continuous, so is |uK |2H0 . Proceeding as in the proof of (2.9) in Theorem 2.1 and applying hypothesis (d), we find that, for t < tmax and t ≤ τ , Z t 2 0 1 K |uK (s)|2H1 ds 2 |u (t)|H0 +θ (T0 ) 0 (6.8)   XZ t ≤ Cλ0 (T0 ) Cdata (T0 ) + δi (s)|uK (s)|pHi0 ds . i

0

Let H(t0 ) be the sup over t ∈ [0, t0 ] of the left side here and fix t¯ < tmax and t¯ ≤ τ . Then H is continuous on [0, t¯] and satisfies the hypothesis in the statement in the first paragraph of this proof, hence its conclusion. Thus Z t 2 0 1 K (6.9) |u (t)| + θ (T ) |uK (s)|2H1 ds ≤ (1 + η)Cλ0 (T0 )Cdata (T0 ) 0 H0 2 0

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132

6. APPLICATIONS TO QUASILINEAR SYSTEMS

for t ≤ τ and t < tmax . In particular if J is chosen so that 12 |wa |2H0 > (1 + η)Cλ0 (T0 )Cdata (T0 ) for a ∈ ∂VJ , then |a(t)| cannot exceed J at any such time t ≤ τ . Corollary A.8 therefore applies to show that tmax ≥ τ as claimed. The argument is considerably simpler for the case that the δi are all zero: the calculus bound in the first paragraph of the proof is avoided and τ is not defined. Instead, we choose T0 ∈ (0, T ) arbitrarily, then define V as above in terms of T0 . Repeating the above estimates, we find that (6.8) holds with the δi terms omitted and therefore that (6.9) holds (with η = 0) for t ∈ [0, min{tmax , T0 }). Corollary A.8 then applies in the same way to show that tmax ≥ T0 and therefore that tmax = T since T0 < T was arbitrary. We will give the remainder of the proof only for the case that the δi are not all zero. The changes required for the improvements stated in the theorem when all δi are zero will be obvious. Since the bound in (6.9) is independent of K, as is τ , we can choose a sequence Ki → ∞ such that uKi converges weakly in L2 ([0, τ ]; H1 ), say to w. We claim that there is a subsequence, still denoted by Ki , and an element u ∈ C([0, τ ]; H0weak ) such that huKi (t), viH0 → hu(t), viH0 uniformly on [0, τ ] for each fixed v ∈ H1 . To see this we let t1 ≤ t2 ∈ [0, τ ], v ∈ H1 and v K its H0 –projection onto HK . Then by(6.6) and hypothesis (c), K hu (t2 ) − uK (t1 ), viH = huK (t2 ) − uK (t1 ), v K iH 0 0 Z t2
≤ |A(uK , v K )| + |F (·, uK ) · v K | t1

Z

t2

≤ const.

|uK |H1 + ψ +

X

t1


|uK |rHi 1 |v|H1

i

≤ ωv (|t2 − t1 |) where ωv is a continuous function on [0, τ ] which is independent of K and which satisfies ωv (0) = 0. (We have applied here the bound (6.9), the assumption that ri < 2 for every i and the fact that |v K |H1 ≤ 2|v|H1 .) Thus for each fixed v ∈ H1 the sequence of maps t 7→ huK (t), viH0 is equicontinuous. We can therefore apply the Ascoli–Arzela theorem together with a diagonal process to arrive at a further subsequence Ki such that, for every v in a countable dense set in H1 , huKi (t), viH0 converges uniformly on [0, τ ]. This same result then holds for all v ∈ H0 because H1 is dense in H0 and the left side above is bounded by const.|v|H0 . The limit is linear in v at each t and is therefore given by hu(t), viH0 for some u(t) ∈ H0 . The above bound then shows that u ∈ C([0, τ ]; H0weak ), and we can apply the argument following (2.11) in the proof of Theorem 2.1 to show that its Lebesgue equivalence class on [0, τ ] is w, which is an element of L2 ([0, τ, ]; H1 ). The proof that u satisfies (6.1) is exactly as in the proof of Theorem 2.1 together with a direct application of hypothesis (e) for the term involving F . Observe also that in the case that the δi are all zero, (6.3) (with η = 0) shows that

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6. A QUASILINEAR SYSTEM IN HILBERT SPACE

133

lim supt→0+ |u(t)|H0 ≤ |ς0 |H0 , so that u(t) → ς0 in the norm of H0 . This completes the proof of (f). To prove (g) we change notation and now denote by uK (t) the H0 – orthogonal projection of u(t) onto HK . Then since each hu(t), ϕk iH0 is absolutely continuous in t, we can put v = ϕk in (6.1), differentiate, sum over k, and then integrate. The result is that Z t Z t K 2 2 1 K 1 K A(u, u ) = 2 |u (0)|H0 + F (·, u) · uK . 2 |u (t)|H0 + 0

0

The equality in (g) then follows in the limit as K → ∞. For example, the Z t integrand in F (s, u(s)) · (uK (s) − u(s)) ds goes to zero almost everywhere 0

because u ∈ H1 for almost all s so that the projection uK (s) converges to u(s) strongly in H1 ; also, this integrand is bounded by C|F (s, u(s))|H1∗ |u(s)|H1 ∈ L2 ×L2 ⊂ L1 by the hypothesis of (g) and the fact that |uK (s)|H1 ≤ 2|u(s)|H1 . This integral therefore goes to zero by the dominated convergence theorem. The resulting equality in (g) shows in particular that |u(t)|2H0 is a continuous function of t and therefore that u ∈ C([0, τ ]); H0 ) in the norm topology. This proves (g). Finally to prove (h) we let u and u0 be solutions of (6.1) in the given regularity class and proceed as above in the proof of (g) to find that Z t 0 2 1 A(u0 − u, u0 − u) 2 |(u − u)(t)|H0 + 0 Z t  = F (·, u0 ) − F (·, u)] · (u0 − u). 0

Then by the hypothesis of (h), Z t 0 2 1 |(u − u)(t)| + |u0 − u|2H1 H0 2 0 Z th i2−s
≤C ζ + |u0 |H1 + |u|H1 |u0 − u|H0 |u0 − u|sH1 0

so that 0

|(u −

u)(t)|2H0

≤C

Z th 0

i
ζ 2 + |u0 |2H1 + |u|2H1 |u0 − u|2H0 .

Gronwall’s inequality now applies to show that u0 (t) = u(t) for every t.  It is of interest to know whether a solution can be extended past the time τ given in the theorem and if not to understand the nature of the obstruction. We will say that if u is a solution of (6.1) on [0, τ ] in the sense of Theorem 6.2(f), then a solution u ˜ of (6.1) with the same initial value ς0 on an interval [0, τ˜] containing [0, τ ] is an extension of u if u and u ˜ agree on

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134

6. APPLICATIONS TO QUASILINEAR SYSTEMS

[0, τ ] (thus u is an extension of itself). The maximum time of existence tmax of the solution u relative to [0, T ) is then the supremum of times τ˜ ∈ [τ, T ) for which such an extension exists on [0, τ˜]. Notice that tmax is an attribute of the particular solution u and not just of the data A, F and ς0 . See in particular Exercise 6.2 for an example in which two different solutions with the same data have different maximal times of existence. Corollary 6.3. Let T, H0 , H1 and A be as in Theorem 6.2. Assume that F satisfies hypotheses (a)–(d) of Theorem 6.2 and that u is a solution of (6.1) in the sense of Theorem 6.2(f) with maximal time of existence tmax relative to [0, T ). Then either tmax = T or Z t   lim sup |u(t)|2H0 + |u|2H1 = ∞. t→t− max

0

PROOF. If tmax < T and the above limit is finite, then |u(t)|H0 is uniformly bounded on [0, tmax ) and |u|2H1 is integrable on (0, tmax ). The first of these shows that if tj → t− max then u(tj ) has weak-H0 subsequential limits, and the second together with hypothesis (c) of Theorem 6.2 shows that Z tmax − A(u(t), v) + F (t, u(t)) · v ≤ C|v|H 1 0

for a constant C independent of v ∈ H1 . Therefore if u(tjk ) * w in H0 we can take the limit in Z tj
k hu(tjk ), viH0 = hς0 , viH0 + − A(u, v) + F (t, u) · v 0

to find that hw, viH0 is independent of the subsequence {tjk }. Since H1 is dense in H0 , all such weak sequential limits agree and therefore u(t) * w weakly in H0 as t → t− max . Thus if we define u(tmax ) = w, then the extended u weak is in C([0, tmax ]; H0 ). Theorem 6.2 can then be applied to show that there is a nontrivial interval I = [tmax , tmax + τ 0 ] and a solution z ∈ C(I; H0weak ) with equivalence class in L2 (I; H1 ) to the problem Z t hz(t), viH0 = hw, viH0 + − A(z, v) + F (t, z) · v) tmax

for v ∈ H1 and t ∈ I. Concatenating u and z we thus obtain an extension u ˜ of u to an interval strictly larger than [0, tmax ], contradicting the definition of tmax . 

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6. NAVIER–STOKES EQUATIONS OF INCOMPRESSIBLE FLOW

§6.2

135

Navier–Stokes Equations of Incompressible Flow

In this section we apply Theorem 6.2 to the Navier–Stokes system for viscous, incompressible flow, this being one of the most widely studied models in applied analysis. We present here versions of the first and most basic results in the subject, those of Leray [29] and Hopf [23] and refer the reader to other texts, particularly those of Galdi [18] and Temam [41] for more extensive presentations and for references to a very large literature. As we will see, verification of hypotheses (a)–(d) of Theorem 6.2 is nearly immediate but the commutation property (e) is more involved. Its proof depends instead on a crucial approximation result given in Theorem 6.4 below, which is an application of an observation of Friedrichs in [17] and which finds application well beyond its use here. The model is formulated as follows (but see Batchelor [4], 3.1–3.3, for example, for a more detailed discussion): let Ω be an open set in Rn , n = 2, 3, not necessarily bounded, regarded as the set occupied by a fluid, and let u(x, t) ∈ Rn be the velocity of the fluid “particle” which at time t is at position x. The so–called no–slip condition u(x, t) = 0 for x ∈ ∂Ω is imposed, and the stipulation that divx u(x, t) = 0 for x ∈ Ω expresses the law of conservation of mass under the assumption that the fluid density is constant (unity, without loss of generality). Internal forces are modeled by considering a hypothetical surface element with unit normal ν and applying elementary scaling and frame indifference considerations to describe the force exerted by the fluid immediately on one side of the surface
element on the fluid immediately on other side by µ(∇x u + ∇tr u) − pI ν, where p is the scalar x pressure and µ is a positive viscosity constant. External body forces gi are included, and standard rational mechanics arguments based on Newton’s law and the divergence theorem then yield the first equation in the following formal system for the unknown functions u and p:

(6.10)

 j P  ut + ujxk uk + pxj = µ∆uj + i gij , (x, t) ∈ Ω × (0, T ),   divx u = 0, (x, t) ∈ Ω × (0, T ),  u = 0, (x, t) ∈ ∂Ω × [0, T ),   u(x, 0) = ς0 (x), x ∈ Ω

where ς0 is the initial velocity, summation over k is understood and ∆ is the Laplace operator. weak form is derived as follows. Let H1 be the subset of  1A suitable n H0 (Ω) whose elements v satisfy divx v = 0 in the sense of weak derivatives in Ω and let H0 be the completion of H1 with respect to the L2 norm, so that H1 is dense in H0 . Then for v ∈ H1 we multiply the first equation above by v j and integrate formally with respect to x and t and apply the divergence theorem. The result is that the pressure term p drops out and

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136

6. APPLICATIONS TO QUASILINEAR SYSTEMS

that u is to satisfy Z Z u(t) · v = ς0 · v Ω Ω Z tZ (6.11) X
+ − µ∇x u · ∇x v + uj uk vxj k + gi · v 0

Ω

i

for Z t ≥ 0 and v ∈ H1 . This has the required form (6.1) with A(w, v) = µ∇x w · ∇x v and Ω Z X
(6.12) F (t, w) · v = wj wk vxj k + gi (·, t) · v . Ω

i

Once a solution u of (6.11) has been constructed, the pressure p is recovered, at least formally, as follows. Taking the xj -derivative in the first equation in (6.10) and summing over j we find that at almost every t, X divx gi . ∆p = −(ujxk uk )xj + i

Thus if n = 3 and if u(t) ∈ H1 for a particular t, then at that t and on Ω, (ujxk uk ) ∈ L2 × L6 ⊂ L3/2 ; and if each gi (t) ∈ [L3/2 (Ω)]3 we can anticipate solutions p(·, t) ∈ W 1,3/2 (Ω). Issues of the boundary condition for p, boundary regularity, uniqueness and regularity in t arise, however, and these are beyond the scope of the present discussion. See [18] Theorem 2.1 or [41] Remark 3.8 for more detailed discussions of the pressure. We begin the analysis with the following result, which will be applied to verify the commutation assumption in Theorem 6.2(e): Theorem 6.4. Let Ω be an open set in Rn , n ≥ 2, and t¯ ∈ (0, ∞). Let {wk }k be a sequence of weakly continuous mappings from [0, t¯] into L2 (Ω) and assume that for every v ∈ L2 (Ω), hwk (t), viL2 (Ω) → 0 uniformly on [0, t¯] as k → ∞ and that the [0, t¯]–equivalence classes of the wk are uniformly (in k) bounded in L2 ([0, t¯]; H 1 (Ω)). Then wk → 0 strongly in L2 ([0, t¯]; L2 (Ω0 )) for every open Ω0 ⊂ Ω whose closure is compact in Ω. PROOF. We first claim that there is a constant Cn depending on n such that if K ⊂ Rn is a cube in Rn of side h > 0 and if w ∈ H 1 (K), then Z Z Z 2 2 −n 2 (6.13) |w| dx ≤ 2h w dx + Cn h |∇x w|2 dx. K

K

K

It suffices to prove this for the restriction w to K of an element of Cc∞ (Rn ) (Theorem A.2). For such w and for x ∈ K, Z Z Z 1 −n −n w = h ∇w(y + s(x − y)) · (x − y) ds dy w(x) − h K K 0 Z Z 1 1/2 ≤ Cn h−n/2 |∇w(y + s(x − y))|2 ds dy . K

0

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6. NAVIER–STOKES EQUATIONS OF INCOMPRESSIBLE FLOW

Thus Z

w(x) dx ≤ 2h K

Z

2 w(y)dy K Z Z Z −n +Cn h

−n

2

K

Z We write

1

Z

1/2

= 0

K

1

|∇w(y + s(x − y))|2 ds dy dx.

0

1

Z +

0

137

and for the first of these reverse the s and 1/2

y integrals and make the change of variables z = y + s(x − y), so that dy |det | = (1 − s)−n ≤ Cn . For the second we write ds dy dx = dy ds dx so dz dx that |det | = s−n ≤ Cn . This proves the claim. dz Now if Ω0 is compactly contained in Ω and if h > 0 is sufficiently small, then Ω0 can be covered by a finite collection of cubes Kih of side h contained in Ω. We let χhi be h−n times the characteristic function of Kih and apply (6.13) to wk to obtain that for almost all t depending on the specific representative, Z XZ k 2 (w ) ≤ (wk )2 Ω0

Kih

i

≤C

hXZ i

wk χhi

2

+ h2

Ω

Z

|∇wk |2

i

Ω

so that Z t¯Z

k 2

(w ) ≤ C 0

Ω0

h X  Z t¯Z i

0

wk χhi

2

i + h2 .

Ω

The integrals on the right go to zero as k → ∞ because each χhi is in L2 (Ω). The right side can therefore be made arbitrarily small by choosing h small then k large depending on h.  We can now give the application of Theorem 6.2 to the Navier–Stokes system (6.11): Theorem 6.5. Let Ω be an open set in R2 or R3 , not necessarily bounded, and let H0 and H1 be the subspaces of [L2 (Ω)]n and [H01 (Ω)]n described above preceding (6.11). Assume that µ > 0 and let ς0 ∈ H0 and (finitely many) gi be given with gi ∈ L2loc ([0, ∞); [Lqi (Ω)]n ) where qi ∈ (1, 2] if n = 2 and qi ∈ [6/5, 2] if n = 3. Then: (a) There is an element u ∈ C([0, ∞); H0weak ) with [0, ∞)–equivalence class in L2loc ([0, ∞); H1 ) satisfying (6.11) for all t ≥ 0 and all v ∈ H1 ; u is strongly continuous at t = 0: u(t) → ς0 in L2 (Ω) as t → 0+ ; and given t¯ ∈ (0, ∞) there is a constant C(t¯) depending on t¯, µ and the qi if n = 2

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138

6. APPLICATIONS TO QUASILINEAR SYSTEMS

such that sup (6.14)

0≤t≤t¯

|u(t)|2[L2 (Ω)]n +

Z 0

t¯

|∇x u|2[L2 (Ω)]n dt

 ≤ C(t¯) |ς0 |2[L2 (Ω)]n +

Z t¯ X 0

 |gi (t)|2[Lqi (Ω)]n dt .

i

(b) In the case that n = 2 there is an element u ∈ C([0, ∞); H0 ) (that is, L2 (Ω) with the norm topology) with [0, ∞)–equivalence class in L2loc ([0, ∞); H1 ) satisfying (6.11) for all t ∈ [0, T ) and all v ∈ H1 . The bound in (6.14) holds, as does the energy equality Z t¯ 2 1 |∇x u|2[L2 (Ω)]2 dt 2 |u(t)|[L2 (Ω)]2 + µ 0

(6.15)

= 21 |ς0 |[L2 (Ω)]2

+

Z t¯Z X 0

Ω

u · gi dx dt.

i

The solution u is unique in the set of functions in C([0, ∞); H0weak ) with equivalence classes in L2loc ([0, ∞); H1 ) which satisfy (6.11). PROOF. We apply Theorem 6.2. First, it is obvious that the requirements on H0 , H1 and A (that is, the hypotheses of Theorem 2.1(b)) are satisfied, as is (a) of Theorem 6.2. We will check that F defined in (6.12) satisfies hypotheses (b)–(d) of Theorem 6.2 for n = 3, the case n = 2 being similar. For n = 3 the space H01 (Ω) is continuously included in L6 (Ω) (Theorem A.3(a)); thus if w ∈ H01 (we abbreviate [H01 (Ω)]n by H01 , etc.) then by interpolation, (6.16)

1/4

3/4

1/4

3/4

|w|L4 ≤ |w|L2 |w|L6 ≤ C|w|L2 |w|H 1 .

Therefore if F is as in (6.12) and w, w0 , v ∈ H01 then
| F (w0 ) − F (w) · v| ≤ (|w0 |L4 + |w|L4 )|w0 − w|L4 |v|H 1 , which goes to zero as w0 → w in H 1 as required in (b) of Theorem 6.2. Next, since the H¨ older conjugate of each qi is in [2, 6], X X
1/2 3/2 |F (t, w)|H1∗ ≤ |w|2L4 + C |gi (t)|Lqi ≤ C |w|L2 |w|H 1 + |gi (t)|Lqi , X which shows that (c) of Theorem 6.2 holds with r = 3/2 and ψ = C |gi |Lqi ∈ L2loc ([0, ∞)). For (d) we note first that if w ∈ H1 then Z Z j k j 2 k 1 (6.17) w w w xk = 2 (|w| )xk w = 0 Ω

Ω

by the zero boundary condition and the divergence–free condition. Thus Z X
−A(w, w) + F (·, w) · w = − µ|∇x w|2 + gi (·) · w Ω

≤

− 12 µ|w|2H 1

X 2 + µ|w|2L2 + C(µ) |gi (·)|Lqi ,

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6. NONLINEARITIES WITH POLYNOMIAL GROWTH

so that (d) holds with all δi = 0 and γ(·) =

X

139

|gi (·)|2Lqi .

Finally to check (e) of Theorem 6.2 we let {wk }, w and t¯ be as in the hypothesis. Then for ϕ ∈ Cc∞ (Ω) with compact support Ω0 ⊂ Ω and for fixed i and j, Z t¯Z Z t¯Z   k i k j i j (w ) (w ) − w w ϕ ≤ C(ϕ) 0

Ω

0

≤ C(ϕ, t¯) sup |wk |H0 + |w|H0 0≤t≤t¯


|wk | + |w| |wk − w|

Ω0




Z t¯ Z 0

|wk − w|2

1/2

Ω0

which goes to zero as k → ∞ by Theorem 6.4. It follows by an application of (6.16) that the double integral on the left goes to zero as k → ∞ for every ϕ ∈ L2 (Ω) and in particular for ϕ = vxi j if v ∈ H1 , as required. We have now checked that all the hypotheses (a)–(e) of Theorem 6.2 hold. The conclusions (a) of the present theorem therefore follow from Theorem 6.2(f), including the strong convergence u(t) → u(0) in L2 as t → 0+ . To prove (b) we let n = 2 and record that, if w ∈ H01 (Ω) then |w|2L4 (Ω) ≤ C(Ω)|w|L2 (Ω) |w|H 1 (Ω) , which is proved in Theorem A.4. It follows that X X |F (·, w)|H1∗ ≤ |w|2L4 + C |gi (·)|Lqi ≤ C(|w|L2 |w|H 1 + |gi (·)|Lqi ), which shows X that hypothesis (g) of Theorem 6.2 holds with r = 1 and ψ = C |gi |Lqi ∈ L2loc ([0, ∞)). To prove the hypothesis in (h) we let w, w0 ∈ H1 with H0 -norms bounded by M and compute Z

0 0 | F (·, w ) − F (·, w) · (w − w)| ≤ |w0 | + |w| |w0 − w||∇x (w0 − w)|
Ω0 0 ≤ C |w |L4 + |w|L4 |w − w|L4 |w0 − w|H 1 h i1/2 3/2 ≤ C(M ) (|w0 |H 1 + |w|H 1 )|w0 − w|L2 |w0 − w|H 1 so that the required bound in (h) of Theorem 6.2 holds with s = 3/2. The statements in (b) of the present theorem then follow from the conclusions (g) and (h) of Theorem 6.2. 

§6.3

Nonlinearities with Polynomial Growth

In this section we consider systems of parabolic equations in which the inhomogeneous terms may depend on the unknown function and are defined globally, with growth at infinity controlled in terms of the dimension of the physical space. Specifically we consider the problem (6.1) in which H0 and H1 are subspaces of [L2 (Ω)]N and [H 1 (Ω)]N respectively, the bilinear

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140

6. APPLICATIONS TO QUASILINEAR SYSTEMS

form A is exactly as in (3.2) with coefficients satisfying the hypotheses of Theorem 3.6, and for w, v ∈ H1 , Z Z
(6.18) F (t, w)·v = g(x, t, w)·v +G(x, t, w) : ∇x v dx+ Γ(x, t)·v dσ. Ω

∂Ω

Thus g and G are nonlinear functions of the unknown, but Γ is not. Then given ς0 ∈ H1 the problem is to find a mapping u on an interval [0, τ ] satisfying Z t Z t (6.19) hu(t), viH0 = hς0 , viH0 − A(s)(u(s), v)ds + F (s, u(s)) · v ds 0

0

for all v ∈ H1 . It will be convenient to regard g and G as measurable functions mapping Ω×[0, T )×RN into RN and RN ×n respectively, but to retain the formulation of Theorem 3.6 in which the solution u is an element of C([0, τ ]; H0weak ) for some τ > 0 with equivalence class in L2 ([0, τ ]; H1 ). We will show in Theorem 6.6 below that, under suitable hypotheses on g and G and for such u, (6.18) defines F as a weakly measurable, locally integrable mapping from [0, τ ] into H1∗ . The following gives the application of Theorem 6.2 to the above problem. The required “estimate” referred to in the discussion preceding Lemma 6.1 is derived as a consequence of growth conditions on g and G (hypothesis (a) below) and the “commutation property” is derived from a H¨ older-type regularity assumption (hypothesis (b) below) via Theorem 6.4. Theorem 6.6. Let n ≥ 2 and N ≥ 1, let T, Ω, H0 , H1 , A, λ and θ be as in the hypotheses of Theorem 3.6, and let Γ ∈ L2loc ([0, T ); L2 (∂Ω)) be given (this and subsequent references to Γ are omitted if Ω = Rn ). Assume the following: X X (a) g and G are finite sums gi and Gi of functions from Ω×[0, T )×RN i

i

into RN and RN ×n respectively, and there is a set E0 ⊂ Ω × [0, T ) of measure zero (and therefore a set E ⊂ [0, T ) of measure zero such that for t ∈ / E, (x, t) ∈ / E0 for almost all x ∈ Ω) such that:

(6.20)

• for (x, t) ∈ / E0 the maps w ∈ RN → gi (x, t, w) and Gi (x, t, w) are continuous; • for all w ∈ RN , the maps (x, t) ∈ Ω × [0, T ) → gi (x, t, w) and Gi (x, t, w) are measurable; • for t¯ ∈ (0, T ) there are measurable functions ϕgij and ϕG i , exponents ¯ αi and βi and a constant CF (t) such that for (x, t) ∈ Ω × [0, T ) − E0 and w ∈ RN , P ( |gi (x, t, w) − j ϕgij (x, t)| ≤ CF (t¯)|w|αi βi ¯ |Gi (x, t, w) − ϕG i (x, t)| ≤ CF (t)|w|

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6. NONLINEARITIES WITH POLYNOMIAL GROWTH

141

satisfying either one of the following sets of conditions (A) or (B) (in which the distinction between ϕgij and ϕG i and their equivalence N classes on Ω × [0, T ) is suppressed and spaces [Lp,q (t)loc (Ω × [0, T ))] are abbreviated Lp,q (t)loc ):  1 ≤ αi < 1 + 4/n,        1 ≤ βi < 1 + 2/n,    2,qij 2n (A) ϕgij ∈ L(t)loc for some qij ∈ (1, 2] if n = 2 and in [ n+2 , 2]     if n ≥ 3, and       ϕG ∈ L2,2 . i (t)loc   1 + 6/n, n ≤ 6,    1 ≤ αi <   (n + 2)/(n − 2), n ≥ 7,        1 + 4/n, n ≤ 4,    1 ≤ βi <   n/(n − 2), n ≥ 5,    g 1,qij ϕij ∈ L(t)loc for some qij ∈ (1, 2] if n = 2 (B)  2n  , 2] if n ≥ 3, and in [ n+2       1,2  ϕG  i ∈ L(t)loc , and       the bound in (6.2) holds for all w ∈ H1 and for some θ0 , λ0 , δi , and γ    as described in Theorem 6.2(d) with F as in (6.18); (b) for each component h of gi or Gi there are ε > 0, q ∈ [2−ε, (2−ε)(1+2/n)] (q ∈ [2 − ε, 2(2 − ε)) if n = 2) and a measurable function χ whose equivalence class is in Ls,s (t)loc where s = 2/(2 − ε), such that for (x, t) ∈ Ω × [0, T ) − E0 and w, w0 ∈ Rn ,   |h(x, t, w0 ) − h(x, t, w) ≤ χ(x, t) + CF (t¯)(|w0 |q + |w|q ) |w0 − w|ε . Then: (c) Given ς0 ∈ H0 , Γ ∈ L2loc ([0, T ); ∂Ω), T0 ∈ (0, T ) and η > 0, there is a τ ∈ (0, T0 ] and an element u ∈ C([0, τ ], H0weak ) with equivalence class in L2 ([0, τ ]; H1 ) satisfying (6.19) for all v ∈ H1 and t ∈ [0, τ ], in which the equivalence class of F (·, u(·)) is in L1 ([0, τ ]; H1∗ ). In addition, • the bound in (6.3) holds in case (B) with θ0 , λ0 , δi , and γ as in the final assumption in (B), and in case (A) with θ0 = 12 θ(τ ), λ0 = λ(τ ) + C1 , δi ≤ C1 , where C1 is a constant depending on n, Ω, CF (τ ), the exponents αi and βi and the number of summands comprising g and G, and

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142

6. APPLICATIONS TO QUASILINEAR SYSTEMS

hX γ(t) = C(Ω, n) |ϕgij (·, t)|2[Lqij (Ω)]N ij

+

X

i 2 2 |ϕG i (·, t)|[L2 (Ω)]N ×n + |Γ(·, t)|[L2 (∂Ω)]N .

i

• If the δi referred to in the previous item are known to be zero, then there is an element u ∈ C([0, T ), H0weak ) whose equivalence class is in L2loc ([0, T ); H1 ) and which satisfies (6.19) for every v ∈ H1 and t ∈ [0, T ) as well as the bound in (6.3) with η = 0, τ = T0 and T0 arbitrary in [0, T ). • More generally, if tmax is the maximal time of existence of any such solution u relative to [0, T ) as defined above preceding the statement of Corollary 6.3, then either tmax = T or Z t h i 2 |u(s)|2[H 1 (Ω)]N ds = ∞. lim sup |u(t)|[L2 (Ω)]N + t→t− max

0

• If um is any specific representative of the measurable counterpart of such a solution u, then g(x, t, um (x, t)) and G(x, t, um (x, t)) are measurable and locally integrable on Ω × (0, τ ), and if Cc∞ (Ω) ⊂ H1 , then the equation

∂ ∂ ∂ η um = aη,ω cωm um m Dxη um − bm Dxη um + ∂t ∂xω ∂xω (6.21) − dm um + g(·, ·, um ) − Div G(·, ·, um ) holds in the sense of distributions on Ω × (0, τ ), each term here being either locally integrable on Ω × (0, τ ) or the distribution derivative ∂ ∂ , , or Div of a locally integrable function. ∂t ∂xω (d) In case (A) there is a solution as described above in (c) which is in C([0, τ ]; [L2 (Ω)]N ) (that is, L2 with the norm topology) and which satisfies the energy equality (6.4) for all t ∈ [0, τ ]. If all δi = 0 and the hypotheses of case (A) hold, then there is a solution u ∈ C([0, T ); H0 ) satisfying the energy equality for all t ∈ [0, T ). (e) Assume in addition to the hypotheses of case (A) that for t¯ ∈ [0, T ) there is a constant C2 (t¯) such that for (x, t) ∈ / E0 with t ≤ t¯ and for w, w0 ∈ RN , |g(x, t, w0 ) − g(x, t, w)| and |G(x, t, w0 ) − G(x, t, w)| are bounded by finite sums of terms of the form C2 (t¯)ξ(x, t)|w0 − w| where (the equivalence 1 ,q2 1 ,2q2 class of) ξ is in Lq(t)loc for g and in L2q (t)loc for G, where q1 and q2 satisfy n/2 < q2 ≤ ∞ and

(p∗ − 2)q20 ≤ q1 ≤ ∞ p∗ − 2q20

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6. NONLINEARITIES WITH POLYNOMIAL GROWTH

143

for n ≥ 3 (q 0 is the H¨ older conjugate of q2 ) and p∗ = 2n/(n − 2)); and for n = 2, q2 ∈ (1, ∞] and q1 ∈ (q20 , ∞]. Then solutions of (6.19) in C([0, τ ]; H0weak ) having equivalence classes in L2 ([0, τ ]; H1 ) are uniquely determined by Γ and ς0 . For the proof we will need the following consequences of the imbedding theorem in Theorem A.3(a), the latter showing in particular that H 1 (Ω) is continuously included in Lp (Ω) for p ∈ [2, ∞) if n = 2 and p ∈ [2, p∗ ] for n ≥ 3: Lemma 6.7. Let Ω be an open set in Rn , n ≥ 2, satisfying the cone condition. (a) If n ≥ 3 and p ∈ [2, 2 + 4/n] or if n = 2 and p ∈ [2, 4) then there is an r ∈ [0, 2] depending on p such that for all w ∈ H 1 (Ω), Z |w|p ≤ C|w|p−r |w|rH 1 (Ω) . L2 (Ω) Ω

The constant C depends on Ω and n if n ≥ 3 and on Ω and p if n = 2. (b) Assume that p satisfies ( 1≤p
n/2 and let T ∈ (0, ∞]; then given ς0 ∈ H2m we construct u : [0, τ ] ⊆ [0, T ) → H2m satisfying

(6.27)

hu(t), vi[L2 (Ω)]N = hς0 , vi[L2 (Ω)]N Z tZ  + − (aη,ω (x, t, u)uxη ) · vxω + g(x, t, u, ∇x u) · v 0 Ω  + G(x, t, u) : ∇x v dx dt

for all v ∈ H1 and t ≤ τ < T . We will assume that there are neighborhoods W and W 0 of zero in RN and RN ×n respectively such that aη,ω and G map Ω × [0, T ) × W into RN ×N and RN ×n respectively and that g maps Ω × [0, T ) × W × W 0 into RN . (A quasilinear boundary term may also be included in (6.27); see Remark 1 below following the statement of Theorem 6.10.) High–order derivatives of aη,ω , g and G will occur, and these are polynomials in the derivatives of the unknown u, complicating the verification of the regularity hypotheses of Theorem 5.5 as well as the description of compatibility conditions. We consider these issues first, then state the main result in Theorem 6.10. Derivatives of Compositions. The following lemma will facilitate the formulation of the main theorem and will also constitute a significant part of its proof: 0,1 Lemma 6.8. Let Ω be a Cbdd domain in Rn , n ≥ 2, not necessarily bounded (Definition 1.7) and I a nontrivial compact interval in R, and let N ≥ 1 and m > n/2. Assume that mappings w(0) = w, w(1) , . . . , w(m) are given taking values in [L2 (Ω)]N and satisfying the following, in which [H k (Ω)]N is abbreviated H k , etc.: 2m−2j w(j) ∈ Lip(I; H 2m−2j−2 ) ∩ wAC 2 (I; H 2m−2j−1 ) ∩ C(I; Hweak )

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6. HIGHER ORDER REGULARITY FOR QUASILINEAR SYSTEMS

149

for j ≤ m − 1, w(m) ∈ L∞ (I; L2 ) ∩ L2 (I; H 1 ), and the derivative of w(j) is the equivalence class over I of w(j+1) for j ≤ m − 1. Assume also that m X (6.28) sup |w(j) (t)|2H 2m−2j ≤ M1 j=0 t∈I

and m Z X

(6.29)

j=0

I

|w(j) |2H 2m−2j+1 ≤ M2 .

(a) There is then a bounded continuous function wc ∈ C(Ω × I) such that wc (·, t) is a member of the Ω–equivalence class w(t) for each t and the calculus derivatives Dxα Dtj wc exist on Ω × I ◦ for 2j + |α| ≤ 2 and have bounded, H¨ older continuous extensions to Ω × I. The images of wc and ∇x w are therefore contained in compact sets K and K 0 respectively. Now assume further that K and K 0 contain the origins in RN and RN ×n respectively. The following results then hold for compositions involving involving w and ∇x w: (b) Let W be a neighborhood of K and assume that a real-valued mapping 2m+1 f ∈ Cbdd (Ω × I ◦ × W ) is given whose derivatives have extensions to Ω × I × K which are Lipschitz with respect to the variable in K. Let f˜(t) be the Ω-equivalence class f˜(·, t, w(t)) as w(t) ranges over its Ωequivalence class. Then there are maps f˜(0) , . . . , f˜(m) such that f˜(0) = f˜, the equivalence class over I of f˜(j) is in L∞ (I; H 2m−2j )∩L2 (I; H 2m−2j+1 ) for j ≤ m, and for j ≤ m − 1, f˜(j) ∈ Lip(I; H 2m−2j−2 ) ∩ wAC 2 (I; H 2m−2j−1 ) ∩ C(I; H 2m−2j ) weak

with derivative the equivalence class over I of f˜(j+1) . In addition m m X X (j) 2 ˜ sup |f (t)|H 2m−2j ≤ sup |(Dtj f )(·, t, 0)|2H 2m−2j (6.30) j=0 t∈I j=0 t∈I + C(M1 + · · · + M12m ) and m Z X

(6.31)

j=0

I

|f˜(j) (t)|2H 2m−2j+1 ≤

m Z X j=0

I

|(Dtj f )(·, t, 0)|2H 2m−2j+1 dt

+ C(1 + M1 + · · · + M12m )M2 where the sums on the right are assumed to be bounded and the constant C depends on the assumed bounds and Lipschitz constants for the derivatives of f on Ω × I × K and on the norm of the imbedding from H m into L∞ (Theorem A.3(a)). Finally, for j ≤ m, f˜(j) is a polynomial in w(1) , . . . , w(j) whose coefficients are multiples of partial derivatives of f evaluated at (·, ·, w(·)), these coefficients being continuous and bounded on

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Ω×I, and in which the corresponding monomials are in L∞ (I; H 2m−2j )∩ L2 (I; H 2m−2j+1 ), modulo equivalence classes over I. (c) Next let W 0 be a neighborhood of K 0 in RN ×n and assume that a real2m (Ω×I ◦ ×W ×W 0 ) is given whose derivatives have valued mapping h ∈ Cbdd continuous extensions to Ω × I × K × K 0 which are bounded, continuous, ˜ and Lipschitz with respect to the variable in K × K 0 . Let h(t) be the Ω-equivalence class h(·, t, w(t), ∇x w(t)) as (w(t), ∇x w(t)) ranges over ˜ (0) , . . . , h ˜ (m) such that its Ω-equivalence class. Then there are maps h (0) (j) 2 ˜ ˜ ˜ h = h, the equivalence class over I of h is in L (I; H 2m−2j ) for j ≤ m, and for j ≤ m − 1, ˜ (j) ∈ wAC 2 (I; H 2m−2j−2 ) ∩ C(I; H 2m−2j−1 ) h weak

with derivative the equivalence class of m−1 X

(6.32)

j=0

˜ (j) (t)|2 2m−2j−1 ≤ sup |h H I

m−1 X

h(j+1) .

In addition,

sup |(Dtj h)(·, t, 0, 0)|2H 2m−2j−1

j=0 t∈I

+ C(M1 + · · · + M12m−1 ) and m Z X

(6.33)

j=0

I

˜ (j) (t)|2 2m−2j ≤ |h H

m Z X j=0

I

|(Dtj h)(·, t, 0, 0)|2H 2m−2j dt + C(1 + M1 + · · · + M12m−1 )M2

where the constant C depends on the assumed bounds and Lipschitz constants for the derivatives of h on Ω × I ◦ × K × K 0 and on the norms of the imbeddings from H m into L∞ and H m−1 into L4 (Theorem A.3(a)). Fi˜ (j) is a polynomial in (w(1) , ∇x w(1) ), . . . , (w(j) , ∇x w(j) ) nally, for j ≤ m, h whose coefficients are multiples of partial derivatives of h evaluated at (·, ·, w, ∇x w), these coefficients being continuous and bounded on Ω × I, and in which the corresponding monomials are in L∞ (I; H 2m−2j−1 ) ∩ L2 (I; H 2m−2j ). PROOF. The H¨older continuity and boundedness statements in (a) follow exactly as in the proof of Theorem 5.5(ci) and the fact that m > n/2, which implies that, in the notation of the latter theorem, 2j0 + κ ≥ 2. We will give the proof of (c), the proof of (b) being similar and somewhat easier. It will be convenient to work at the level of measurable counterparts, applying Theorems A.23 and A.24 and Corollary A.25 to transfer information between the Bochner space statements of the theorem and their weak derivative formulations. ˜ m (·, ·) is the Ω × I equivalence class The reader can easily check that h h(·, ·, wm , ∇x wm ) as wm ranges over its equivalence class. We can therefore fix specific representatives and apply the chain rule ([19], Lemma 7.5) to check by induction that

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151

(wm ,∇x wm )

˜ m (·, ·) = Dj Dα h (·, ·, (0, 0)) + (Dj Dα h (·, ·, (·)) Dtj Dxα h t x t x (0,0) (6.34) X

j1 α1 +σ1
jl αl +σl wm + ∗ Dt Dx wm · · · D t D x
where j1 + · · · + jl ≤ j, α1 + · · · + αl ≤ α, σi ∈ {0, 1}, and ∗ denotes a partial derivative of h evaluated at (·, ·, (wm , ∇x wm )) which by hypothesis is bounded a.e. in Ω × I provided 2j + |α| ≤ 2m (the sum is absent if j = |α| = 0). We need to bound the terms on the right in L∞,2 for j ≤ m − 1 and |α| ≤ 2m − 2j − 1, and in L2,2 for j ≤ m and |α| ≤ 2m − 2j. Bounds for the first term on the right are recorded in (6.32) and (6.33), and the second term is bounded pointwise by C(|wm | + |∇x wm |) by the assumed Lipschitz continuity, and is therefore bounded in L∞,2 and L2,2 as required. Concerning the product Π in the sum on the right in (6.34) we first note that if 2ji + |αi | + σi ≤ m, then the corresponding factor Dtji Dxαi +σi wm is 1/2 in L∞,∞ because in this case Dtji Dxαi +σi wm (·, t) ∈ H m with bound M1 1/2 for almost all t by (6.28), and therefore is in L∞ with bound CM1 by Theorem A.3(a). Now to prove (6.32) we suppose that 2j + |α| ≤ 2m − 1 and observe that there is then at most one i for which 2ji + |αi | + σi ≥ m + 1, so that Π ∈ L∞,∞ · · · L∞,2 ⊆ L∞,2 with the bound C(M1 + . . . + M12m−1 )1/2 , proving (6.32). To prove (6.33) we suppose that 2j + |α| ≤ 2m. If the condition 2ji + |αi | + σi ≤ m fails for at most one value of i, then at most one factor in Π fails to be in L∞,∞ , and therefore Π ∈ L∞,∞ · · · L2,2 ⊆ L2,2 with the bound in (6.33). Otherwise l = 2 and 2ji + |αi | + σi = m + 1 for i = 1, 2. In this case Dtji Dxαi +σi wm (·, t) ∈ H m−1 ⊂ L4 for almost all t with bound 1/2 CM1 , and |Π|2L2,2 ≤ CM1 M2 . This proves (6.33), and the corresponding bounds in (6.30) and (6.31) in (b) are proved in a similar way. The final statements in (b) and (c) regarding f (j) and h(j) are easily deduced from Theorems A.23 and A.24 as in the proof of Corollary A.25.  Compatibility Conditions. If solutions of (6.27) are to be obtained in the regularity class of Theorem 5.5 then the compatibility conditions(5.23) must be satisfied. It is important that these be formulated solely in terms of the ςj appearing in (5.23), independent of properties of a putative solution u. To develop a suitable notation we first illustrate the assertion in (a) above concerning polynomial structure, say of f˜(2) : f˜(2) = ftt + 2ftw · w(1) + (fww w(1) ) · w(1) + fw · w(2) where subscripts denote partial derivatives evaluated at (·, ·, w), fw and ftw are N × 1 and fww is N × N . The expression on the right is indeed a polynomial in w(1) , w(2) and will be denoted by f2 (·, t, w(t))[w(1) , w(2) ]. More generally f˜(j) (t) will be denoted fj (·, t, w(t))[w(1) (t), . . . , w(j) (t)], so that fj is a mapping from Ω × I × W into the set of polynomials of degree j over RN

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(the brackets are omitted if j = 0). Thus whereas the notation f˜(j) suppresses the dependence on w, the more satisfactory notation fj (·, ·, w)[w1) , . . . , w(j) ] makes clear that w is an argument of fj and that the mapping fj itself is independent of w. Applying this notation to the components of aη,ω and G we obtain matrices denoted (aη,ω )j and Gj whose components are mappings from [0, T ) × Ω × W into the set of polynomials of degree j on RN , and to the components of g to obtain N × 1 arrays gj of mappings from [0, T ) × Ω × W × W 0 into the set of polynomials of degree j on RN × RN ×n . Now suppose that u is a solution of (6.27) in the sense of Theorem 5.5 on an interval [0, τ ]. That is, u = u(0) , . . . , u(m) satisfy the regularity conditions stated in Theorem 5.5(a) and for v ∈ H1 and t ∈ [0, τ ], Z tZ hu(t), viL2 (Ω) = hς0 , viL2 (Ω) + 0



−(˜ aη,ω uxη ) · vxω

Ω

 ˜ : ∇x v dx ds + g˜ · v + G ˜ (we have replaced where a ˜η,ω (x, s) = a(x, s, u(x, s)) and similarly for g˜ and G the mapping w in Lemma 6.8 by u). The compatibility conditions (5.23) must then be satisfied: if ςj ≡ u(j) (0) then for j ≤ m − 1 and v ∈ H1 , Z h X j  
j hςj+1 , viL2 = − (˜ aη,ω )(i) (x, 0)(ςj−i )xη · vxω i Ω i=0 i ˜ (j) (x, 0) : ∇v dx. + g˜(j) (x, 0) · v + G In light of the above discussion these conditions can be be written Z h X j  
j hςj+1 , viL2 = − aη,ω i (·, 0, ς0 )[ς1 , . . . , ςi ](ςj−i )xη · vxω i Ω i=0   (6.35) + gj (·, 0, ς0 , ∇x ς0 )[(ς1 , ∇x ς1 ), . . . , (ςj , ∇ςj )] · v   i + Gj (·, 0, ς0 )[ς1 , . . . , ςj ] : ∇x v dx (again, the brackets are omitted if i or j is zero) which, while somewhat more complicated, involve only the initial values ς0 , . . . , ςm and the partial derivatives of aη,ω , g and G evaluated at ς0 or (ς0 , ∇ς0 ), as intended. It will be convenient in what follows to adopt the following terminology: Definition 6.9. Fix an ensemble (5.4) as in Definition 5.1 and a nontrivial interval I in R. A mapping w : I → H2m is m-regular on I with respect to the spaces H0 , . . . , H2m+1 if there are mappings w(0) = w, . . . , w(m) satisfying the following four conditions: for j = 0, . . . , m − 1, weak w(j) ∈ Lip(I; H2m−2j−2 ) ∩ wAC 2 (I; H2m−2j−1 ) ∩ C(I; H2m−2j )

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6. HIGHER ORDER REGULARITY FOR QUASILINEAR SYSTEMS

153

and w(m) ∈ C(I; H0 ); for j = 0, . . . , 2m the equivalence class over I of w(j) is in L2 (I; H2m−2j+1 ); the derivative of w(j) for j < m is the equivalence class over I of w(j+1) ; and (w(m) )∗ ≡ hw(m) , ·iH0 ∈ wAC 2 (I; H1∗ ). For example, the solution described in Theorem 5.5 is m-regular on [0, t¯] for every t¯ ∈ (0, T ) with respect to the particular spaces Hj = H1 ∩ [H j (Ω)]N , the final condition being a consequence (5.25) with j = m. Observe that the functions considered in Lemma 6.8 satisfy all but this final condition and thus narrowly fail to be m-regular. We can now state the main result of this section: Theorem 6.10. Fix an ensemble (5.4) and constant CBC as in Definition 5.1 with m > n/2, let W and W 0 be neighborhoods of the origins in RN and RN ×n respectively and fix t¯ > 0. Assume the following: 2m+1 (a) Mappings aη,ω and G in Cbdd (Ω × (0, t¯) × W ) are given taking values N ×N N ×n in R and R respectively with derivatives having continuous extensions to Ω × [0, t¯] × K for every compact K ⊂ W which are bounded, continuous and Lipschitz with respect to the variable in K. And a mapping 2m (Ω × (0, t¯) × W × W 0 ) is given taking values in RN whose g in Cbdd derivatives have continuous extensions Ω × [0, t¯] × K × K 0 for all compact K ⊂ W and compact K 0 ⊂ W 0 which are bounded, continuous and Lipschitz with respect to the variable in K × K 0 . The derivatives and Lipschitz constants included here are assumed to be bounded by a constant Csys (K, K 0 ). (b) The following regularity conditions are satisfied (in which j ≥ 0 in each instance and Lp,2 denotes [Lp,2 (Ω × [0, t¯])]r×s for various p, r and s):

(6.36)

Dtj Dxα gm (·, ·, 0, 0)

(6.37)

Dtj Dxα Gm (·, ·, 0)



L∞,2 , j ≤ m − 1, 2j + |α| ≤ 2m − 1, L2,2 , j ≤ m, 2j + |α| ≤ 2m;



L∞,2 , j ≤ m, 2j + |α| ≤ 2m, L2,2 , j ≤ m, 0 ≤ 2j + |α| ≤ 2m + 1;



L∞,2 , j ≤ m, 2j + |α| ≤ 2m, L2,2 , j ≤ m, 0 ≤ 2j + |α| ≤ 2m + 1.

∈

∈

and for (j, α) 6= (0, 0),

(6.38)

Dtj Dxα aη,ω m (·, ·, 0)

∈

The norms of the derivatives in (6.36)–(6.38), although independent of K and K 0 , are assumed to be bounded by the aforementioned Csys (K, K 0 ) for any compact subsets K and K 0 of W and W 0 . (c) There is a positive number θ such that for all (x, t, w) ∈ Ω × [0, t¯] × W , ω,η (6.39) (M tr aη,ω ≥ 2θ(t¯)|M |2 m (x, t, w)M )

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154

6. APPLICATIONS TO QUASILINEAR SYSTEMS

for all N × n matrices M . (d) Initial data (ς0 , . . . , ςm ) ∈ H2m × . . . × H0 is given satisfying the compatibility conditions (6.35), and the continuous counterparts of ς0 and ∇x ς0 (see Lemma 6.8(a)) take values in compact sets K and K 0 a positive distance ε from ∂W and ∂W 0 respectively (see also Remark 2 below concerning this hypothesis). (j) (e) There is an m-regular function udata : [0, t¯] → H2m such that: udata (0) = ςj for j ≤ m, (udata , ∇x udata ) takes values in W × W 0 on Ω × [0, t¯], and |udata (t) − ς0 |2H 2m ≤ δ for t ∈ [0, t¯] where δ > 0 is chosen so that if z ∈ H 2m with |z|2[H 2m (Ω)]N ≤ δ then |z|[L∞ (Ω)]N , |∇x z|[L∞ (Ω)]N ×n ≤ 21 ε (see Remark 3 below concerning this hypothesis).

There is then an m-regular solution u to the problem (6.27) on an interval [0, τ ], described in terms of constants Csolve and Ctime as follows: • Csolve and Ctime depend on n, N, m and on upper bounds for CBC , 0 ), where K Csys (Kε/2 , Kε/2 ε/2 is the closure of K + Bε/2 (0) and simX 0 , and on upper bounds for θ −1 , ε−1 and ilarly for Kε/2 |ςj |2H2m−2j j

(Csolve and Ctime are otherwise independent of the particular solution u and its data); and

(6.40)

n δ − Ctime |ςm − ζ|2L2 o −1 τ ≥ min t¯, Ctime , sup . |ζ|2H1 ζ∈H1 • u is m-regular on [0, τ ] with respect to the spaces H0 , . . . , H2m+1 ; the compositions g(·, ·, u, ∇x u), G(·, ·, u) and aη,ω (·, ·, u) satisfy the hypotheses of Theorem 5.5; and u is a solution of (6.27) on [0, τ ] and satisfyies the conclusions in (a)–(c) of Theorem 5.5 with Z

 η,ω A(t)(z, v) = (a (x, t, u)zxη ) · vxω dx Ω Z   F (t) · v = g(x, t, u, ∇x u) · v + G(x, t, u) : ∇x v dx Ω
L(t)[z] = aη,ω (x, t, u(x, t))zxη xω and if Ω 6= Rn ,   L∂ (t)[z] = ν ω T aη,ω (·, t, u(·, t))zxη where ν is the unit outer normal on ∂Ω and T is the trace operator.

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155

• The following bounds are saisfied: Z τ m   X sup |u(j) (t)|2H 2m−2j + |u(j) (t)|2H 2m−2j+1 dt (6.41)

j=0

0≤t≤τ

≤ Csolve

0

m hX

|ςj |2H 2m−2j +

m−1 X

|γjg |2H 2m−2j−1 + |γjG |2H 2m−2j


i

0

j=0

  where γjg = gj (·, 0, ς0 , ∇x ς0 ) (ς1 , ∇x ς1 ), . . .], γjG = Gj (·, 0, ς0 ) ς1 , . . .] (see (6.35)) and [H k (Ω)]N is abbreviated H k etc. • For j = 0, . . . , m, u(j) ∈ C([0, τ ]; H2m−2j ), that is, H2m−2j with the norm topology. The solution u is unique in the following sense: if u0 ∈ C([0, τ 0 ]; H0 ) with equivalence class in L2 ([0, τ 0 ]; H1 ) and with (u0m , ∇x u0m ) taking values in W × W 0 a.e., and if (6.27) holds for all v ∈ H1 and t ∈ [0, τ 0 ] with u replaced by u0 , then u0 = u on their common domain (see also Remark 4 below). Remarks: Z tZ f (x, t, u) · v dσdt may be included in

(1) A quasilinear boundary term 0

∂Ω

(6.27) under suitable hypotheses on f : the proof requires that if w is in the regularity class under consideration and if S 1 and S 2 are the functionals defined in Theorem A.26, then S 1 (f (·, ·, w)) and S 2 (f (·, ·, w)) should satisfy the same regularity conditions as those satisfied respectively by g(·, ·, w, ∇x w) and G(·, ·, w). This allows for absorption of the boundary integral into the integral over Ω, exactly as in the proofs of Theorems 4.2 and 5.5. It suffices that f have derivatives up to order 2m + 1 on a neighborhood of ∂Ω × [0, t¯] × W which are bounded, continuous and Lipschitz with respect to the variable in W . Details and proofs are left to the interested reader. (2) If either W or W 0 is the entire space, then the corresponding condition in (d) on the distance from the image of ς0 or ∇x ς0 to ∂W or ∂W 0 is omitted. If both, then the dependence of constants on ε−1 is omitted as is the sup on the right side of (6.40). (3) Hypothesis (e) is obviously necessary for the validity of the theorem since the solution itself, if one exists, satisfies (e). We list here four important cases in which a suitable udata is known to exist; a construction for the general case of the theorem remains elusive, however: • Ω = Rn : the proof is given in Exercise 6.4; • ςj ∈ H2m+1 for all j: in this case the obvious Taylor polynomial in t satisfies the required conditions;

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• there is a time-independent bilinear form A on H1 ×H1 satisfying the hypotheses of Theorem 5.5 with corresponding differential operators L and L∂Ω as defined in section 3.1 such that Li ςj and L∂Ω Li ςj satisfy essential and natural boundary conditions for certain i and j; the construction of udata for this case is by induction on m and is outlined in Exercise 6.5; • n = 2 or 3, m = 2, aη,ω = aη,ω (x, t), g = g(x, t, u), and G is absent: the proof, which is outlined Exercise 6.6, is somewhat lengthy and depends on the relative strength of the Sobolev imbeddings H k → Lp in low dimensions. (4) Concerning the final statement of the theorem, we leave to the reader to check that the given hypotheses on u0 are sufficient to insure that the terms in (6.27) are well-defined with u0 in place of u. PROOF. We give the proof for the case that the boundaries of W and W 0 are nonempty, the minor modifications required for the other cases being obvious. Throughout the proof we abbreviate [H k (Ω)]N by H k , etc., and Lp ([0, τ ]; H k ) by Lp (H k ) when the value of τ is clear from context. Let C0 be the term in brackets on the right side of (6.41): C0 =

m X j=0

|ςj |2H 2m−2j +

m−1 X


|γjg |2H 2m−2j−1 + |γjG |2H 2m−2j ,

0

and define M1 = 2(C0 + δ). Note that C0 and δ are bounded in terms of the same quantities as Csolve and Ctime by Lemma 6.8 and hypothesis (b); the same is therefore true for M1 . Step 1. Preliminary bounds. Let τ ∈ (0, t¯] and w = w(0) , . . . , w(m) be as in the hypotheses of Lemma 6.8 on I = [0, τ ] with w(j) (0) = ςj for j = 0, . . . , m. Assume that (6.28) holds with the above M1 , that (6.29) holds with some finite M2 and that the continuous representatives of w and 0 ∇x w take values in Kε/2 and Kε/2 respectively. Define a ˜η,ω (t) = a(·, t, w(t)), ˜ = G(·, t, w(t)) for t ∈ I. Then: g˜(t) = g(·, t, w(t)), ∇x w(t)) and G(t) There are maps (˜ aη,ω )(0) , . . . , (˜ aη,ω )(m) with (˜ aη,ω )(0) (t) = aη,ω (·, t, w(t)) and for which the associated bilinear forms A˜(0) , . . . , A˜(m) defined by Z
A˜(j) (t)(z, v) = (˜ aη,ω )(j) (t)zxη · vxω Ω

satisfy the hypotheses of Theorem 2.7 and therefore of Theorem 5.5 on I. The proof consists in first applying Lemma 6.8(b) to a(·, ·, w)−a(·, ·, 0) to show that for 0 < 2j + |α| ≤ 2m, Dxα a ˜(j) (·) is bounded in L∞ ([0, τ ]); L2 ) (modulo equivalence class) by a constant depending on the same quantities as for

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6. HIGHER ORDER REGULARITY FOR QUASILINEAR SYSTEMS

157

0 ) and on M , and then applying these bounds in Lemma 5.3. Csys (Kε/2 , Kε/2 1 The details are straightforward and left to the reader. It follows that solutions of corresponding linear problems (2.3) as described in Theorem 2.7 satisfy the R t¯
bound in (2.28) and that the multiplier C(t¯) exp 0 β 2 , which we denote by 12 Csolve , is as described in the statement of the present theorem and does not depend on M2 . Now define 0 M2 = Csolve (Kε/2 , Kε/2 )C0

and assume that w also satisfies (6.29) with this value of M2 . We claim that: ˜ (0) , . . . , G ˜ (m) with g˜(0) (t) = There are mappings g˜(0) , . . . , g˜(m) and G (0) ˜ g(·, t, w(t), ∇x w(t)) and G (t) = G(·, t, w(t)) and for which the associated functionals F˜ (0) , . . . , F˜ (m) defined by F˜ (j) (t) · v =

Z

˜ (j) (t) : ∇x v g˜(j) (t) · v + G




Ω

satisfy the hypotheses of Theorem 2.7 and therefore of Theorem 5.5. In particular,

m−1 X

m X

|˜ g (j) (t)|2L∞ (H 2m−2j−1 ) +

j=0

(6.42) +

˜ (j) (t)|2 ∞ 2m−2j |G L (H )

j=0 m  X

(j)

|˜ g

(t)|2L2 (H 2m−2j )

˜ (j) (t)|2 2 2m−2j+1 + |G L (H )



j=0

≤ C, and as a consequence m−1 X

(6.43)

|F˜ (j) |2L∞ (V2m−2j−2 ) +

j=0

m X

|F˜ (j) |2L2 (V2m−2j−1 )

j=0

≤

m−1 X


|γjg |2H 2m−2j−1 + |γjG |2H 2m−2j + Cτ 1/2 ,

0

where γjg and γjG are as defined following (6.41) and C is a generic positive constant which depends on the same quantities as Csolve . ˜ (j) and For the proof we first note that the statements concerning g˜(j) and G the bounds in (6.42) are immediate from Lemma 6.8 and that for (6.43) it

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6. APPLICATIONS TO QUASILINEAR SYSTEMS

suffices to prove that m−1 X

˜ (j) |2 ∞ 2m−2j−1 + |˜ |˜ g (j) |2L∞ (H 2m−2j−2 ) + |G g (j) |2L2 (H 2m−2j−1 ) L (H )



0

+ |(g (m) )∗ |2L2 (H∗ ) +

m X

1

˜ (j) |2 2 2m−2j |G L (H )

0

≤

m−1 X


|γjg |2H 2m−2j−1 + |γjG |2H 2m−2j + Cτ 1/2 .

0

We label the terms on the left (i )-(v ). First, (6.42) shows that (iii ) and (v ) are bounded by Cτ . To bound (iv ) we fix v ∈ H1 and t ∈ [0, τ ] and observe that the highest order termZ in the expansion (corresponding to
k(x, t, w(t), ∇x w(t)) · ∇x w(m) (t) v (6.34)) of h˜ g (m) (t), viL2 has the form Ω

where k is a partial derivative of g of order one. We apply the divergence theorem and the fact that k and Dx2 w(t) are bounded in L∞ (Ω) (since Dx2 w(t) ∈ H 2m−2 ⊆ H m ⊂ L∞ ) to bound this integral by   C |w(m) (t)|L2 (∂Ω) + |w(m) (t)|L2 (Ω) |v|H1   1/2 1/2 ≤ C |w(m) (t)|L2 (Ω) |w(m) (t)|H 1 (Ω) + |w(m) (t)|L2 (Ω) |v|H1 (we have applied the trace theorem for W 1,1 integrands to |w(m) |2 ). Thus 1/2 |(˜ g (m) )∗ (t)|H1∗ ≤ C(1+|w(m) (t)|H1 ) and (iv ) is bounded by Cτ 1/2 . We record ˜ from (6.42): this result for later reference, including a bound for G (6.44)

˜ (m) |2 2 2 ≤ Cτ 1/2 . |(˜ g (m) )∗ |2L2 (H∗ ) , |G L (L ) 1

Next we examine term (i ). By considering actions on test functions we find that for j ≤ m − 1 and |α| ≤ 2m − 2j − 2 and for all t ∈ [0, τ ], |Dxα g˜(j) (t)|L2 ≤ |gj (·, 0, ς0 , ∇x ς0 )[ς1 , . . . ,ςj ]|H 2m−2j−2 Z t + |Dxα g˜(j+1) |L2 . 0

The integral here is bounded by over α we then obtain

Cτ 1/2

by (6.42). Squaring and summing

|˜ g (j) (t)|2H 2m−2j−2 ≤ |γjg |2H 2m−2j−2 + Cτ ˜ (j) in term (ii ). where γjg is as in (6.41). A similar argument applies to G Summing over j we then find that terms (i )–(v ) together are bounded by the right side of (6.43) and therefore that (6.42) holds as claimed. Step 2: Construction of an invariant set. Now fix τ ∈ (0, t¯] and let Sτ denote the set of mappings w : [0, τ ] → H2m which are m-regular on [0, τ ] with respect to the spaces H0 , . . . , H2m+1 and which satisfy the following:

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6. HIGHER ORDER REGULARITY FOR QUASILINEAR SYSTEMS

(6.45)

w(j) (0) = ςj , j = 0, . . . , m,

(6.46)

0 on Ω for all t ∈ [0, τ ], (w(t), ∇wx (t)) ∈ Kε/2 × Kε/2

(6.47)

m X

159

sup |w(j) (t) − ςj |2H 2m−2j (Ω) ≤ δ

j=0 t∈[0,τ ]

and

(6.48)

m Z X j=0

0

τ

|w(j) (t)|2H 2m−2j+1 (Ω) dt ≤ M2 .

We will show that for τ sufficiently small, Sτ is invariant for simple iteration applied to (6.27) as described below in Step 3. Observe first that, by the choice of δ, (6.47) implies (6.46), which is therefore redundant. Also, the triangle inequality, (6.47) and the definitions of C0 and M1 show that if w ∈ Sτ then (6.49)

m X

sup |w(j) (t)|2H 2m−2j ≤ M1 .

j=0 t∈[0,τ ]

Finally, by reducing t¯ if necessary, we may assume that udata ∈ Sτ for every τ ≤ t¯, so that all such Sτ are nonempty. ˜ Now let w ∈ Sτ , define a ˜η,ω (t), g˜(t)) and G(t) as in Step 1 and consider the following linear problem for an unknown u : [0, τ ] → H2m : Z tZ  η,ω
hu(t), viL2 = hς0 , viL2 + a ˜ (s)uxη (s) · vxω (6.50) 0 Ω  ˜ + g˜(s) · v + G(s) : ∇x v dx ds for all v ∈ H1 and t ∈ [0, τ ]. We have already checked in Step 1 that a ˜, ˜ satisfy the regularity conditions required for the application of g˜ and G Theorems 2.7 and 5.5, and the compatibility conditions (5.23) hold because w satisfies (6.45). Theorem 5.5 therefore applies to show that there is a unique m-regular solution u defined on [0, τ ]. We will prove that if τ is sufficiently small consistent with (6.40) then u ∈ Sτ . First, Theorem 5.5 shows that u satisfies (6.45), and (6.46) follows from (6.47); we therefore need only check (6.47) and (6.48). For the latter we recall the definitions of Csolve and F˜ (j) from Step 1 and apply (2.28) and (6.43) to obtain that

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160

6. APPLICATIONS TO QUASILINEAR SYSTEMS

m  X j=0

(6.51)

(j)

sup |u [0,τ ]

(t)|2H 2m−2j

≤ 12 Csolve

m hX

τ

Z

|u(j) |2H 2m−2j+1

+ 0

|ςj |2H 2m−2j +

j=0

m−1 X



|F (j) |2L∞ (V2m−2j−2 )

j=0

+

m X

|F (j) |2L2 (V2m−2j−1 )

i

j=0

≤

1 2 Csolve

C0 + Cτ

1/2




≤ Csolve C0 =M2

if τ is sufficiently small, consistent with (6.40). In particular, u satisfies (6.48). There remains to prove that u satisfies (6.47). To do this we let βj (t) = (j) u (t) − ςj and apply the strong form (5.26) of (6.50) to obtain that for t ≤ τ , j ≤ m − 1 and all v ∈ H1 , j   X j ˜(j−i) (6.52) hβj+1 , viL2 + A (t)(βi , v) = Hj · v i i=0

where the A˜(i) are as in Step 1 and j   h X j ˜(j−i) A (·)(ςi , v) Hj (t) · v = − i i=0 (6.53) Z + Ω


i t ˜ (j) (·) : ∇x v . g˜(j) (·) · v + G 0

We checked in Step 1 that the A˜(i) satisfy the hypotheses of Theorems 2.7 and 5.5 and therefore also of Lemma 2.9. Thus from (2.32), for each t, (6.54)

m−1 X

m−1   X |βj |2H 2m−2j ≤ C |βm |2L2 + |Hj |2V2m−2j−2 .

j=0

j=0

The contribution to |Hj |V2m−2j−2 in (6.53) from A˜(j−i) is bounded as follows. If t ≤ τ and v ∈ H1 then Z tZ t
(j−i) ˜ (6.55) A (t)(ςi , v) 0 = (˜ a(j−i+1 )η,ω (s))(ςi )xη ) · vxω ds, 0

so that t (j−i) (t)(ςi , ·) 0 A

Ω

Z V2m−2j−2

≤C

τ

|˜ a(j−i+1 )η,ω (s))(ςi )xη )|H 2m−2j−1

0

(recall Definition 5.1). We checked in Step 1 that the bounds (6.30) and (6.31) apply to the components of a ˜ − a(·, ·, 0), so that in particular a ˜j−i+1 ∈ L2 (H 2m−2j+2i−1 ) with norm bounded by C; and ςj−i ∈ H 2m−2j by hypothesis (and is constant in time). Their product is therefore in L2 ([0, τ ]; H 2m−2j−1 )

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6. HIGHER ORDER REGULARITY FOR QUASILINEAR SYSTEMS

161

by Theorem A.3(d), and the contribution to |Hj |V2m−2j−2 of the term in question is therefore bounded by Cτ 1/2 . Applying (6.42) in a similar way ˜ (j) terms in (6.53) and substituting into (6.54), we to bound the g˜(j) and G then obtain that at each t, m−1 X

(6.56)


|βj |2H 2m−2j ≤ C |βm |2L2 + τ .

j=0

To bound βm we apply (5.25) of Theorem 5.5 to write Z t
(m) hu (t), viL2 = hςm , viL2 + − A˜(0) (u(m) , v) + Φ · v ds 0

where Φ(s) · v = −

m−1 X i=0

 m ˜(m−i) A (s)(u(i) , v) i Z
˜ (m) (s) : ∇x v . + g˜(m) (s) · v + G Ω

Now if ζ ∈ H1 is an L2 -approximation of u(m) then hu(m) (t) − ζ, viL2 = hςm − ζ, viL2 Z t   + − A˜(0) (u(m) − ζ, v) + Φ(s) · v − A˜(0) (ζ, v) ds 0

so that by Theorem 2.1, sup |u

(m)

t∈[0,τ ]

(t) −

ζ|2L2



≤ C |ςm −

ζ|2L2

Z + 0

τ

 |Φ|2H1∗ + τ |ζ|2H1 .

˜ (m) are bounded The contributions to the integral on the right from g˜(m) and G 1/2 by Cτ by (6.44), and an argument similar to that given above for A˜(j−i) following (6.55) shows that the contribution from A˜(m−i) is bounded by Cτ . Triangulating βm = u(m) − ςm via ζ, we then obtain from (6.54) that m X

  sup |βj (t)|2H 2m−2j ≤ C |ςm − ζ|2L2 + τ |ζ|2H1 + τ 1/2 ,

j=0 t≤τ

which is less than δ for appropriate choice of ζ and for τ small consistent with (6.40) (observe that the j = m term is included on the left). This proves that u satisfies the condition in (6.47) and completes the proof that u ∈ Sτ . We have now established that for all positive τ sufficiently small consistent with (6.40), the function w 7→ u defined by (6.50) maps Sτ to Sτ .

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162

6. APPLICATIONS TO QUASILINEAR SYSTEMS

Step 3: Iteration. Now let τ be as above and define a sequence {wk } by taking w0 = udata ∈ Sτ and wk+1 ∈ Sτ the solution of Z tZ h
k+1 k+1 hw (t), viL2 = hς0 , viL2 + a ˜η,ω k (s)wxη (s) · vxω (6.57) 0 Ω  ˜ k (s) : ∇x v dx ds + g˜k (s) · v + G η,ω (·, ·, w k ) as in Step 2, and similarly for g ˜k. where a ˜η,ω ˜k and G k (·, ·) = a k+1 k Subtracting the equations satisfied by w and w we find that their difference ∆wk+1 satisfies Z tZ  k+1 ) · vxω + ∆gk · v h∆w (t), viL2 = − (aη,ω (x, t, wk )∆wxk+1 η 0 Ω  + ∆Gk : ∇x v − (∆(˜ ak )η,ω wxkη ) · vxω dx dt

for v ∈ H1 . The bounds in Theorem 2.1(b) (or Theorem 3.6) together with the assumed Lipschitz continuity of a, g, and G and the fact that ∇x wk ∈ L∞ ([0, τ ]; L∞ ) then show that Z tZ k+1 2 |∆w (t)|L2 + |∇x ∆wk+1 |2 0 Ω Z tZ h i
≤C |∆wk | + |∇x ∆wk | |∆wk+1 | + |∆wk ||∇x ∆wk+1 | . 0

Ω

Applying Gronwall’s inequality we find that if Z τ E k+1 = sup |wk+1 (t) − wk (t)|2L2 + |wk+1 (t) − wk (t)|2H 1 0

t∈[0,τ ]

and if τ is sufficiently small consistent with (6.40) then E k+1 ≤ λE k for some λ ∈ (0, 1). The sequence wk therefore converges in the norm implied by E. That is, wk (t) converges in L2 (Ω), say to w(t), uniformly for t ∈ [0, τ ], so that w ∈ C([0, τ ]; L2 ); and the equivalence classes over [0, τ ] of the wk converge in L2 ([0, τ ]; H1 ) to that of w. It follows from the uniform bound in (6.48), translated to the measurable counterparts of the wk , that for 2j + |α| ≤ 2m + 1 the weak derivatives Dtj Dxα wm of the measurable counterpart of w exist on Ω × (0, τ ) and inherit the bounds X |Dtj Dxα wm |2L∞,2 ≤ M1 , 2j+|α|≤2m

X (6.58)


|Dxα Dtj wm − ςj |2L∞,2 ≤ δ,

2j+|α|≤2m

X

|Dtj Dxα wm |2L2,2 ≤ M2

2j+|α|≤2m+1

where Lp,2 = [Lp,2 (Ω × [0, τ ])]N . For example, to prove the second of these we fix a particular representative wm and let {ϕi }i ⊂ Cc∞ (Ω) be countable and

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6. HIGHER ORDER REGULARITY FOR QUASILINEAR SYSTEMS

163

dense in ZL2 (Ω). There is then a null set E such if t ∈ / E then t is a Lebesgue
j point of Dxα Dt wm (·, t) − ςj ϕi for all j and α with 2j + |α| ≤ 2m and all Ω

i. Thus if ψ h (t) is a smooth nonnegative function supported on an interval of length h in [0, τ ] containing t and integrating to one, then the latter integral is equal to Z Z ± lim wm Dxα Dtj (ϕi ψ h ) − Dxα ςj · ϕi h→0 Ω×(0,τ ) Ω Z Z
α j k h = lim lim Dx Dt wm (ϕi ψ ) − Dxα ςj · ϕi h→0 k→∞ Ω×(0,τ ) Z τZ

= lim lim

h→0 k→∞ 0

Ω

  Dxα (wk )(j) − ςj ϕi ψ h .

Ω

ϕj,α

Therefore if elements ∈ {ϕi }i are chosen for each j and α, then Z X
Dxα Dtj wm (·, t) − ςj ϕj,α 2j+|α|≤2m Ω

τ

Z

= lim lim

X

≤ limk→∞

|(w )

h→0 k→∞

2j+|α|≤2m 0 m X k (j)

Z Ω

  Dxα (wk )(j) − ςj ϕj,α ψ h

− ςj |2L∞ (H 2m−2j )

1/2  X

≤ δ 1/2

|ϕj,α |2L2

1/2

j,α

j=0

X

|ϕj,α |2L2

1/2

j,α

by (6.47). This proves the second bound in (6.58); the others are proved in a similar way. The bounds in (6.58) are exactly the conditions stated in Corollary A.25(a); there are therefore mappings w(0) = w, . . . , w(m) satisfying (b) of that corolweak ) with |w(·) − ς |2 lary. In particular, w ∈ C([0, τ ], H2m 0 H2m ≤ δ on [0, τ ], 0 so that (w, ∇x w) takes values in K × K . Thus w satisfies the hypotheses in Step 1 and therefore its conclusions: if a ˜η,ω (·, ·) = aη,ω (·, ·, w) and simi˜ then a ˜ satisfy the regularity hypotheses of larly for g˜ and G, ˜η,ω , g˜, and G Theorem 5.5. We can also infer that Z tZ  η,ω
hw(t), viL2 = hς0 , viL2 + a ˜ (s)wxη (s) · vxω (6.59) 0 Ω  ˜ + g˜(s) · v + G(s) : ∇x v dx ds for all v ∈ H1 . The proof consists in taking limk→∞ in (6.57), the details being nearly identical to those leading to the bound for E k+1 above. In particular w is a solution of (6.27), but at this point only in the sense of the basic existence result Theorem 2.1 (or Theorem 3.6). To prove that w is m-regular we have to check the compatibility conditions w(j) (0) = ςj for

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164

6. APPLICATIONS TO QUASILINEAR SYSTEMS

j = 0, . . . , m − 1. These hold because (wk )(j) (0) = ςj for all k and w(j) and 1/2 (wk )(j) are in Cbdd ([0, τ ]; H1 ) with modulus independent of k, which follows from (6.58). Thus both the compatibility and the regularity hypotheses of Theorem 5.5 are satisfied and we can therefore conclude that w is m-regular, w ∈ Sτ and that the other descriptions given in Theorem 5.5 and listed in the statement of the present theorem hold (with u = w). The penultimate statement of the theorem, that u(j) ∈ C([0, τ ]; H2m−2j ), is proved by checking the hypotheses of Theorem 2.7(g). For example, the requirement that F˜ (j) ∈ C([0, τ ]; V2m−2j−2 ) for j ≤ m − 1 follows by ˜ respectively to obtain applying (6.33) and (6.31) of Lemma 6.8 to g˜ and G (j+1) 2 (j+1) 2 2m−2j−2 ˜ that g˜ ∈ L ([0, τ ]; H ) and G ∈ L ([0, τ ]; H 2m−2j−1 ) so that (j) 2 2m−2j−2 (j) ˜ ∈ wAC 2 (([0, τ ]; H 2m−2j−1 ), which g˜ ∈ wAC ([0, τ ]; H ) and G are somewhat stronger than is required. The hypothesis on a ˜η,ω is proved in a similar way and the ellipticity condition follows because the hypotheses of Theorem 5.5, which we have checked in Step 1, include those of Theorem 4.2. Finally, to prove uniqueness we let u0 be as in the statement and subtract the equations satisfied by u and u0 to obtain that, with the obvious meanings, Z tZ h∆u(t), viL2 = − (aη,ω (x, t, u0 )∆uxη ) · vxω + ∆g · v 0 Ω
+ ∆G : ∇x v − (∆aη,ω uxη ) · vxω dx dt for v ∈ H1 and t in the common domain. The bounds in Theorem 2.1(b) (or Theorem 3.6) together with the assumed Lipschitz continuity of a, g and G and the fact that ∇x u ∈ L∞ ([0, τ ]; L∞ ) then apply to show that Z tZ Z tZ
2 2 |∇x ∆u| ≤ C |∆u(t)|L2 + |∆u|2 + |∆u||∆ux | 0

Ω

0

Ω

from which we conclude via Gronwall’s inequality that u0 = u on the common domain.  We conclude this section with a result describing the maximal time of existence of a solution of (6.27). Recall Corollary 6.3, which applies to problems for which uniqueness may fail and for which the maximal existence time is a property of a particular solution, not just of the data. The opposite is true here, although the proof is similar: the solution is shown to extend past a time for which the stated blowup criteria fail. The main point of interest in the present case is the derivation of a time-independent lower bound away from zero for the approximation term involving ζ in (6.40). Corollary 6.11. Fix an ensemble (5.4) as in Definition 5.1 with m > n/2, let T ∈ (0, ∞] and assume that sets W and W 0 and mappings a, g, and G are given satisfying hypotheses (a), (b) and (c) of Theorem 6.10 on [0, t¯] for every t¯ ∈ (0, T ), and finally that hypotheses (d) and (e) concerning initial data hold. Let tmax be the supremum of the set of times τ ∈ (0, T ) for

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6. HIGHER ORDER REGULARITY FOR QUASILINEAR SYSTEMS

165

which a corresponding m-regular solution u of (6.27) exists on [0, τ ] with (u, ∇x u) taking values in W × W 0 . Then at least one of the following is true: tmax = T , Z t m   X sup |u(j) (s)|2H 2m−2j + |u(j) |2H 2m−2j+1 = ∞, lim t→t− max j=0

0≤s≤t

0

or lim dist(I(u(t)), ∂W ) ∧ dist(I(∇x u(t)), ∂W 0 )

t→t− max

exists and is zero (I here denotes the image of Ω under the mapping u(t) or ∇x u(t); the term involving ∂W or ∂W 0 is omitted if W or W 0 is the whole space, and the third alternative is omitted altogether if both.) PROOF. If all three alternatives fail then tmax ∈ (0, T ), there is a number M < ∞ such that the sum in the second alternative is bounded by M for every t < tmax , and there is a sequence of times ti → t− max such that the minimum in the third alternative is bounded below by a positive number ε for every t = ti (because the negation of the third alternative is that this term has at least one nonzero limit point). We will show that Theorem 6.10 guarantees a positive existence time τ which is independent of i for each of the problems (6.27) posed at initial time ti with initial data (u(0) (ti ), . . . , u(m) (ti )). These problems will then be solvable on [ti , ti + τ ], and this together with the uniqueness result in Theorem 6.10 will then yield a solution on [0, tmax + τ ), thus contradicting the maximality of tmax . First note that the above bounds in terms of M and ε show that there are compact subsets K and K 0 of W and W 0 such that the continuous representatives of u(0) (ti ) and ∇x u(0) (ti ) take values in K and K 0 for all i. Therefore if we choose t¯ ∈ (tmax , T ) then t¯, M and ε determine the constants Csys and Ctime (see their descriptions preceding (6.40)) independently of i. We thus conclude from (6.40) that there is a positive existence time τ which applies to each of the problems posed at initial time ti provided that for every i there is an element ζi ∈ H1 with (6.60)

Ctime |u(m) (ti ) − ζi |2L2 ≤ δ/2 and |ζi |2H 1 ≤ M 0

for some constant M 0 . To construct ζi we note that u(m) is m-regular on [0, t] for every t < tmax and consequently (u(m) )∗ is in wAC 2 ([0, tmax ); H1∗ ) with derivative in L2loc ([0, tmax ); H1∗ ). The action of this derivative on an element v ∈ H1 is given by the right side of (5.26) with j = m − 1, applied to the problem (6.27). The bounds associated with the constant M can then be applied to show that the derivative of (u(m) )∗ is in fact in L2 ([0, tmax ]; H1∗ ). We denote this derivative by hz(·), ·iH1 and conclude that z ∈ L2 ([0, tmax ]; H1 ). Now let {ϕk } be an L2 -orthonormal basis as described in Theorem A.10: each ϕk is in H1 and if v ∈ H1 then its L2 -projection onto the first J elements

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166

6. APPLICATIONS TO QUASILINEAR SYSTEMS

is bounded in the H1 norm by 2|v|H1 and converges to v in H1 as J → ∞. The relation Z ti (m) hu (ti ), ϕk iL2 = hςm , ϕk iL2 + hz(t), ϕk iH1 dt 0

then shows that Z ∞ ∞ X X (m) 2 2 hu (ti ), ϕk iL2 ≤ 2 hςm , ϕk iL2 + 2tmax J+1

J+1

0

∞ tmax X

hz(t), ϕk i2H1 dt.

J+1

The integrand on the right goes to zero for almost all t as J → ∞ and is dominated by 2|z(t)|2H1 , which is integrable. This integral therefore goes to zero. We can therefore choose J sufficiently large that the first inequality in (6.60) holds with ζi the L2 -orthogonal projection of u(m) (ti ) onto {ϕ1 , . . . , ϕJ }; the second will then hold for a constant M 0 determined by J. 

§6.5

Global Existence and Stability of Steady-States

In this section we consider m-regular solutions of (6.27) with initial data close to that of a known global solution when the functions g and G enforce a certain dissipativity. A typical example is that in which the known solution is a steady-state, that is, a time-independent solution of the associated differential equations and boundary conditions. The main result is given in Theorem 6.12 in which u ≡ 0 is a steady-state and the dissipativity assumptions guarantee that solutions corresponding to nearly initial data converge to zero as t → ∞ (if T = ∞). More general stability problems can then by treated by applying Theorem 6.12 to the difference between a known global solution and a solution with nearby initial data with the conclusion that the latter is also global and coalesces with the former as t → ∞. A steady-state for which this holds is therefore an “attracting rest point” or a “stable steady-state ” of the underlying differential equations and boundary conditions. We will give only a brief, informal discussion (below following the proof of Theorem 6.12) of the application to these more general settings, the details typically being specific to the specific problem as in the example in Exercise 6.9. To motivate the analysis let us suppose that g and G are as in (6.27) and that g(x, t, 0, 0) = 0 and G(x, t, 0) = 0 for all x and t so that u ≡ 0 is a solution of (6.27) with ς0 = 0. Then letting P ∈ RN ×n denote the variable in W 0 we can write (6.61)

g(x, t, w, P ) = glin (x, t, w, P ) + gquad (x, t, w, P )

where glin consists of the linear terms in the Taylor expansion of g(x, t, ·, ·) about (w, P ) = (0, 0) and gquad is the quadratic remainder term. Specifically, (6.62)

k k i k il glin (x, y, w, P ) = gw i (x, t, 0, 0)w + gP il (x, t, 0, 0)P ,

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6. GLOBAL EXISTENCE AND STABILITY OF STEADY-STATES

167

where subscripts denote partial derivatives and summation over repeated indices is understood, and h Z 1Z 1 i k k i l (6.63) gquad (x, t, w, P ) = gw (x, t, s s w, s s P )ds ds i wl 1 2 1 2 1 2 w w +. . . 0

0

2m+2 and similarly for G, Glin , and Gquad . Observe that if g ∈ Cbdd on a given 2m+1 2m+3 2m domain, then glin ∈ Cbdd and gquad ∈ Cbdd ; similarly, if G ∈ Cbdd then 2m+2 2m+1 Glin ∈ Cbdd and Gquad ∈ Cbdd . In particular, gquad and Gquad will satisfy the regularity conditions in Lemma 6.8(c) and (b) respectively. We will impose sign conditions on the derivatives defining glin and Glin sufficient to insure that the bilinear form Z   (6.64) B(t)(z, v) ≡ glin (x, t, z, ∇x z) · v + Glin (x, t, z) : ∇x v dx Ω

(which is bilinear in (z, v)) is dissipative in a certain sense, and then to combine B with the bilinear form Z A(t, u)(z, v) ≡ (aη,ω (·, t, u)zxη ) · vxω Ω

to write (6.27) in the form t  − A(s, u)(u, v) + B(s)(u, v) ds hu(t), viL2 = hς0 , viL2 + 0 Z tZ   + gquad (x, s, u, ∇x u) · v + Gquad (x, t, u) : ∇x v dx ds

Z

(6.65)

0

Ω

in which the −A + B term is strongly dissipative and the gquad and Gquad terms are of secondary importance. An explicit expression for B is given below in (6.68) followed by a brief discussion of the required conditions on the partials of g and G. These ideas are formalized in the following theorem in which hypotheses (a) and (b) are routine regularity conditions, (c) guarantees that u ≡ 0 is a steady-state solution, (d) is the important dissipativity hypothesis and (e) is the assumption that a, g and G vary slowly in time relative to the dissipation. Observe that the assertion of global existence of solutions with small, consistent initial data is made only subject to hypothesis (e) of Theorem 6.10 (recall the discussion in Remark 3 following the statement of that theorem). Theorem 6.12. Fix an ensemble (5.4) as in Definition 5.1 with m > n/2, let T ∈ (0, ∞] and assume the following: (a) Mappings aη,ω are given satisfying the hypotheses in Theorem 6.10(a) for all t¯ < T and with W replaced by the ball Br0 , and there is a positive θ such that (6.39) holds for all t ∈ (0, T ). (b) Mappings g and G are given satisfying the hypotheses in Theorem 6.10(a) for all t¯ with W and W 0 replaced by balls Br0 and Bs0 and m replaced by m + 1.

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The derivatives and Lipschitz constants included in these hypotheses are assumed to be bounded by a constant Csys which is independent of t¯, and Br0 and Bs0 are the balls of positive radii r0 and s0 centered at the respective origins. (c) g(x, t, 0, 0) = 0 and G(x, t, 0) = 0 for all (x, t) ∈ Ω × [0, T ). (d) There are constants θ0 ∈ (0, θ) and γ > θ − θ0 such that if B is the bilinear form defined in (6.64) then B(t)(z, z) ≤ θ0 |∇x z|2[L2 (Ω)]N ×n − γ|z|2[L2 (Ω)]N . (e) The number ζ defined by X j ζ 1/2 ≡ sup |Dt a(η,ω) (x, t, 0)| + sup|Dtj g(x, t, 0, 0)| η,ω

+ sup |Dtj G(x, t, 0)|, where each sup is over (x, t) ∈ Ω × [0, T ) and j = 1, . . . , m, satisfies 2ζ < 2λ ≡ γ − (θ − θ0 ). Then given ε ∈ (0, r0 ∧ s0 ) there are positive numbers κ−1 and Csolve , upper bounds for which depend on n, m, N and on upper bounds for CBC , Csys , λ−1 , ε−1 and θ−θ0 )−1 such that the following holds: Given initial data (ς0 , . . . , ςm ) ∈ H2m × . . . H0 for which • the compatibility conditions (6.35) are satisfied • (ς0 , ∇x ς0 ) takes values in Br0 −ε × Bs0 −ε m X • |ςj |2H 2m−2j (Ω) ≤ κ j=0

• hypothesis (e) of Theorem 6.10 holds for some t¯ > 0, the corresponding m-regular solution of (6.27) described in Theorem 6.10 has maximal existence time tmax = T relative to Br0 × Bs0 in the sense of Corollary 6.11. That is, there is an m-regular solution u of (6.27) with these data satisfying the conclusions of Theorem 6.10 on [0, τ ] for every τ < T . In particular, the bound in (6.41) holds with Csolve replaced by Csolve e−λτ . Thus if T = ∞, then u(j) → 0 in H 2m−2j (Ω) as t → ∞ for all j = 0 . . . , m. PROOF. It will be convenient to denote by kwk2τ the squared norm on the left side of (6.41) with w in place of u. The first step is to reexamine R t¯ 2 the multiplier C(t¯)e 0 β on the right side of (2.28) in the application of Theorem 2.7 to the bilinear form A(·, w) − B(·): Step 1: Dissipation Estimate. We claim the following: there is a constant M0 ∈ (0, 1] such that if w is m-regular on an interval [0, t¯] ⊂ (0, T ) with ˜ ∈ kwk2t¯ ≤ M0 , then (w, ∇x w) takes values in Br0 − 1 ε × Bs0 − 1 ε and if A(t) 2

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B(H1 × H1 ) is the bilinear form Z ˜ (aη,ω (x, t, w)zxη ) · vxω dx − B(t)(z, v) A(t)(z, v) = Ω

where B is as in (6.64), then A˜ satisfies the regularity hypotheses of Lemma 5.3 and therefore of Theorem 5.5. In addition there is a constant Csolve which is independent of t¯ such that if z is an m-regular solution of (2.3) with A and u replaced by A˜ and z and with inhomogeneous term F and compatible 0 ) then initial data (ς00 , . . . , ςm m hX

¯

kzk2t¯ ≤ Csolve e−λt (6.66)

|ςj0 |2H2m−2j +

j=0

m−1 X

sup |F (j) (t)|2V2m−2j−2

¯ j=0 0≤t≤t m Z t¯ X

+

j=0

0

i |F (j) |2V2m−2j−1 .

The first statement above holds because ∇x w(t) ∈ H 2m−1 ⊂ H m ⊂ L∞ 1/2 with norm bounded by CM0 , which can be taken arbitrarily small, and Rt 2 similarly for w. For the second we examine the multiplier C(t¯)e 0 β (s)ds on the right side of (2.28) in Theorem 2.7, first applied to the shifted bilinear form A˜λ = A˜ − 2λh·, ·iL2 . Note that the hypothesis in (d) of the present theorem and the definition of λ show that A˜λ satisfies the boundedness and coerciveness conditions in (a) of Theorem 2.7, which is therefore applicable. The description in (f) of that theorem shows that in the present case C(t¯) is independent of t¯ and therefore may be designated Csolve . To estimate β, which is defined in (ci ) of Theorem 2.7, we fix i ∈ {1, . . . , m} and (i) (z, v) ∈ H2i × H1 and bound |A˜λ (z, v)| under the assumption that kwkτ is (i) small. The dominant term in A˜λ (z, v) is seen to be Z
Dti aη,ω (x, t, w)zxη · vxω Ω Z h w i = (Dti aη,ω )(x, t, 0)zxη + (Dti aη,ω )(x, t, ·) 0 zxη · vxω Ω XZ
+ O |w(i1 ) | · · · |w(il ) ||∇x z||∇x v| Ω

where i1 + · · · + il = i is as in (6.34). The contribution of the first term on the
P right is bounded by supΩ×[0,T )) η,ω |Dtj a(η,ω) (x, t, 0)| |z|H1 |v|H1 and the second can be absorbed into the third by the assumed Lipschitz continuity of Dti aη,ω . The summand in the third term is bounded by Z 1/2 |Π|2 |∇x z|2 |v|H1 Ω

where Π is the product of the |w(ij ) |. Consider two cases: if 2i ≥ m + 1 then |∇x z|L∞ ≤ |∇x z|H m ≤ C|∇x z|H 2i−1 ; it follows that |Π|L2 ≤ Ckwkt¯, as in the

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derivation of (6.30) from (6.28), because we can take kwkt¯ ≤ 1. The above term is therefore bounded in this case by Ckwkt¯ |z|H 2i |v|H 1 . In the second case 2i ≤ m and therefore each 2ij ≤ m and w(ij ) ∈ H 2m−2ij ⊆ H m ⊂ L∞ , and the same bound holds. Treating the Dti glin and Dti Glin terms in the (i) integral representing A˜λ (z, v) in a similar way we conclude that (i)

|A˜λ (z, v)| ≤ (ζ 1/2 + Ckwkt¯)|z|H 2i |v|H1 and therefore that β ≤ (ζ 1/2 + Ckwkt¯). Consequently if kwk2t¯ ≤ M0 then the Rt 2 ¯ multiplier C(t)e 0 β (s)ds on the right side of (2.28), applied to A˜λ , is bounded by Csolve e(ζ+CM0 )t and the corresponding multiplier for A˜ is bounded by Csolve e(ζ+CM0 −2λ)t (Exercise 2.1), which is bounded by Csolve e−λt if M0 is chosen sufficiently small. This proves the claim. Step 2: Estimates for gquad and Gquad . We claim the following: The functions gquad and Gquad defined in (6.63) satisfy the hypotheses in Lemma 6.8(c) and (b) respectively with W × W 0 replaced by Br0 × Bs0 and with m as in the present theorem. And if w is as in Step 1 and F˜ : [0, T ) → H1∗ is defined by Z
˜ F (t) · v = gquad (x, t, w, ∇x w) · v + G(x, t, w)quad : ∇x v Ω

then F˜ satisfies the regularity hypotheses (2.26) and (2.27) of Theorem 2.7 and there is a constant CF independent of t ∈ (0, T ) and w such that (6.67)

m−1 X j=0

|F˜ (j) |2L∞ ((0,t);V2m−2j−2 ) +

m X

|F˜ (j) |2L2 ((0,t);V2m−2j−1 ) ≤ CF kwk4t .

j=0

The regularity statements here are obvious from (6.63) for gquad and the corresponding representation for Gquad . To prove (6.67) we apply Lemma 6.8(c) with h = gquad , with M1 and M2 in (6.30) and (6.31) bounded by kwk2t ≤ 1 and with l ≥ 2 in the sum in (6.34). The proof for Gquad is similar. These observations prove the claim. We now complete the proof of the theorem as follows. Fix the constants Csolve ≥ 1 and CF from Steps 1 and 2 and choose κ > 0 so that 2 (4Csolve CF )κ < 1 and the roots s± of Csolve CF s2 − s + Csolve κ are real and positive with s− < M0 . It then follows that κ < s− and that if s1 ∈ (s− , s+ ∧ M0 ) then the graph of the line y = s in the s–y plane lies strictly below that of y = Csolve (CF s2 + κ) for s ∈ (0, s− ) and strictly above for s ∈ (s− , s1 ]. Now let u be the solution of (6.27) of Theorem 6.10 with initial data as described in the hypotheses of the present theorem and assume that the maximal time of existence tmax of u with respect to Br0 × Bs0 is strictly less than T . Let s(t) ≡ kuk2t , which is increasing and continuous in t by the penultimate conclusion of Theorem 6.10. Then s(0) ≤ κ by hypothesis and so s(0) ≤ κ < s− < s1 . There is therefore a first time t1 < T such that s(t) < s1 on [0, t1 ) and s(t1 ) = s1 (otherwise s(t) < s1 ≤ M0 for

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t ∈ [0, tmax ) and by Step 1 all three alternatives in Corollary 6.11 would be precluded, contradicting the maximality of tmax ). The bounds (6.66) and (6.67) therefore apply on [0, s1 ] and show that s1 = kuk2t1 ≤ Csolve (κ + CF s21 ), which violates the last conclusion in the preceding paragraph. This proves that tmax = T .  Before considering the application of Theorem 6.12 to more general problems we examine the dissipativity assumption in (d) of the theorem. We have that for z ∈ H 1 (Ω), Z h  B(t)(z, z) = gw (x, t, 0, 0)z) · z) + gPk il (x, t, 0, 0) Ω (6.68)  i + Gilwk (x, t, 0) zxi l z k where gw is the N × N Jacobian matrix of g with respect to w, superscripts denote components and subscripts denote partial derivatives. Thus (d) holds if gw (·, ·, 0, 0) is uniformly negative definite on Ω × [0, T ), say (gw z) · z ≤ −C −1 |z|2 for z ∈ RN , and the partial derivatives of g and G in the second and third terms in the integrand on the right above are small relative to C. The latter holds in particular if g is independent of P and G of w. The application of Theorem 6.12 to more general settings is straightforward in principle if not in detail. Suppose that u ˜ is a known, global m-regular solution of (6.27) with initial data (˜ ς0 , . . . , ς˜m ), as in Theorem 6.10. Suppose also that nearby compatible initial data (ς0 , . . . , ςm ) is given and that u is the corresponding local-in-time solution of Theorem 6.10. We can then consider the application of Theorem 6.12 to the difference z = u − u ˜, which satisfies Z tZ  hz(t), viL2 = hς0 − ς˜0 iL2 + − (˜ aη,ω (x, t, z)zxη ) · vxω 0 Ω  + h(x, t, z, ∇x z) · v + H(x, t, z) : ∇x v dx dt for v ∈ H1 . Here a ˜η,ω (x, t, z) = aη,ω (x, t, u ˜(x, t) + z) and explicit expressions for h and H are easily computed. These expressions show first that h(x, t, 0, 0) = 0 and H(x, t, 0) = 0, so that z ≡ 0 is a steady-state solution, and second that the dissipativity hypothesis in (d) holds if the Jacobian matrix gw (·, ·, u ˜ , ∇x u ˜) is uniformly negative definite and if the partial derivaη,ω tives aw (·, ·, u ˜), gP (·, ·, u ˜, ∇x u ˜) and Gw (·, ·, u ˜) are small, as in the paragraph above. These smallness conditions hold in particular if a and G are independent of w and g is independent of P . If Theorem 6.12 can be shown to apply then the conclusion will be that u is also globally defined and coalesces with u ˜ as t → ∞. This application of Theorem 6.12 requires two additional hypotheses, however, which cannot be overlooked: first, the domains of regularity of the nonlinear functions aη,ω , g and G must be respected, and

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second, the continuous differentiability hypotheses in (a) and (b) will require a high degree of regularity of the known solution u ˜, as in Theorem 5.5(c) and Theorem 4.2(c). A representative example is outlined in Exercise 6.9.

§6.6

Invariant Regions for Quasilinear Systems

In this section we establish sufficient conditions for a region Σ in u-space to be invariant for m-regular solutions of (6.27). This means that if the initial function ς0 takes values in Σ then so too does the solution u(·, t) for as long as it is defined. The existence of such a set is of particular interest in applications in which the solution loses its physical meaning outside Σ, such as when its components represent nonnegative concentrations or densities. Also, the existence of an invariant set can serve as the basis for precluding the blow-up alternative in Corollary 6.11, as will be seen in Theorem 6.14 in the next section. The key idea is illustrated by the same question for systems of ordinary differential equations u(t) ˙ = g(u(t)), where g : W ⊆ RN → RN and an initial condition u(0) = ς0 ∈ W is given. If Σ = {u ∈ W such that Φ(u) < 0} C1

where Φ is and ∇u Φ(u) · g(u) < 0 for u ∈ ∂Σ, then Σ is invariant. The proof is that if ς0 ∈ Σ and if there is a minimal time t0 > 0 at which u ∈ ∂Σ, d then Φ(u(t)) t0 < 0, contradicting the minimality of t0 . dt We will extend this idea to m-regular solutions of systems (6.27) with the restriction that g is linear in ∇x u and Hj = [(H j ∩ H01 )(Ω)]N . In this case the divergence theorem can be applied to the term involving G, bringing the underlying differential equation to the form
(6.69) ut = aη,ω (x, t, u)uxη xω + g1 (x, t, u) + g2i (x, t, u)uxi where g1 is RN -valued and each g2i is RN ×N -valued. The corresponding weak form is therefore Z tZ h hu(t), viL2 = hς0 , viL2 + −(aη,ω (x, t, u)uxη ) · vxω 0 Ω (6.70)
i + g1 (x, t, u) + g2i (x, t, u)uxi · v . The following theorem finds wide application in spite of the complexity of its statement. In many cases invariance is more easily established directly by adapting its proof rather than by checking its hypotheses, as in the first three examples given at the end of this section. On the other hand the theorem can be quite useful for identifying candidates for invariant sets when these are not otherwise suggested by the application, as in the fourth example. Theorem 6.13. Fix an ensemble (5.4) as in Definition 5.1 with m > n/2 and H1 = [H01 (Ω)]N and let T ∈ (0, ∞] and W be a neighborhood of the

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2m and aη,ω with origin in RN . Let mappings g1 and g2i with components in Cbdd 2m+1 components in Cbdd be given, all on (Ω × (0, T ) × W ), with g1 taking values in RN and g2i and aη,ω taking values in RN ×N . Assume that the derivatives of these mappings up to the given orders have continuous extensions to Ω × [0, t¯] × K for every compact K ⊂ W and t¯ ∈ (0, T ) which are bounded, continuous and Lipschitz with respect to the variable in K, and finally that the aη,ω satisfy the positivity condition (6.39) for every t¯ ∈ (0, T ). Suppose also that a C 2 mapping Φ : W → R is given, taking both positive and negative values in W and satisfying Φ(0) < 0 and ∇u Φ 6= 0 on Φ−1 (0). Define Σ = {w ∈ W : Φ(w) < 0}, ∂Σ = {w ∈ W : Φ(w) = 0} and Σ = Σ ∪ ∂Σ. Assume that the following conditions hold: (a) There are mappings λη,ω : Ω × (0, T ) × W → R, 1 ≤ η, ω ≤ n, such that

• for t ∈ (0, T ), λη,ω (·, t, ·) ∈ C 1 (Ω × W ) • for all η and ω, (aη,ω )tr ∇u Φ = λω,η ∇u Φ on Ω × (0, T ) × W • if S is a nonnegative symmetric n×n matrix, then the scalar λη,ω S η,ω is nonnegative on Ω × (0, T ) × W • for all (constant) N × n matrices M ,
ω,η M tr Φ00 aη,ω M ≥0 on Ω × (0, T ) × W . (b) Either (i) g1 · ∇u Φ < 0 on Ω × (0, T ) × ∂Σ or (ii) g1 · ∇u Φ ≤ 0 on Ω × (0, T ) × ∂Σ and there is a (constant) N × N matrix P such that (P u) · ∇u Φ < 0 on ∂Σ. (c) There are maps µi : Ω × (0, T ) × ∂Σ such that (g2i )tr ∇u Φ = µi ∇u Φ on Ω × (0, T ) × ∂Σ. Let u be an m-regular solution of (6.70) on an interval [0, tmax ) taking values in W and with compatible initial data ς0 and let uc be its continuous representative (recall the description in Lemma 6.8(a)). Under hypothesis (bi): if ς0 (x) ∈ Σ for all x ∈ Ω, then uc (x, t) ∈ Σ for all (x, t) ∈ Ω × [0, tmax ). Under either hypothesis (bi) or (bii): if ς0 (x) ∈ Σ for all x ∈ Ω, then uc (x, t) ∈ Σ for all (x, t) ∈ Ω × [0, tmax ). PROOF. We denote the continuous counterpart uc by u. Recall from Lemma 6.8(a) that u and its (calculus) derivatives ut , uxj and uxj xk have H¨older continuous extensions to every Ω × [0, t¯]. Also, Theorem 6.10 invokes (5.35) in Theorem 5.5(c) to show that (6.69) holds pointwise in Ω × (0, tmax ) and therefore on Ω × [0, tmax ). We will prove the second statement in the conclusion of the theorem, that Σ is invariant; the proof of the first is similar (see Exercise 6.10).

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Assume first that (bi) holds and let u be as in the statement with u(x, 0) ∈ Σ for all x. Define t¯ = sup{t < tmax : u(x, s) ∈ Σ for all (x, s) ∈ Ω × [0, t]} (notice that t = 0 is a competitor in the above set). We have to show that t¯ = tmax . Suppose to the contrary that 0 ≤ t¯ < tmax . There is then a sequence {(xk , tk )} ∈ Ω × (t¯, tmax ) such that tk → t¯+ and ϕ(xk , tk ) ≡ Φ(u(xk , tk )) > 0 for all k. Note that there is a compact set K ⊂ Ω such that ϕ < 0 on (Ω − K) × [0, t1 ]. This follows because u(·, t) = 0 on ∂Ω, u(x, t) → 0 as |x| → ∞ if Ω is unbounded, Φ(0) < 0 and ϕ is H¨ older continuous. Thus xk ∈ K for all k, so that, modulo a subsequence, xk converges to an element x ∈ K ⊂ Ω. It follows that ϕ(x, t¯) = 0 and we claim further that ϕt (x, t¯) ≥ 0. This is obvious if t¯ > 0 and in any case follows from the observation that the inequalities ϕ(xk , t¯) ≤ 0 < ϕ(xk , tk ) show that there is a τk ∈ (t¯, tk ) with ϕt (xk , τk ) > 0, and (xk , τk ) → (x, t¯). Noting also that the C 2 function ϕ(·, t¯) has a relative maximum at x, we then conclude that at (x, t¯), ϕ = 0, ϕt ≥ 0, ϕxj = 0 for all j and the second derivative matrix ϕ00 is nonpositive. On the other hand we can take the dot product of both sides of (6.69) with ∇u Φ(u(x, t¯)) to find that the term on the left, which is ϕt , is nonnegative, the term involving g1 is strictly negative by hypothesis (bi), and the terms involving g2i are all zero by hypothesis (c). The remaining term is
∇u Φ · (aη,ω uxη ) xω − Φui uj ujxω (aη,ω )ik ukxη
iω η,ω
iη =(λη,ω ϕxη )xω − Φ00 ∇x u a ∇x u
η,ω
ω,η =λη,ω ϕ00 − (∇x u)tr Φ00 aη,ω ∇x u , which is nonpositive by the last item in hypothesis (a). These results contradict equality in (6.69), however, so that t¯ = tmax , as required. Now suppose that hypothesis (bii) holds in place of (bi) and let u and t¯ be as in the second paragraph of this proof. Suppose again that t¯ < tmax and let (xk , tk ) be as above with tk → t¯+ and ϕ(xk , tk ) > 0. Consider the problem (6.70) posed at initial time t¯ but with g1 replaced by h1 (x, t, w) = g1 (x, t, w) + δ(t − t¯)2(m+1) P w, with initial data (u(·, t¯), . . . , Dtm u(·, t¯)) and δ ∈ (0, 1). Hypotheses (a)–(e) of Theorem 6.10 are then satisfied (note in particular that u itself provides the required element udata in (e)). There is therefore an m-regular solution uδ defined on an interval [t¯, t¯ + tδmax ), and the description in (6.40) shows that there is a positive lower bound τ for tδmax independent of δ. The result just proved with hypothesis (bi) in effect applies to this modified system, so that uδ takes values in Σ on Ω × [t¯, t¯ + τ ]. On the other hand, it is easy to show that uδ (·, t) → u(·, t) as δ → 0, say in L2 (Ω), for t ∈ [t¯, t¯ + τ ] (see the last paragraph in the proof of Theorem 6.10, for example). We conclude that Φ(u(x, t)) ≤ 0 on Ω × [t¯, t¯ + τ ], which contradicts the supposition

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that Φ(u(xk , tk )) > 0 for k large. It follows that t¯ = tmax , as required.  We consider several representative examples: Example 1. N = 1. In this case aη,ω , g1 , g2i and Φ are scalar-valued and, apart from the regularity hypotheses preceding (a), conditions (a) and (c) hold automatically: the third item in (a) follows from the spectral representation of S and Remark 2 following Theorem 3.6, and the fourth is equivalent to (6.39). Thus if zero is an interior point of a compact subset [u, u] of W and Φ(u) = (u − u)(u − u), then (bii) holds with P = −I provided g1 (·, ·, u) ≤ 0 ≤ g1 (·, ·, u). The theorem then applies to show that [u, u] is invariant for m-regular solutions. That is, the standard “maximum principle” holds for scalar equations (6.69) satisfying the hypotheses of the theorem. Example 2. Example 1 generalizes to the reaction-diffusion system (6.71)

uit + wi (x, t, u) · ∇x ui = (di (x, t, u)uixj )xj + F i (x, t, u)

posed on Ω × [0, T ) with initial data ς0 , as in the theorem. Here u, wi and F take values in RN and the di are scalar-valued. In a typical application ui represents the concentration of the i-th species in a mixture which in the absence of spatial dependence undergoes chemical reactions governed by an ordinary differential equation u˙ = F . On the other hand if the i-th species is assumed to convect in a region Ω along the integral curves of a known vector field wi and to undergo molecular diffusion at the rate di , then u˙ i is replaced by the convective time-derivative on the left in (6.71) and both diffusion and reaction are represented on the right. The more common boundary condition in this setting, that the normal derivative of ui is zero on ∂Ω, is not accommodated in Theorem 6.13, however. Instead we renormalize the nominal range of ui to be the interval [−1, 1] and take H1 = H01 (Ω) (which includes the case that Ω = Rn ). The corresponding weak form is then (6.70) with aη,ω = δηω dη I (Kronecker δ here), g1 = F and g2i the diagonal N × N matrix whose diagonal entries are the components of −wi . It is then easy to check that if the regularity hypotheses preceding (a) of Theorem 6.13 are satisfied and Φ(u) = (ui − 1)(ui + 1) for a fixed i, then (a), (bii) and (c) hold with P = −I provided F i (·, ·, ei ) ≤ 0 ≤ F i (·, ·, −ei ) where ei is the i-th standard basis vector in RN . If this condition holds for all i, then the set [−1, 1]N is invariant for m-regular solutions, consistent with the interpretation of its components as normalized concentrations. Example 3. Here we consider a simplification of the Navier-Stokes system considered in section 6.2 in which the motion of a fluid of constant density

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6. APPLICATIONS TO QUASILINEAR SYSTEMS

occupying a region Ω ⊆ Rn is described by its velocity field u(x, t) (so that n = N ), which is to satisfy uit + ∇x ui · u = ε(x, t, u)uixj )xj + F i (x, t, u) together with the condition that u = 0 on ∂Ω. The weak form is then exactly as in (6.70) with H1 = [H01 (Ω)]n , aη,µ = δηµ εI, g1 = F, and g2i = −ui I. We leave to the reader to check that if the regularity hypotheses preceding (a) are satisfied and if Φ(u) = 12 (|u|2 − R2 ) where R > 0 is constant, then (a), (bii) and (c) hold with P = −I provided F (·, ·, u) · u ≤ 0 for |u| = R. If so, then the ball of radius R centered at the origin of u-space is invariant for m-regular solutions. Example 4. The p-system is the set of equations vt − ux = 0 ut + p(v)x = 0 for unknown functions v and u of x ∈ R and t ≥ 0, where p is a given function of v (the reader need not be concerned that we have assumed that n ≥ 2 throughout the text). These equations model the flow of a compressible fluid or gas occupying a thin tube along the real axis for which v, u and p represent specific volume (the reciprocal of density), velocity and pressure, and x is the so-called Lagrangean coordinate, which is constant along fluidparticle trajectories. (Thus the time derivatives on the left above correspond to the convective derivatives on the left sides in Examples 2 and 3.) It is generally assumed that p ∈ C 2 ((0, ∞)) with p0 < 0 < p00 . See [8] Sect. I.18 for the underlying rational mechanics and [39] Sect. 17§A for the fairly rich mathematical theory, disjoint from that of this text, of this deceivingly simple system. The p-system can be “regularized,” both for theoretical and computational purposes, by the addition of so-called artificial viscosity terms εvxx and εuxx to the right sides, where ε is a positive p constant. Making this change and replacing v by v + 1 and writing c(v) = −p0 (v + 1), we obtain the system        0 −1 v v v + =ε (6.72) u t u x u xx −c2 (v) 0 for which the origin of v–u space is in the domain of regularity. The eigen ±c vectors of the transpose of the matrix on the left are , so that if 1 Z v Φ(v, u) = u − c(s)ds v0

where v0 ∈ (−1, 0) is fixed, then ∇(v,u) Φ is an eigenvector, and hypotheses (a) and (c) of Theorem 6.13 hold with W = {(v, u) : v > −1}. The term g1 is

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6. A GLOBAL EXISTENCE RESULT FOR SYSTEMS IN R2 AND R3

177

zero in this example, so we check condition (bii) with P = −I: if Φ(v, u) = 0 then at (v, u), Z v    v ∇(v,u) Φ · P = c(v)v − u = c(v)v − c(s)ds. u v0 If v > v0 then the integral here is bounded below by c(v)(v − v0 ), since c is decreasing, and the right side here is therefore negative. For v ∈ (−1, v0 ) the derivative of the right side with respect to v is c0 (v)v, which is positive, so that its value is bounded above by its value c(v0 )v0 at v = v0 , which again is negative. The (n = 1 version of) Theorem 6.13 together with a similar analysis for the second eigenvector therefore shows that the set Z v Z v 2 {(v, u) ∈ R : v ≥ v0 and − c(s)ds ≤ u ≤ c(s)ds} v0

v0

is invariant for (6.72).

§6.7

A Global Existence Result for Systems in R2 and R3

In this section we prove global existence of 2-regular solutions of the quasilinear system (6.70) in two and three space dimensions when the aη,ω depend only on x and satisfy the symmetry conditions in section 3.4 and when the solution in question is known to take values in an invariant set S in which the derivatives of aη,ω , g1 and g2i are pointwise bounded. The proof consists in precluding the two blow-up alternatives of Corollary 6.11 by chaining in a particular order three a priori bounds established in previous sections (Theorems 3.7, 4.2 and 5.5) and by exploiting the strength of the Sobolev imbeddings H k → Lp in low dimensions. The reader can check that the required invariant region hypothesis of the following theorem is satisfied in all four examples in the previous section. The first three of these, posed with initial conditions taking values in S, therefore have global solutions if n = 2 or 3 and if the other relatively routine conditions of the theorem below are met. The solution u to be found is to satisfy Z tZ h hu(t), viL2 = hς0 , viL2 + −(aη,ω (x)uxη ) · vxω (6.73) 0 Ω
+ g1 (x, t, u) + g2i (x, t, u)uxi · v for v ∈ H1 . The global existence result is as follows: Theorem 6.14. Fix an ensemble (5.4) as in Definition 5.1 with n = 2 or 3 and m = 2, let T ∈ (0, ∞] and let W be a neighborhood of the origin in RN . Assume the following: (a) Mappings aη,ω , 0 ≤ η, ω ≤ n, from Ω into the set of N × N matrices are given, satisfying the hypotheses of Theorem 5.5, that is, the positive

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6. APPLICATIONS TO QUASILINEAR SYSTEMS

definiteness condition (5.5), any one of the sets of regularity conditions (5.8), (5.10) or (5.11)–(5.13) (in which all t-derivatives are now zero) and the symmetry conditions (aη,ω )tr = aω,η for all η, ω. (b) g1 and g2 are C 4 mappings from Ω × (0, T ) × W into RN and RN ×n respectively and g1 satisfies (6.36) with m = 2. (c) There is a closed set S ⊂ W which is star-shaped with respect to 0 ∈ S such that (i ) S is invariant for (6.73) in the sense that if u is a 2-regular solution on [0, t¯] with u(x, 0) ∈ S for all x ∈ Ω, then u(x, t) ∈ S for all (x, t) ∈ Ω × [0, tmax ) where tmax is its maximal time of existence with respect to W and T in the sense of Corollary 6.11. (ii ) For every t¯ ∈ (0, T ) there is a constant Csys (t¯) which bounds all the derivatives and Lipschitz constants of aη,ω , g1 and g2 referred to above in (a) and (b) on Ω × [0, t¯] × S, as well as θ−1 (which is now independent of t; see (5.5)) and the L∞,2 (Ω × [0, t¯]) and L2,2 (Ω × [0, t¯]) norms of g1 (x, t, 0) in (6.36) with m = 2. Let initial data (ς0 , ς1 , ς2 ) for (6.73) be given satisfying the compatibility conditions (6.35) with m = 2 and with ς0 taking values in S. Then if the corresponding solution described in Theorem 6.10 exists (which is true if hypothesis (e) of that theorem is satisfied) then it is global: its maximal time of existence with respect to W and T in the sense of Corollary 6.11 is T . PROOF. We give the proof for n = 3, the proof for n = 2 being similar. Spaces [Lp (Ω)]N will be abbreviated Lp , etc., and the bound 1/4

3/4

|w|L4 ≤ C(Ω)|w|L2 |w|H 1 ,

(6.74)

which follows from the imbedding H 1 → L6 for N = 1 (Theorem A.3(a)) via interpolation, will be applied throughout, as will the Bochner-Sobolev space description in Theorem 5.5(a). Let u be as in the statement and tmax its maximal time of existence with respect to W and T . We will assume that tmax < T and prove that Z tmax 2   X (j) 2 (6.75) sup |u (t)|H 4−2j + |u(j) |2H 5−2j dt < ∞ j=0

0≤t 0 and for j ∈ Z let χεj be the characteristic function of the set I ∩ [jε, (j + 1)ε]. Then by (A.19) i i XZ hZ XZ hZ u(t)v(·, jε)dµ χεj (t)dt = um (x, t)v(x, jε)dµ χεj (t)dt, j

I

X

j

I

X

and the result follows in the limit as ε → 0. 

Bochner Space Differentiability and Absolute Continuity Next we let Ω be an open set in Rn and examine a concept of spatial differentiability of an element u ∈ Lp (I; Lq (Ω)) in terms of the existence of weak derivatives on Ω × I ◦ of its measurable counterpart um . We begin with the following formulation: Definition A.18. Let u ∈ Lp (I; Lq (Ω)) and w ∈ Lr (I; Ls (Ω)) where Ω is an open set in Rn , I is a nontrivial interval in R, p, r ∈ [1, ∞] and q, s ∈ (1, ∞], and let α be a multi-index. Then w = Dxα u in the Bochner space sense if for every pair of representatives ϕ : I → Lq (Ω) and ψ : I → Ls (Ω) of u and w there is a null set E ⊂ I such that for t ∈ I − E, ψ(t) = Dxα ϕ(t) in the sense of weak derivatives on Ω. The following equivalent formulation in terms of measurable counterparts is applied throughout the text: Theorem A.19. Let u ∈ Lp (I; Lq (Ω)) and w ∈ Lr (I; Ls (Ω)) where Ω is an open set in Rn , I is a nontrivial interval in R, p, r ∈ [1, ∞] and q, s ∈ (1, ∞], and let α be a multi-index. Then w = Dxα u in the Bochner space sense if and only if wm = Dxα um in the sense of weak derivatives in Ω × I o . If so, then (Dxα u)m = Dxα (um ). PROOF. First assume that w = Dxα u in the Bochner space sense and let |α| v ∈ Cc (Ω × I o ). Then from Corollary A.17 and Fubini’s theorem, each applied twice, we have Z Z hZ i wm v = wm (x, t)v(x, t)dx dt Ω×I Z hZ Z h ZI Ω i i |α| = w(t)v(·, t)dx dt = (−1) u(t)Dxα v(·, t)dx dt I Ω I Ω Z hZ Z i |α| |α| α = (−1) um (x, t)Dx v(x, t)dx dt = (−1) um Dxα v, I

which proves that wm =

Ω

Dxα um

Ω×I

in the weak sense on Ω ×

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I o.

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SELECTED TOPICS IN ANALYSIS

Now assume that wm = Dxα um in the weak sense on Ω × I o , let ϕ : I → Lq and γ : I → Ls be representatives of u and w respectively and let {vj }j ⊂ |α| |α| Cc (Ω) be a countable set with the property that if v ∈ Cc (Ω) then there 0 0 is a sequence vji → v in Ls such that Dxα vji → Dxα v in Lq (the vj may be 0 0 taken to be elements of a set in Cc∞ (Rn ) which is dense in W |α|,q ∨s (Rn ) multiplied by suitable cutoff functions, for example; see Theorem A.2). There is then a null Z set E ⊂ I such that Z each t0 ∈ I − E is a Lebesgue point of the ϕ(t)Dxα vj and t →

maps t → Ω

γ(t)vj . Fix t0 ∈ I − E and let ψ ε : I → R

Ω

be a nonnegative function supported on an interval of length ε contained in I and containing t0 and whose integral over I equals one. Then applying Corollary A.17 and Fubini’s theorem, each twice, we obtain Z Z hZ i ϕ(t0 )Dxα vj dx = lim ϕ(t)Dxα vj dx ψ ε (t)dt ε→0 I Ω Z h ZΩ i = lim um (·, t)Dxα vj dx ψ ε (t)dt ε→0 I Ω Z Z α ε |α| = lim um Dx (vj ψ ) = (−1) lim wm v j ψ ε ε→0 Ω×I ε→0 Ω×I Z = . . . = (−1)|α| γ(t0 )vj dx. Ω

This proves that, for almost all t0 ∈ I, Dxα ϕ(t0 ) = γ(t0 ) in the sense of weak derivatives on Ω and therefore that Dxα u = w in the Bochner space sense of Definition A.18.  We now consider the relationship between the derivative of an absolutely continuous mapping u ∈ wAC(I; Lq (Ω)) in the sense of Definition 2.2 and the weak time-derivative of its measurable counterpart um : Theorem A.20. Let Ω be an open set in Rn , I a nontrivial interval in R and q ∈ (1, ∞]. (a) If u ∈ L1loc (I; Lq (Ω)) then its equivalence class on I contains an element uAC ∈ wAC(I; Lq (Ω)) with derivative (uAC )0 if and only if the t-derivative Dt um exists in the weak sense on Ω×I ◦ and is in L1,q (t)loc (Ω×I). 0 If so, then ((uAC ) )m = Dt um . (b) Let H1 be a closed subspace of H 1 (Ω) which is dense in L2 (Ω) and let 2 ∗ um ∈ L∞,2 (t)loc (Ω×I). Assume that there is an F ∈ Lloc (I; H1 ) such that for every v ∈ H1 and ψ ∈ Cc1 (I) (having one-sided derivatives at endpoints of I if any) Z Z
um (x, t)v(x) ψt (t) = F (t) · v ψ(t) dt. Ω×I

I

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THE SPACES Lp,q

209

Then um is the measurable counterpart of an element u ∈ C(I; L2weak (Ω)) for which u∗ ∈ wAC 2 (I; H1∗ ) with derivative F (recall that for z ∈ L2 , z ∗ ∈ H1∗ is defined by z ∗ · v = hz, viL2 .) PROOF. To prove (a) we first assume that uAC exists as described in the statement and let ψ ∈ Cc1 (Ω × I ◦ ). Then by Fubini’s theorem and Corollary A.17, Z Z Z Z Z   um ψt = u ψt dx dt = uAC ψt dx dt Ω×I ◦ I Ω I Ω Z Z  = lim h−1 uAC (x, t)[ψ(x, t + h) − ψ(x, t)]dx dt h→0 I Z ΩZ  −1 = − lim h [uAC (x, t) − uAC (x, t − h)]ψ(x, t)dx dt h→0 I Ω Z Z t Z  = − lim h−1 (uAC )0 (s)ψ(·, t) ds dt h→0

I

t−h Ω s+h  Z

Z Z

 = − lim h−1 (uAC )0 (s)ψ(·, t) dt ds h→0 I s Ω Z Z Z  0 =− ((uAC ) )m (s)ψ(·, s) ds = − ((uAC )0 )m ψ. I

Ω×I ◦

Ω

This proves that ((uAC )0 )m is the t-derivative of um in the weak sense on Ω × I ◦. Now suppose that the weak derivative Dt um exists in the weak sense as described in the statement of the theorem and let {ϕj }j ⊂ Cc1 (Ω) be 0 countable and dense in Lq (Ω) where q 0 ∈ [1, ∞) is the H¨ older conjugate of q. Abusing notation slightly, we let u denote a specific representative of its equivalence class. There is then a null set E ⊂ I such that if t0 ∈ I − E thenZ|u(t0 )|Lq (Ω) < ∞ and t0 is a Lebesgue point of each of the maps u(t)ϕj dx. Fix t0 ∈ I ◦ − E and observe that, for any t ∈ I,

t→ Ω

Z tZ Z Dt um ϕj dx ds ≤ C(t)|ϕj |Lq0 (Ω) . u(t0 )ϕj dx + Ω

t0

Ω

It follows that there is an element of Lq (Ω), which we denote by uAC (t), such that Z tZ Z Z (A.23) uAC (t)v dx = u(t0 )v dx + Dt um v dx ds Ω

Ω

t0

Ω

q0

for all v ∈ L (Ω). Apparently uAC ∈ wAC(I; Lq (Ω)) with derivative satisfying ((uAC )0 )m = Dt um , which justifies the notation. To complete the proof we need to show that uAC (t) = u(t) for almost all t. Let t1 ∈ I ◦ − E and assume without less of generality that t1 > t0 . For h > 0 define ψ h ∈ Cc1 (I ◦ ) by taking ψ = 0 in the complement of [t0 , t1 ],

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SELECTED TOPICS IN ANALYSIS

ψ = −1 on [t0 + h, t1 − h], and ψ monotone in the remaining two sets. Then by Corollary A.17, for all j, Z Z Z  [u(t1 ) − u(t0 )]ϕj dx = lim u(t)ϕj dx ψth (t)dt h→0 I Ω Z ZΩ = lim um ϕj ψth = − lim Dt um ϕj ψ h =

h→0 Ω×I ◦ Z t1  Z

h→0 Ω×I ◦



Z [uAC (t1 ) − u(t0 )]ϕj dx

Dt um ϕj dx dt =

t0

Ω

Ω

by (A.23). It follows that uAC (t1 ) = u(t1 ) for all t1 ∈ I ◦ − E, so that uAC and u agree almost everywhere on I as elements of Lq (Ω). The proof of (b) is nearly identical. We fix a specific representative um , a set {ϕj } which is dense in H1 and a corresponding null set E as in the proof of (a), and repeat the argument above to construct an element G ∈ wAC 2 (I; H1∗ ) whose derivative is F and which agrees with um (·, t)∗ for t ∈ I − E. Then for v ∈ H1 , I 0 compact in I and t ∈ I 0 − E, |G(t) · v| = |u∗m (·, t) · v| ≤ |um (·, t)|L∞,2 (Ω×I 0 ) |v|L2 . The bound on the right extends to all t ∈ I 0 by the absolute continuity of G, and this together with the fact that H1 is dense L2 shows that for each t ∈ I there is an element u(t) ∈ L2 (Ω) such that G(t) = u(t)∗ . Also, u ∈ C(I; L2weak (Ω)) because G(t) · v is continuous in t for v ∈ H1 and H1 is dense in L2 ; and the proof that um is the measurable counterpart of u is essentially identical to that of the final step in the proof of (a).  A similar result for mappings into Lq (∂Ω) can be derived by applying Theorem A.20 in the coordinate domains associated to ∂Ω. To do this we recall the notation for Lipschitz domains in Definition 1.7(a) and the definition (1.6) of the operators Ti and define the composition operator τi mapping functions f : ∂Ω ∩ Ui → R into functions τi f : Bi → R by the relation (τi f )(˜ y ) = f (Ti (˜ y )). The operator τi then maps σ-equivalence classes (σ is defined in Theorem 3.1) on ∂Ω into Lebesgue equivalence classes on q q Bi and extends as an isometry from p L (∂Ω ∩ Ui ) onto L (Bi ) (provided the measure in the latter space is 1 + |∇ψi (˜ y )|2 d˜ y ). For mappings f of q an interval I ⊆ R into L (∂Ω ∩ Ui ) we define (τi f )(t) ≡ τi (f (t)). Again τi extends as an isometry of Lp (I; Lq (∂Ω ∩Ui )) onto Lp (I; Lq (Bi )) and preserves weak absolute continuity. All the above statements hold as well for the inverse mapping τi−1 , mutatis mutandi. The following theorem gives a useful characterization of weakly absolutely continuous mappings into Lq (∂Ω): Theorem A.21. Let u ∈ L1loc (I; Lq (∂Ω)) where I is a nontrivial interval in R, Ω is a Lipschitz domain in Rn and q ∈ (1, ∞]; and let Ui , Ti , Bi and ψi be as in Definition 1.7 and (1.6). Let τi be as above and define vi = (τi u)m ,

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THE SPACES Lp,q

211

which is an element of L1,q (Bi × I 0 ) for every compact interval I 0 ⊆ I. Then the Lebesgue equivalence class u contains an element uAC ∈ wAC(I; Lq (∂Ω)) if and only if for each i the weak derivative Dt vi with respect to t ∈ I exists in the weak sense on Bi × I ◦ and i1/q hX (A.24) |Dt vi |qLq (Bi ) ∈ L1loc (I) i

p (the measure on Bi is 1 + |∇ψi (˜ y )|2 d˜ y ) or sup |Dt vi |L∞ (Bi ) ∈ L1loc (I)

(A.25)

i

according as q < ∞ or q = ∞. If so, then for each i, [(τi uAC )0 ]m = Dt [τi (um )] as elements of L1,q (t)loc (Bi × I). The proof of (A.24) for the “only if” statement requires the additional hypothesis that there is a nonnegative integer i0 such that no Ui intersects more than i0 of the other covering sets Ui . PROOF. First note that the statements in (A.24) and (A.25) are meaningful by Fubini’s theorem because vi is measurable on the product domain Bi × I. If the weak derivatives Dt vi exist as in the statement then by Theorem A.20 there are functions viAC in the equivalence class τi u which are in wAC(I; Lq (Bi )) with derivatives vi0 ∈ L1loc (I; Lq (Bi )). We then define X X χi · τi−1 viAC and w = χi · τi−1 vi0 uAC = i

i

where χi is the characteristic function of Wi (which is defined in Theorem 3.1(a)), and then check that uAC is in the equivalence class u and that uAC ∈ wAC(I; Lq (∂Ω)) with derivative w. The details are straightforward and so are left to the reader. To prove the converse we suppose that there is a function uAC ∈ wAC(I; Lq (∂Ω)) which is in the Lebesgue equivalence class of u on I. Then viAC ≡ τi uAC is in wAC(I; Lq (Bi )) and viAC is in the equivalence class on I of τi u, whose measurable counterpart (τi u)m is vi . Theorem A.20 therefore applies to show that the weak derivatives Dt vi exist in the weak sense on Bi × I ◦ . The statements in (A.24) and (A.25) then follow easily. Note, however, that since the covering sets {Ui } are not disjoint, the proof of (A.24) requires an application of the following general fact: if {Ei } is a countable collection of sets in an arbitrary σ-algebra on which there is a nonnegative measure µ, and if no Ei intersects more than i0 of the other Ei , then X µ(Ei ) ≤ (1 + i0 )µ(∪i Ei ). i



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The spaces Lp (I; W k,q (Ω)) and wAC(I; W k,q (Ω)) We conclude this section by defining the above two spaces and by characterizing their elements in terms of the weak derivatives of their measurable counterparts; this will give succinct and useful versions of Theorems A.19 and A.20. Notice that Definitions 1.3 and 2.2 do not apply here because we cannot readily identify W k,q (Ω) with the dual of a separable Banach space X. It can, however, be identified with a closed subspace of such a space, as follows. Let Ω be an open set in Rn , p ∈ [1, ∞], q ∈ (1, ∞], q 0 the H¨older conjugate of q and k a positive integer. Let K be the number of multi-indices of length less than or equal to k and define the norm of h X i1/s an element v = {vα : |α| ≤ k} ∈ [Ls (Ω)]K by |vα |sLs (Ω) or by |α|≤k

max|α|≤k |vα |L∞ (Ω) according as s ∈ [1, ∞) or s = ∞. Then since q 0 < ∞, 0 X ≡ [Lq (Ω)]K is a separable space and its dual X ∗ can be identified with [Lq (Ω)]K in the obvious way. (An easy exercise based on H¨older’s inequality shows that the norm of an element in [Lq (Ω)]K is the same as the norm in X ∗ of the corresponding linear functional.) The spaces Lp (I; [Lq (Ω)]K ) and wAC(I; [Lq (Ω)]K ) are therefore included in Definitions 1.3 and 2.2, and the spaces of interest can be defined as closed subspaces: Definition A.22. Let Ω, p, q, q 0 , k and K be as above and let I be a nontrivial interval in R. Then Lp (I; W k,q (Ω)) is the set of Lebesgue equivalence classes over I of functions ϕ : I → W k,q (Ω) which are weakly measurable X Z in the sense that the mappings t → Dxα ϕ(·, t) vα dx are measurable |α|≤k

Ω

q0

on I for all v = {vα } ∈ [L (Ω)]K and such that the equivalence class of mappings t → |ϕ(t)|W k,q (Ω) is in Lp (I). The space Lploc (I; W k,q (Ω)) is the set of elements u ∈ Lploc (I; Lq (Ω)) such that u ∈ Lp (I 0 ; W k,q (Ω)) for every compact interval I 0 ⊆ I. The space wAC(I; W k,q (Ω)) is the the set of elements in wAC(I; [Lq (Ω)]K ) which take values in W k,q (Ω) for all t ∈ I. The reader will see that the measurability condition in this definition is Z α equivalent to the requirement that t → Dx ϕ(·, t) v dx is measurable on I 0

Ω

for all v ∈ Lq (Ω) and |α| ≤ k. Observe also that if u ∈ wAC(I; W k,q (Ω)), then u0 ∈ L1loc (I; [Lq (Ω)]K ), but there is no a priori stipulation that u0 must be an element of L1loc (I; W k,q (Ω)). This is always true, however, as the reader can easily verify. Theorem A.19 can now be applied to give useful alternative characterizations of the spaces Lp (I; W k,q (Ω)). The proof consists in straightforward checking of details and is therefore omitted.

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THE SPACES Lp,q

213

Theorem A.23. Let Ω be an open set in Rn , I a nontrivial interval in R, u ∈ Lp (I; Lq (Ω)) where p ∈ [1, ∞] and q ∈ (1, ∞], and k a positive integer. The following three statements are then equivalent: (a) u ∈ Lp (I; W k,q (Ω)). (b) For every multi-index α of length less than or equal to k the derivative Dα u exists in the Bochner space sense of Definition A.17 and is an element of Lp (I; Lq (Ω)). (c) For every multi-index α of length less than or equal to k the derivative Dxα um with respect to x ∈ Ω exists in the weak sense on Ω × I ◦ and is an element of Lp,q (Ω × I). In the same way we can apply Theorem A.20 to give a useful characterization of absolutely continuous mappings into W k,q (Ω) in terms of their measurable counterparts. Again, the proof is elementary and so is omitted. Theorem A.24. Let u ∈ L1loc (I; Lq (Ω)) where Ω is an open set in Rn , I is a nontrivial interval in R and q ∈ (1, ∞]. Then the Lebesgue equivalence class u contains an element uAC ∈ wAC(I; W k,q (Ω)) if and only if for every α with |α| ≤ k the derivatives Dxα um and Dt Dxα um with respect to x ∈ Ω and t ∈ I exist in the weak sense in Ω × I ◦ and are elements of L1,q (Ω × I 0 ) for every compact interval I 0 ⊆ I. The following specific application of Theorems A.23 and A.24 is particularly useful for transferring information about solutions of the problems considered in Chapters 5 and 6 between their Bochner–Sobolev space descriptions, which have an intuitive dynamical systems appeal, and the measurable counterpart descriptions, for which the calculus of weak derivatives is especially convenient: 2 n Corollary A.25. Let w ∈ L∞ loc (I; L (Ω)) where Ω is an open set in R and I is a nontrivial interval in R, let wm be its measurable counterpart and let m ≥ 1. The following two statements are then equivalent:

(a) For j = 0, . . . , m the weak derivatives Dtj Dxα wm on Ω × I ◦ are in 2,2 L∞,2 (t)loc (Ω × I) for 2j +|α| ≤ 2m and in L(t)loc (Ω×I) for 2j +|α| ≤ 2m+1. (b) There are maps w(0) = w, w(1) , . . . , w(m) such that the equivalence class 2m−2j ) ∩ L2 (I; H 2m−2j+1 ) for j ≤ m (we over I of w(j) is in L∞ loc (I; H loc k k abbreviate H (Ω) = H ); and for j ≤ m − 1, 2m−2j w(j) ∈ Liploc (I; H 2m−2j−2 ) ∩ wAC 2 (I; H 2m−2j−1 ) ∩ C(I; Hweak )

and the derivative of w(j) is the the equivalence class of w(j+1) . PROOF: The proof is a straightforward iteration of the results of Theorems A.23 and A.24 with the the exception of the weak continuity of w(j)

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214

SELECTED TOPICS IN ANALYSIS

into H 2m−2j in (b). To prove this we first note that w(j) ∈ H 2m−2j for almost all t with the bound independent of t, hence for all t, because weak H 2m−2j –subsequential limits of sequences w(j) (ti ) are unique by the absolute continuity assumed in (b). The same two facts then show that if ti → t0 in I then the weak H 2m−2j –limit of w(j) (ti ) is w(j) (t0 ), as required. 

§A.6

A Representation Theorem for Boundary Integrals

The following result is useful in a variety of contexts and in particular was applied in chapters 4 and 5 to express integrals over the boundary of a spatial domain as a sum of integrals over its interior. Parts (a)–(d) give Bochner–Sobolev space formulations of the relationship between the respective kernels and part (e) is a reformulation in terms of weak derivatives. Theorem A.26. 0,1 (a) Let Ω be a Cbdd domain in Rn , n ≥ 2, not necessarily bounded, 0 q ∈ (1, ∞] and q its H¨ older conjugate. Then there exist operators S 1 ∈ L(Lq (∂Ω), Lq (Ω)) and S 2 ∈ L(Lq (∂Ω), Lq (Ω)]n ) such that for 0 f ∈ Lq (∂Ω) and v ∈ W 1,q (Ω), Z Z  1  (A.26) f v dσ = (S f ) v + (S 2 f ) · ∇x v dx. ∂Ω

The norms of tion 1.7.

S1

Ω

and

S2

depend on n, q and the constant MΩ0,1 in Defini-

0,1 Now assume either that Ω is a Cbdd domain in Rn , n ≥ 2, in which case we k set k = 1, or that Ω is a Cbdd domain with k ≥ 2. Let l ∈ {0, . . . , k − 1}, q ∈ (1, ∞] with H¨ older conjugate q 0 , p ∈ [1, ∞] and I a nontrivial interval in R.

(b) If f ∈ Lq (∂Ω) with hf iαLq , ∂Ω < ∞ for every |α| ≤ l (see (5.6)) then S 1 fX and the components of S 2 f are in W l,q (Ω) with norms bounded by C hf iαLq , ∂Ω where the constant C depends on n, q, l and either MΩ0,1 |α|≤l

or MΩk , according as k = 1 or k > 1. (α,0)

(c) If f ∈ Lp (I; Lq (∂Ω)) and if hf iLq , ∂Ω ∈ Lp (I) for every |α| ≤ l (see (5.7)), then S 1 f and X the components of S 2 f are in Lp (I; W l,q (Ω)) with (α,0) norms bounded by C hf iLq , ∂Ω where C is as in (b). |α|≤l

(d) If f ∈ wAC p (I; Lq (∂Ω)) with derivative f 0 and if for every |α| ≤ l (α,0) supI hf iαLq , ∂Ω < ∞ and hf 0 iLq , ∂Ω ∈ Lploc (I), then S 1 f and the components

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A REPRESENTATION THEOREM FOR BOUNDARY INTEGRALS

215

of S 2 f are in wAC p (I; W l,q (Ω)) with derivatives S 1 f 0 and the components of S 2 f 0 respectively. (α,i)

(e) If f ∈ L1 (I; Lq (∂Ω)) and if for some i ≥ 0 and all |α| ≤ l, hf iLq , ∂Ω ∈ Lp (I), then Dti Dxα (S 1 f )m and the components of D ti Dxα (S 2 f )m are in X (α,i) Lp,q (Ω × I) for |α| ≤ l with norms bounded by C hf iLq , ∂Ω p |α|≤l

L (I)

where C is as in (b) and Dt and Dx denote weak derivatives on Ω × I ◦ . Observe that by the trace theorem, Theorem 3.2, the left side of (A.26) 0 defines the action of an element of the dual of W 1,q (Ω) on v, so that for the case p = p0 = 2, say, the representation (A.26) is already known: the self-duality of H 1 implies that there is an element F in H 1 (Ω) such that Z

Z f v dσ = hF, viH 1 =

(A.27) ∂Ω

(F v + ∇F · ∇x v)dx Ω

for all v ∈ H 1 (Ω). The important point of (b) is therefore the regularity of F , or in the general case, of S 1 f and S 2 f . For example, since the right side of (A.27) defines an element of B(H1 × H1 ), exactly of the the type considered in Chapter 4, the application of Theorem 4.2, were its proof independent of Theorem A.26, would show that F ∈ H 2 (Ω) so that (A.27) would hold with S 1 f = F and S 2 f = ∇x F . Observe also that there may be multiple choices for S 1 f and S 2 f : again in the case W l,q = H 1 there will be solutions γ ∈ H 2 ∩ H01 of the weak form of the eigenvalue equation for the Laplace operator, ∆γ = λγ. Thus if the pair (F, ∇F ) satisfies (A.27) then so too does the pair (F + λγ, ∇F + ∇γ). The other important point of the theorem is therefore that specific, explicit choices S 1 f and S 2 f are made so that S 1 f and S 2 f are in the appropriate Bochner spaces in the time–dependent case. PROOF of Theorem A.26. We will omit the proof of (a)–(b) for the (0,1) 1 Cbdd case, this proof being essentially the same as that of (b) for Cbdd domains but with the ψi in Definition 1.7 having L∞ rather than bounded continuous derivatives. This distinction plays no role in the proof. The notations of Definition 1.7 will be applied freely throughout, C will denote a k domain generic constant depending on n, q, k and l, and Ω will be a Cbdd with k ≥ 1. To prove (a) and (b) we let v be the restriction to Ω of an element of 0 Cc∞ (Rn ), the latter being dense in W 1,q , and a nondecreasing C ∞ function χ = χ(s) on R which is zero for s ≤ ε0 /2 and identically one for s ≥ ε0 (ε0 is defined in Definition 1.7(b)). Let {ϕi } be the partition of unity constructed in Theorem 1.8 with X = ∂Ω subordinate to the cover {Ui }. Then by

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216

SELECTED TOPICS IN ANALYSIS

Theorem 3.1, Z XZ f v dσ = ∂Ω

i

XZ

=

i

h

f v ϕi dσ

∂Ω∩Ui

(f ◦ Ti )(˜ y)

p 1 + |∇ψi (˜ y )|2

Bi

Z

ψi (˜ y)

×

  i Dyn χ(yn ) (vϕi ) ◦ Ri )(˜ y , yn ) dyn d˜ y.

0

Carrying out the Dyn -differentiation, rearranging and writing the double integral as an integral over Ω ∩ Ui , we find that Z XZ (A.28) f v dσ = (Fi1 v + Fi2 · ∇v) dx, ∂Ω

i

Ω∩Ui

where explicit expressions for Fi1 and Fi2 in terms of f, ψi , Ti , Ri and χ, which the reader can easily compute, show by inspection that Fi1 and the components of Fi2 depend linearly on f and extend to elements of W l,q (Ω) with supports contained in the support of ϕi and with |Fi1 |W l,q (Ω) , |Fi2 |[W l,q (Ω)]n ≤ C|f ◦ Ti |W l,q (Bi ) .

(A.29)

Now if q ∈ (1, ∞) and |α| ≤ l, then for specific representatives Fi1 and for almost all x ∈ Ω, i2 i2 i2 X q  X q X α 1 α 1 D Fi (x) ≤ |D Fi (x)| ≤ C(i0 , q) |Dα Fi1 (x)|q i1

i1

i1

by the last hypothesis in Definition 1.7 and the fact that the support of ϕi , hence of Fi1 , is contained in Ui . Integrating, summing over α and applying (A.29) we thus find that i2 X q Fi1 i1

W l,q (Ω)

≤C

i2 X

|f ◦ Ti |qW l,q (B ) , i

i1

which X goes to zero as i1 , i2 → ∞ by the assumption in (b). This proves that Fi1 converges in W l,q (Ω) to an element F 1 ≡ S 1 f . A similar argument i X applies to Fi2 ≡ F 2 ≡ S 2 f , and the representation (A.26) then follows i

we apply the immediately from (A.28). For the case that q = ∞X Xfact that 1 each x ∈ Ω has a neighborhood in which the sums Fi and Fi2 have no more than i0 nonzero summands; the representations for F 1 and F 2 then follow easily. This proves (a) and (b).

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A REPRESENTATION THEOREM FOR BOUNDARY INTEGRALS

217

To prove (c) we write Si1 f in place of Fi1 and re-examine the computation leading to (A.28) to find that, for v ∈ Cc∞ (Rn ), Z Z 1 (A.30) (Si f ) v dx = f L1i v dσ Ω

∂Ω

L1i

(Rn )

where is a linear map from Cc to Cc (∂Ω ∩ Ui ). Thus if f temporarily denotes a specific representative of its equivalence class, then since the right side Z above is measurable in t, so is the left, and hence so is the sum defining (S 1 f ) v dx. This proves that S 1 f is weak-∗ measurable as a map into Ω

Lq (Ω), hence so are its Bochner-space derivatives Dxα (S 1 f ) described in (b). It then follows from the Lp bound assumed in (c) that S 1 f ∈ Lp (I; W l,q (Ω)) with the required bound, and a similar argument applies to S 2 f to complete the proof of (c). To prove (d) we let f and f 0 be as in the hypotheses so that by (b) and (c) S 1 f is a bounded map from I into W l,q (Ω) and S 1 f 0 ∈ Lploc (I; W l,q (Ω)). Let v ∈ Cc∞ (Rn ), |α| ≤ l and t1 , t2 ∈ I. Then applying (A.30) twice, we obtain that Z t2 X Z t2 X Z t2 (S 1 f )(Dxα v) dx = (Si1 f )(Dxα v)dx = f (L1i Dxα v) dσ t1

Ω

=

XZ

t1

i

Z t2Z = t1

i

t2 Z

t1

Ω

f 0 (L1i Dxα v) dσdt =

∂Ω

i

X Z t2Z i

t1

∂Ω

t1

(Si1 f 0 )(Dxα v)dx dt

Ω

(S 1 f 0 )(Dxα v)dx dt.

Ω

This proves that the Bochner space derivative Dxα (S 1 f ) is a weakly absolutely continuous map into Lq (Ω) with derivative Dxα (S 1 f 0 ). A similar argument applies to S 2 f , completing the proof of (d). We give the proof of (e) for the case q < ∞, the case q = ∞ being similar. Reexamining again the computation that led to (A.28) and (A.29), we find that we can write Fi1 = gi · (f ◦ Ti ◦ Π ◦ Ri−1 ) k−1 where if k ≥ 2 (resp. k = 1) then gi ∈ Cbdd (Ui ) (resp. L∞ (Ui )) with compact support in Ui and with derivatives up to order k − 1 bounded independently of i; and Π is the projection Π(˜ y , yn ) = (˜ y , 0). Now fix α with |α| ≤ l and define    Hi = Dxα gi · Dti (f ◦ Ti )m ◦ Π ◦ Ri−1

Choosing particular representatives, we then have that for almost all t ∈ I, X 1/q |Hi |Lq (Ω) ≤ C |Dyβ˜ Dti [(f ◦ Ti )m ]|qLq (Bi ) |β|≤l

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for all i. The sum on the right is finite by hypothesis so that, as in the proof j0 X of (b) above, Hi converges strongly in Lq (Ω) as j0 → ∞ for all such t and 1

X

the limit H is in Lp (I; Lq (Ω) with norm bounded by C

(α,i)

hf iLq , ∂Ω . The

|α|≤l

same bound therefore applies to |Hm |Lp,q (Ω×I) . To complete the proof we have to show that Hm = Dt Dxα (S 1 f )m . To do this let ϕ ∈ Cc∞ (Ω × I ◦ ) and compute Z XZ 1 i α (S f )m Dt Dx ϕ = (Fi1 )m Dti Dxα ϕ Ω×I

(Ω∩Ui )×I

i

=(−1)i+|α|

XZ

=(−1)i+|α|

XZ

i

i

(Ω∩Ui )×I

(Ω∩Ui )×I

   Dxα gi · Dti (f ◦ Ti )m ◦ Π ◦ Ri−1 ϕ Hi ϕ = (−1)i+|α|

Z Hϕ, Ω×I

Dti Dxα (S 1 f )m

so that = Hm ∈ Lp,q (Ω × I) with the required bound. A similar argument applies to S 2 f and completes the proof of (e). 

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A REPRESENTATION THEOREM FOR BOUNDARY INTEGRALS

219

Notes and References

The material of this text depends to a considerable extent on the foundation laid by the remarkable development of mathematical analysis in the first half of the twentieth century. The Lebesgue measure, weak differentiation, Hilbert and Banach spaces, Sobolev spaces and spectral theory have all proved to be essential concepts and structures for modern applied analysis. An excellent exposition of the history of this development is given in Naumann [31]. Our primary source for the details of the theory of Sobolev spaces has been Adams [1], more compact treatments being given in Ziemer [46], Gilbarg and Trudinger [19] and Evans [12]. The brief account in Chapter 4 of elliptic regularity in Sobolev spaces follows a standard approach included in many texts, most of which amplify the material in important ways. Those of Ladyzhenskaya and Uralceva [27], Bers, John and Schechter [5], Neˇcas [33], Grisvard [20], Gilbarg and Trudinger [19] and Evans [12] have emerged as standards. The definitive work on C k,α regularity of solutions of elliptic systems of arbitrary order is given in Douglas and Nirenberg [10] and Agmon, Douglis and Nirenberg [2] and [3]. These elementary expositions of elliptic systems typically reflect an elegant and efficient application of fundamental concepts in Hilbert space theory to the construction of solutions via the basic theory of Sobolev spaces. The exposition in Chapters 2, 3 and 5 of the present text seeks to duplicate this approach for time–dependent problems. In particular, the abstract Hilbert space formulations in chapter 2 allow for application to problems well beyond the second order systems considered in chapters 3 and 5 (Exercises 3.2 and 3.3 for example). More standard approaches to parabolic equations are given in Ladyzhenskaya et. al. [26], Friedman [15] and [16], Lieberman [30] and Evans [12]. In particular [26] includes a treatment of parabolic systems and has considerable overlap with Chapter 5 of the present text. Sections 3.2 and A.4 develop material first encountered by the author in the original edition of Neˇcas [33]: Rademacher’s theorem and the characterization of the dual of C0 (X) provide derivations of the basic facts of calculus on hypersurfaces which are quite straightforward and yet sufficiently general to accommodate boundaries with corners and edges. This material deserves greater visibility in the curriculum. The spaces Lp,q introduced in sections 3.2 and A.5 have received little attention heretofore but seem natural, perhaps even necessary in the present context. Indeed, one may regard the solution of the basic problem (3.1) as a mapping from the time axis into the set of actions on the test function space, with the Bochner space representation and its measurable counterpart simply being alternative representations of this mapping. Both are useful. In particular, the measurable counterpart representation defined in terms of the spaces Lp,q avoids the pitfalls associated with the Sierpinski example and

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also simplifies the calculus required for the treatment of nonlinear problems, as in section 6.4. Theorems 6.4 and 6.5 in section 6.2 are standard versions of the first and most basic results in the mathematical theory of the Navier–Stokes equations for incompressible flow, those of Leray [29] and Hopf [23]. Similar expositions are given in the texts of Temam [41], Galdi [18] and Tsai [44], among many others, each extending the basic theory in a particular direction. The key step in the construction is Theorem 6.4, which provides the required strong local convergence of approximate solutions. This result depends in turn on the inequality (6.13) which first appears perhaps in Friedrichs [17]. A very substantial research effort proceeding from this basic theory ensued in the second half of the twentieth century, generating a considerable literature. The survey articles of Temam [42] and Lemari´e-Rieusset [28] review these developments, but a corresponding review for the subsequent decades has yet to appear. Section 6.5 and Exercise 6.9 give the most basic result and a representative example in the very broad research area concerned with the asymptotic behavior of solutions of quasilinear systems. A basic reference is that of Henry [22], in which a theory of analytic semigroups of operators is developed and which includes a number of applications, including to parabolic systems of reaction-diffusion equations. There is a considerable literature addressing the stability question of Section 6.5 when the underlying known solution is a one-dimensional traveling wave. Representative results and examples are given in Sattinger [37] and [38], Jones [25], Texier and Zumbrun [43] and Zumbrun [47]. A completely different approach seeks to characterize the ensemble of asymptotic states corresponding to the entire set of allowable initial data for a given system. Representative discussions are given in Constantin et al. [7] and Eden et al. [11]. Section 6.6 follows closely the paper of Chueh, Conway and Smoller [6], which extended earlier work of Weinberger [45] on invariant regions. [6] also gives partial results on the necessity of hypotheses (a)–(c) of Theorem 6.13 and provides further examples. The numerous citations of [6] speak to its applicability, although in many cases invariance is more easily established directly by adapting the proof of the theorem rather than by checking its hypotheses.

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[17] Kurt Friedrichs, Spektraltheorie halbbeschr¨ ankter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann. 109 (1934), no. 1, 685–713. [18] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Second Edition, Springer Monographs in Mathematics, Springer, New York, 2011. [19] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer-Verlag, Berlin, 1983. [20] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. [21] Paul R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1957) edition. [22] Daniel B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, 1981. ¨ [23] Eberhard Hopf, Uber die Anfangswertaufgabe f¨ ur die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213–231. [24] Kelsey Houston-Edwards, Numbers Game, Sci. Amer. Sept. (2019), 35-40. [25] Christopher K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc. 286 (1984), no. 2, 431–469. [26] O. A Ladyˇ zenskaja, V. A. Solonnikov, and N. Ural0ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968. [27] Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. [28] P. G. Lemari´ e-Rieusset, Recent developments in the Navier-Stokes problem 431 (2002), xiv+395. [29] Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248. [30] Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [31] J. Naumann, Notes on the prehistory of Sobolev spaces, Bol. Soc. Port. Mat. 63 (2010), 13–55. English language version available at http://edoc.hu-berlin.de/series/mathematikpreprints/2002-2/PDF/2.pdf. emoires de L’Acad´ emie Royale des [32] M. Navier, Sur les Lois du Mouvement des Fluides, M´ Sciences, Paris. Tome VI (1823), 390-440. [33] Jindˇrich Neˇ cas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner. [34] H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. [35] Walter Rudin, Functional analysis, Second, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. [36]

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Index

C(I; ·), 4 C(I; Hweak ), 4 Cc∞ , 3 λ (I; ·), 4 C λ (I; ·), , Cloc k , 2 Cbdd k domain, 7 Cbdd Cck , 3 0,(λ,λ0 )

Cbdd

um , 206 u∗ , 17, 25, 26 uAC , 208 wAC(I; W k,q (Ω)), 212 wAC, wAC p , 17 Div, 81 adjoint argument, 46

,3

k,λ Cbdd ,2 k,λ Cbdd domain, Hk, 3

biharmonic operator, 9 bilinear form, 3 Bochner space, 5 bootstrapping, 98 boundary conditions Dirichlet, 61 essential, 29, 61, 81 natural, 61 Neumann, 61 boundary measure, 63–64, 198

7

H0k , 3 I ◦ , 66 Lp (I; W k,q (Ω)), 212 Lp (I; ·), Lploc (I; ·), 5 Lp,2 (Ω × I), 66 (Ω × I), 66 Lp,2 (t)loc Lp,q (X × I), 202 Lp,q (X × I), 202 (t)loc Lip(I; ·), Liploc (I; ·), 4 W k,p , 3 X∗, 3 [H k ]N , 3

coercive, 25 compatibility conditions, 29, 41 cone condition, 6 continuum hypothesis, 65 difference quotients, 87–88 Dirichlet condition, 61 distribution derivative, 3 Div, 60 divergence theorem, 64, 200 Duhamel integral, 52

N

[W k,p] , 3 [·]I,V , [·], 5 Ik0 , 188 Ik , 188 Il , 79 (α,j) h·iLq ,∂Ω , 103 hu, viH , 3 h·iα Lq ,∂Ω , 103 hϕiα Lq ,∂Ω , 79 B(X1 , X2 ), 3 L(X1 , X2 ), 3 ⊥, 62 | · |X , 3 || · ||X1 ,X2 , 3 a ∨ b ≡ max{a, b}, 64 a ∧ b ≡ min{a, b}, 34 tmax , 192 uc , 115

elliptic systems, 77–97 energy equality, 12, 13, 44, 45, 74 energy estimates, 73 existence linear ode’s, 192 linear parabolic systems, 67, 73 nonlinear ode’s, 191 problems in Hilbert space, 12, 24, 26, 41, 44 FitzHugh–Nagumo equations, 185 Fourier series, 52 Fourier transform, 9, 46 225

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226

global regularity, 77 H¨ older continuity, 4 heat equation, 46 incompatible data, 40–43, 122–125 interior regularity, 77 interpolation spaces, 58

INDEX

weak ∗-measurability, 4 absolute continuity, 16 derivative, 3 measurability, 4 weak form, 1, 41, 112 Young’s inequality, 75

Lax-Milgram theorem, 24 Leibniz rule, 19 Lipschitz continuity, 4 domain, 7 domain, uniformly, 7 maximal time of existence ode’s, 192 maximum principle, 175 mollifier, 26 multipliers, 85 Neumann condition, 61 normal vector field, 63–64, 198 ordinary differential equations, 190–193 parabolic, 63 partition of unity, 8 product rule, 18, 196 Rademacher’s theorem, 197 Radon measure, 64, 187 regularity parabolic systems, 111–125 problems in Hilbert space, 24–43 quasilinear systems, 148–166 Rellich-Kondrachov Theorem, 189 segment condition, 6 Sierpinski example, 65 Sobolev imbeddings, 187 space, 3 spectral gap, 58 spectral measures, 193–197 spectral representations, 52 strong form, 20, 21, 41, 74, 112 symmetric systems, 43–55, 72–75 trace theorem, 64, 200 uniformly parabolic, 63 uniqueness linear ode’s, 192 linear parabolic systems, 67, 73 nonlinear ode’s, 191 problems in Hilbert space, 12, 24, 26, 44 vector-valued Lp spaces, 5

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224 223 222 221

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This monograph presents a systematic theory of weak solutions in Hilbert-Sobolev spaces of initial-boundary value problems for parabolic systems of partial differential equations with general essential and natural boundary conditions and minimal hypotheses on coefficients. Applications to quasilinear systems are given, including local existence for large data, global existence near an attractor, the Leray and Hopf theorems for the Navier-Stokes equations and results concerning invariant regions. Supplementary material is provided, including a self-contained treatment of the calculus of Sobolev functions on the boundaries of Lipschitz domains and a thorough discussion of measurability considerations for elements of Bochner-Sobolev spaces. This book will be particularly useful both for researchers requiring accessible and broadly applicable formulations of standard results as well as for students preparing for research in applied analysis. Readers should be familiar with the basic facts of measure theory and functional analysis, including weak derivatives and Sobolev spaces. Prior work in partial differential equations is helpful but not required.

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