Linear and Quasi Linear Evolution Equations in Hilbert Spaces: Exploring the Anatomy of Integers 0821875760, 978-0-8218-7576-6

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Linear and Quasi-linear Evolution Equations in Hilbert Spaces

Pascal Cherrier Albert Milani

Graduate Studies in Mathematics Volume 135

American Mathematical Society

Linear and Quasi-linear Evolution Equations in Hilbert Spaces

Linear and Quasi-linear Evolution Equations in Hilbert Spaces Pascal Cherrier Albert Milani

Graduate Studies in Mathematics Volume 135

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 35L15, 35L72, 35K15, 35K59, 35Q61, 35Q74.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-135

Library of Congress Cataloging-in-Publication Data Cherrier, Pascal, 1950– Linear and quasi-linear evolution equations in Hilbert spaces / Pascal Cherrier, Albert Milani. p. cm. — (Graduate studies in mathematics ; v. 135) Includes bibliographical references and index. ISBN 978-0-8218-7576-6 (alk. paper) 1. Initial value problems. 2. Differential equations, Hyperbolic. 3. Evolution equations. 4. Hilbert space. I. Milani, A. (Albert) II. Title. III. Title: Linear and quasilinear evolution equations in Hilbert spaces. QA378.C44 2012 515.733—dc23 2012002958

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

17 16 15 14 13 12

We dedicate this work to our wives, Annick and Claudia, whose love and support has sustained us throughout its redaction. We bow with respect to the memory of our Teachers, Thierry Aubin and Tosio Kato, who have been a continuous source of inspiration and dedication.

Magna non sine Difficultate

Contents

Preface

ix

Chapter 1. Functional Framework

1

§1.1. Basic Notation

1

§1.2. Functional Analysis Results

4

§1.3. H¨older Spaces

7

§1.4. Lebesgue Spaces

9

§1.5. Sobolev Spaces §1.6. Orthogonal Bases in

13 H m (RN )

§1.7. Sobolev Spaces Involving Time Chapter 2. Linear Equations

51 60 77

§2.1. Introduction

77

§2.2. The Hyperbolic Cauchy Problem

78

§2.3. Proof of Theorem 2.2.1

81

§2.4. Weak Solutions

104

§2.5. The Parabolic Cauchy Problem

107

Chapter 3. Quasi-linear Equations

119

§3.1. Introduction

119

§3.2. The Hyperbolic Cauchy Problem

122

§3.3. Proof of Theorem 3.2.1

131

§3.4. The Parabolic Cauchy Problem

145

vii

viii

Contents

Chapter 4. Global Existence

153

§4.1. Introduction

153

§4.2. Life Span of Solutions

155

§4.3. Non Dissipative Finite Time Blow-Up

159

§4.4. Almost Global Existence

171

§4.5. Global Existence for Dissipative Equations

175

§4.6. The Parabolic Problem

214

Chapter 5. Asymptotic Behavior

233

§5.1. Introduction §5.2. Convergence

233 uhyp (t)

→

usta

234

§5.3. Convergence upar (t) → usta

241

§5.4. Stability Estimates

244

§5.5. The Diffusion Phenomenon

278

Chapter 6. Singular Convergence

293

§6.1. Introduction

293

§6.2. An Example from ODEs

295

§6.3. Uniformly Local and Global Existence

301

§6.4. Singular Perturbation

305

§6.5. Almost Global Existence

326

Chapter 7. Maxwell and von Karman Equations

335

§7.1. Maxwell’s Equations

335

§7.2. von Karman’s Equations

343

List of Function Spaces

361

Bibliography

365

Index

375

Preface

1. In these notes we develop a theory of strong solutions to linear evolution equations of the type (0.0.1)

ε utt + σ ut − aij (t, x) ∂i ∂j u = f (t, x) ,

and their quasi-linear counterpart (0.0.2)

ε utt + σ ut − aij (t, x, u, ut , ∇u) ∂i ∂j u = f (t, x) .

In (0.0.1) and (0.0.2), ε and σ are non-negative parameters; u = u(t, x), t > 0, x ∈ RN , and summation over repeated indices i, j, 1 ≤ i, j ≤ N , is understood. In addition, and in a sense to be made more precise, the quadratic form RN  ξ → aij (· · · ) ξ i ξ j is positive definite. We distinguish the following three cases. (1) ε > 0 and σ = 0. Then, (0.0.1) and (0.0.2) are hyperbolic equations; in particular, when ε = 1, they reduce to (0.0.3)

utt − aij ∂i ∂j u = f , and when aij (· · · ) = δij (the so-called Kronecker δ, defined by δij = 0 if i = j, and δij = 1 if i = j), (0.0.3) further reduces to the classical wave equation

(0.0.4)

utt − Δ u = f .

(2) ε = 0 and σ > 0. Then, (0.0.1) and (0.0.2) are parabolic equations; in particular, when σ = 1, they reduce to (0.0.5)

ut − aij ∂i ∂j u = f , ix

x

Preface

and when aij (· · · ) = δij , (0.0.5) further reduces to the classical heat equation (0.0.6)

ut − Δ u = f .

(3) ε > 0 and σ > 0. Then, (0.0.1) and (0.0.2) are dissipative hyperbolic equations; in particular, when ε = σ = 1, they reduce to (0.0.7)

utt + ut − aij ∂i ∂j u = f , and when aij (· · · ) = δij , (0.0.7) further reduces to the so-called telegraph equation

(0.0.8)

utt + ut − Δ u = f .

We prescribe that u should satisfy the initial conditions (or Cauchy data) (0.0.9)

u(0, x) = u0 (x) ,

ε ut (0, x) = ε u1 (x) ,

where u0 and u1 are given functions on RN , and the second condition is vacuous if ε = 0 (that is, in the parabolic case we only prescribe the initial condition u(0, x) = u0 (x)). Our purpose is to show that the Cauchy problems (0.0.1) + (0.0.9) and (0.0.2) + (0.0.9) are solvable in a suitable class of Sobolev spaces; we call the corresponding solutions strong. By this, we mean that the solutions we seek should be functions t → u(t), which are valued in a Sobolev space H r := H r (RN ), and possess a sufficient number of derivatives, either classical or distributional, so that equations (0.0.1) and (0.0.2) hold for (almost) all t and all x. More precisely, when ε > 0 we seek for solutions of (0.0.1) and (0.0.2) in the space (0.0.10)

C 0 ([0, T ]; H s+1 ) ∩ C 1 ([0, T ]; H s ) ∩ C 2 ([0, T ]; H s−1 ) ,

for some T > 0, where s ∈ N is such that s > N2 + 1; this condition implies that strong solutions are also classical. When ε = 0, we seek instead for solutions of (0.0.1) and (0.0.2) in the space (0.0.11)

{u ∈ C([0, T ]; H s+1 ) | ut ∈ L2 (0, T ; H s )} .

In addition, we want to show that the Cauchy problems (0.0.1) + (0.0.9) and (0.0.2) + (0.0.9) are well-posed, in Hadamard’s sense, in these spaces; that is, that their solutions should be unique and depend continuously on their data f , u0 and u1 (of course, the latter only for ε > 0). Finally, we also consider equations with lower order terms, i.e., (0.0.12)

ε utt + σ ut − aij ∂i ∂j u = f + bi ∂i u + c u ,

in particular in the linear case, as well as equations in the divergence form (0.0.13)

ε utt + σ ut − ∂j (aij ∂i u) = f + bi ∂i u + c u .

Preface

xi

In the quasi-linear case, equations (0.0.2) will in general have only local solutions; that is, even if the source term f is defined on a given interval [0, T ], or on all of [0, +∞[, the solution will be defined only on some interval [0, τ ], with τ < T , and cannot be extended to all of [0, T ]. 2. Our main goal is to develop a unified treatment of equations (0.0.1) and (0.0.2), both in the hyperbolic (either dissipative, or not) and the parabolic case, following a common constructive method to solve either problem. In the linear case, of course, a unified theory for both hyperbolic and parabolic equations (0.0.1), in a suitable framework of Hilbert spaces, has been presented by Lions and Magenes in their three-volume treatise [101, 102, 103], where they introduced a variety of arguments and techniques to solve fairly general kinds of initial-boundary value problems. The main reason we seek a unified treatment of equations (0.0.1) and (0.0.2) in the quasi-linear case is that this allows us to compare the solutions to the hyperbolic and the parabolic equations, in a number of ways. In particular, when (0.0.2) admits global solutions (that is, defined on all of [0, +∞[), we wish to study their asymptotic behavior as t → +∞. We assume that the coefficients aij in (0.0.2) depend only on the first-order derivatives ut and ∇u, and are interested in the following questions. The first is that of the convergence of the solutions of (0.0.2) to the solution of the stationary equation (0.0.14)

− aij (0, ∇v) ∂i ∂j v = h .

The second, when ε and σ > 0, is the comparison of the asymptotic profiles of the solutions of the dissipative hyperbolic equation (0.0.2) to those of the solutions of the parabolic equation, corresponding to ε = 0. The third question, related to (0.0.2), is the singular perturbation problem, concerning the convergence, as ε → 0, of solutions uε of the dissipative hyperbolic equation to the solution u0 of the parabolic equation. 3. Linear hyperbolic equations of the type (0.0.12) and (0.0.13), in particular when σ = 0, have been studied by many authors, who have considered the corresponding Cauchy problem in different settings. An elementary introduction to both kinds of equations can be found in Evans’ textbook [47]; for more advanced and specific results, renouncing to any pretense of a comprehensive list, we refer, e.g., to Friedrichs [51], Kato [72], Mizohata [122], and Ikawa [63], who resort to a solution method based on a semigroup approach, complicated by the fact that the coefficients aij depend on t. The semigroup method has later been successfully applied to quasi-linear equations; see, e.g., Okazawa [130], Tanaka [153, 154, 155], and, for a more abstract approach, Beyer [15]. Other methods can be seen, e.g., in Racke [136], and Sogge [151], based respectively on the Cauchy-Kovaleskaya and the Hahn-Banach theorems. In the solution theory we present in Chapter

xii

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2, we prefer to follow the so-called Faedo-Galerkin method, which is a generalization of the method of separation of variables, and which explicitly constructs the solution to (0.0.12) as the limit of a sequence of functions, each of which solves an approximate version of the problem, determined by its projection onto suitable finite-dimensional subspaces. The results we establish for (0.0.1) when ε > 0 are not specifically dependent on the fact that the equation is hyperbolic; in fact, the Faedo-Galerkin method can be readily adapted to obtain strong solutions of the linear parabolic equation. For general references to parabolic equations, both linear and quasi-linear, we refer, e.g., to Ladyzenshkaya, Solonnikov and Ural’tseva [86], Amann [6], Pao [131], Lunardi [107], Lieberman [96], and Krylov [83, 84], where these equations are mostly studied in the H¨ older spaces C m+α/2,2m+α(Q). For the numerical treatment of equation (2.1.1), we refer, e.g., to Meister and Struckmeier [112]. 4. The Cauchy problem for quasi-linear hyperbolic equations such as (0.0.2), as well as their counterpart in divergence form (0.0.13), has been studied by many authors, who have provided local (and, when possible, global, or, at least, almost global) solutions with a number of methods, including a nonlinear version of the Galerkin scheme and various versions of the Moser-Nash algorithm. Renouncing again to any pretense of a comprehensive list, we refer, e.g., to the classical treatise by Courant and Hilbert [38], as well as the more recent works by Kato [72], John [66], Kichenassamy [74], Racke [136], Sogge [151], H¨ormander [57], as well as Lax [88], Li Ta-Tsien [91], and Li Ta-Tsien and Wang Li-Ping [93]. In the above context, local solution means a solution defined on some interval [0, τ ]; almost global solution means a solution defined on a prescribed interval [0, T ], of finite but arbitrary length, possibly subject to some restrictions on the size of the data, depending on T ; global solution means a solution also defined on arbitrary intervals [0, T ], but with restrictions on the size of the data, if any, independent of T (thus, these solutions are defined on the entire interval [0, +∞[). Finally, we also consider global bounded solutions; that is, global solutions which remain bounded as t → +∞. In Chapter 3, we present a solution method based on a linearization and fixed point method, introduced by Kato [70, 71, 72], in which we apply the results for the linear theory, developed in Chapter 2. As for the linear case, the results we establish for (0.0.2), when ε > 0, are not specifically dependent on the fact that the equation is hyperbolic; in fact, the linearization and fixed point method can be adapted to obtain local, strong solutions of the quasi-linear parabolic equation

(0.0.15)

ut − aij (t, x, u, ∇u) ∂i ∂j u = f ,

Preface

xiii

as well as of the analogous equation in divergence form. Moreover, these methods can also be applied to other types of evolution equations, such as the so-called dispersive equations considered in Tao [156], and Linares and Ponce [97]; these include, among others, the Schr¨odinger and the Kortewegde-Vries equations. 5. Not surprisingly, many more results are available on semi-linear hyperbolic and parabolic equations; that is, equations of the form (0.0.16)

utt − Δ u = f (t, x, u, Du) ,

(0.0.17)

ut − Δ u = f (t, x, u, ∇u) .

Among the many works on this subject, we limit ourselves to cite Strauss [150], Todorova and Yordanov [159], Zheng [168], Quittner and Souplet [133], Cazenave and Haraux [24], and the references therein. Most of these results concern the well-posedness of the Cauchy problem for (0.0.16) or (0.0.17) in a suitable weak sense; strong solutions are then obtained by appropriate regularity theorems, and the asymptotic behavior of such weak solutions can be studied in terms of suitable attracting sets in the phase space; see, e.g., Milani and Koksch, [119]. In fact, we could try to develop a corresponding weak solution theory for hyperbolic and parabolic quasi-linear equations in the conservation form (0.0.18)

ε utt + σ ut − div[a(∇u)] = f ,

where a : RN → RN is monotone. However, there appears to be a striking difference between the hyperbolic and the parabolic situation. For the latter, i.e., when ε = 0 in (0.0.18), existence, uniqueness and well-posedness results for weak solutions, at least when a is strongly monotone, are available; see, e.g., Lions [99, ch. 2, §1], and Br´ezis [19]. In contrast, when ε > 0 the question of the existence of even a local weak solution to equation (0.0.18) (that is, in the space (0.0.10) with s = 0) is, as far as we know, totally open (unless, of course, a is linear). 6. To our knowledge, there are not yet satisfactory answers to the question of finding sharp life-span estimates for problem (0.0.2), at least in the functional framework we consider. On the other hand, rather precise results have long been available, at least for more regular solutions of the homogeneous equation; that is, when f ≡ 0 and u0 ∈ H s+1 ∩ W r+1,p , u1 ∈ H s ∩ W r,p , for suitable integers s N2 + 1, r < s, and p ∈ ]1, 2[. In this case, the situation also depends on the space dimension N ; more precisely, one obtains global existence of strong solutions if N ≥ 4, and also if N = 3 if the nonlinearity satisfies an additional structural restriction, known as the null condition. The proof of these results is based on supplementing the direct energy estimates used to establish local solutions, with rather

xiv

Preface

refined decay estimates of the solution to the linear wave equation. We refer to Racke [136], and John [66], for a comprehensive survey of the results of, among many others, John [65], Klainerman [75, 76], Klainerman and Ponce [81], as well as Klainerman [77] and Christodoulou [31], for the null condition. 7. The theory of quasi-linear evolution equations has many important applications. A non exhaustive list would include fluid dynamics (see, e.g., Majda [108], and Nishida [126]); general relativity, and, specifically, the so-called Einstein vacuum equations (Klainerman and Christodoulou [78], and Klainerman and Nicol`o [80]); wave maps (Shatah and Struwe [145], and Tao [156]); von Karman type thin plate equations (Cherrier and Milani [26, 27, 29], Chuesov and Lasiecka [33]); control and observability theory (Li [94]). Other applications, specifically of dissipative hyperbolic equations (0.0.2) with σ > 0 and ε small, include models of heat equations with delay (Li [92], Cattaneo [21], Jordan, Dai and Mickens [67], Liu [104]), where ε is a measure of the delay or heat relaxation time; Maxwell’s equations in ferromagnetic materials (Milani [114]), where ε is a measure of the displacement currents, usually negligible; simple models of laser optic equations (Haus [54]), where ε is related to measures of low frequencies of the electromagnetic field; traffic flow models (Schochet [140]), where ε is a measure of the drivers’ response time to sudden disturbances (which, one hopes, should be small); models of random walk systems (Hadeler [53]), where ε is related to the reciprocal of the turning rates of the moving particles; construction materials with strong internal stress-strain relations, measured by parameters related to the reciprocal of ε (see, e.g., Banks et al. [9, 10] for the case of a one-dimensional elastomer); and models of time-delayed information propagation in economics (Ahmed and Abdusalam [2]). 8. These notes have their origin in a series of graduate courses and seminars we gave at Fudan University, Shanghai, at the Universit´e Pierre et Marie Curie (Paris VI), the Technische Universit¨ at Dresden, and the Pontificia Universidad Cat´ olica of Santiago, Chile. Some of the material we cover is relatively well known, although many results, in particular on hyperbolic equations, seem to be somewhat scattered in the literature, and often subordinate to other topics or applications. Other results, in particular on the diffusion phenomenon for quasi-linear hyperbolic waves, appear to be new. Our intention is, in part, to provide an introduction to the theory of quasi-linear evolution equations in Sobolev spaces, organizing the material in a progression that is as gradual and natural as possible. To this end, we have tried to put particular care in giving detailed proofs of the results we present; thus, if successful, our effort should give readers the necessary basis to proceed to the more specialized texts we have indicated above. In this sense, these notes are not meant to serve as an advanced PDEs textbook;

Preface

xv

rather, their didactical scope and subject range is restricted to the effort of explaining, as clearly as we are able to, one possible way to study two simple and fundamental examples of evolution equations (hyperbolic, both dissipative or not, and parabolic) on the whole space RN . In addition, we also hope that these notes may serve as a fairly comprehensive and selfcontained reference for researchers in other areas of applied mathematics and sciences, in which, as we have mentioned, the theory of quasi-linear evolution equations has many important applications. We should perhaps mention explicitly the fact that, given the introductory level of these notes, we have limited ourselves to present only those results that can be obtained by resorting to one of the most standard methods for the study of equations (0.0.1) and (0.0.2); namely, that of the a priori, or energy, estimates. Of course, this choice forces us to neglect other methods that are more specific to the type of equation under consideration and which are extensively studied in more specialized texts. For example, we do not cover, but only mention, the theory of H¨older solutions of parabolic quasi-linear equations (0.0.15) (see, e.g., Krylov [83]), or the theory of weak solutions to quasilinear first-order hyperbolic systems of conservation laws, as in (3.1.8) of Chapter 3 (see, e.g., Alinhac [5], or Serre [143, 144]), and we do not even mention other very specific and highly refined techniques which have been developed and are being developed for the study of these equations, such as, to cite a few, the theory of nonlinear semigroup (see, e.g., Beyer [15]), the methods of pseudo-differential operators (see, e.g., Taylor [157]) and of microlocal analysis (see, e.g., Bony [17]). On the other hand, one can perhaps be surprised by the extent of the results one can obtain, by means of the one and same technique; that is, the energy method. As we have stated, this method has, among others, the advantage of allowing us to present our results in a highly unified way, and to show that, even today, classical analysis allows us to deal in a simple way, by means of standard and well-tested techniques, with relevant questions in the theory of PDEs of evolution, which are still the subject of considerable study. 9. The material of these notes is organized as follows. In Chapter 1 we provide a summary of the main functional analysis results we need for the development of the theory we wish to present. In Chapter 2 we develop a strong solution theory for the Cauchy problem for the linear equation (0.0.12), with existence, uniqueness, regularity, and well-posedness results for both the hyperbolic equation (ε = 1, σ = 0) and the parabolic one (ε = 0, σ = 1). In Chapter 3 we construct local in time solutions to the quasi-linear equations (0.0.2) and (0.0.15), by means of a linearization and fixed-point technique, in which we apply the results on the linear equations we established in the previous Chapter. Again, we give existence, uniqueness, regularity, and well-posedness results for both types of equation. In

xvi

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Chapter 4 we study the question of the extendibility of these local solutions to either a prefixed finite but arbitrary time interval [0, T ] (almost global and global existence), or to the whole interval [0, +∞[ (global and global bounded existence). We present an explicit example of blow-up in finite time for solutions of the quasi-linear equation (0.0.3) in one dimension of space, as well as some global and almost global existence results for either equation, when the data u0 , u1 and f are sufficiently small. We also present a global existence result for the parabolic equation (0.0.15), for data of arbitrary size. In Chapter 5 we consider the asymptotic behavior, as t → +∞, of global, bounded, small solutions of (0.0.2), both dissipative hyperbolic (ε = 1) and parabolic (ε = 0), and we prove some results on their convergence to the solution of the stationary equation (0.0.14). In the homogeneous case f ≡ 0, we also establish some stability estimates, on the rate of decay to 0 of the corresponding solutions. We also give a result on the diffusion phenomenon, which consists in showing that, when f ≡ 0, solutions of the hyperbolic equation (0.0.7) (both linear and quasi-linear) asymptotically behave as those of the parabolic equation (0.0.5) of corresponding type. In Chapter 6 we consider a second way in which we can compare the hyperbolic and the parabolic problems; namely, we consider (0.0.2) as a perturbation, for small values of ε > 0, of the parabolic equation (0.0.2), with ε = 0. Denoting by uε and u0 the corresponding solutions, we study the problem of the convergence uε → u0 as ε → 0, on compact time intervals. We consider either intervals [0, T ] or [τ, T ], τ ∈ ]0, T [; that is, including t = 0 or not. In the former case, the convergence is singular, due to the loss of the initial condition on ut , and we give rather precise estimates, as t and ε → 0, on the corresponding initial layer . We mention in passing that the estimates we establish on the difference uε − u0 allow us also to deduce a global existence equivalency result between the two types of equations, in the sense that a global solution to the parabolic equation, corresponding to data of arbitrary size, exists, if and only if global solutions to the dissipative hyperbolic equation also exist, corresponding to data of arbitrary size, and ε is sufficiently small. We conclude the chapter with a global result for equation (0.0.2), with data of arbitrary size, when ε is sufficiently large. Lastly, in Chapter 7, we present two applications of the theory developed in the previous chapters. In the first example, we consider a model for the complete system of Maxwell’s equations, in which the use of suitable electromagnetic potentials allows us to translate the first-order Maxwell’s system into a second-order evolution equation of the type (0.0.2). In this model, the parameters ε and σ can be interpreted as a measure, respectively, of the displacement and the eddy currents; in some situations, such as when the equations are considered in a ferro-magnetic medium, displacement currents are negligible with respect to the eddy ones, and this observation leads to

Preface

xvii

the question of the control of the error introduced in the model when the term ε utt is neglected. In related situations, one is interested in periodic phenomena, with relatively low frequencies, thus leading to the question of the existence of solutions on the whole period of time. It is our hope that these questions may be addressed, at least to some extent, by the results of the previous chapters. In the second example, we consider two systems of evolution equations, of hyperbolic and parabolic type, relative to a highly nonlinear elliptic system of von Karman type equations on R2m , m ≥ 2. These equations generalize the well-known equations of the same name in the theory of elasticity, which correspond to the case m = 1, and model the deformation of a thin plate due to both internal and external stresses. For both types of systems we show the existence and uniqueness of local in time strong solutions, which again can be extended to almost global ones if the initial data are small enough. Even though these systems do not fit exactly in the framework of second-order evolution equations for which our theory is developed, their study allows us to show that the unified methods we present can be applied to a much wider class of equations than those of the form (0.0.2). 10. Finally, we mention that an analogous unified theory could be constructed for initial-boundary value problems for equation (0.0.2), in a subdomain Ω ⊂ RN , with u subject to appropriate conditions at the boundary ∂Ω of Ω, assumed to be adequately smooth. The type of results one obtains is qualitatively analogous, but in a different functional setting for both the data and the solutions. Indeed, the data have to satisfy a number of so-called compatibility conditions at {t = 0} × ∂Ω, which are different in the hyperbolic and parabolic cases, and the integrations by parts that are usually carried out in order to establish the necessary energy estimates (see Chapter 2) would involve boundary terms that do not appear when Ω = RN . For example, in our papers [116, 117] we considered the simple case where equation (0.0.2) is studied in a bounded domain, with homogeneous Dirichlet boundary conditions; other results can be found, e.g., in Dafermos and Hrusa [40]. To discuss this topic in a meaningful degree of detail would require a whole new book; here, we limit ourselves to a reference to the above-mentioned papers, and to the literature quoted therein, for a brief overview of the technical issues typically encountered in this situation. Acknowledgments. In the preparation of these notes, we have benefitted from the generous support of a number of agencies, including grants from the Fulbright Foundation (Pontificia Universitad Cat´ olica of Santiago, Chile, 2006), the Alexander von Humboldt Stiftung (Institut f¨ ur Analysis, Technische Universit¨ at Dresden, 2008), and the Deutsche Forschungsgemeinschaft (TU-Dresden, 2010). We are grateful to the departments of mathematics of these institutions for their kind hospitality. We are also greatly

xviii

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indebted to Professor A. Negro of the University of Turin, Italy, to Professors G. Walter and H. Volkmer of the University of Wisconsin-Milwaukee, to Professor R. Picard of the TU-Dresden, and to Professor Zheng SongMu of Fudan University, Shanghai, for their constant encouragement and important suggestions. Last, but not least, we owe special gratitude to Ms. Barbara Beeton and Ms. Jennifer Wright Sharp of the American Mathematical Society Technical Support team, for their invaluable help in solving all the TEX-nical and typographical problems involved in the production of the final version of this book.

Pascal Cherrier Universit´e Pierre et Marie Curie, Paris Albert Milani University of Wisconsin - Milwaukee

Chapter 1

Functional Framework

In this chapter we introduce the functional framework in which we develop the results on strong solutions of the linear and quasi-linear evolution equations we wish to present. After introducing notations and conventions used throughout these lectures, we briefly review some of the functional analysis background we need. The results of this chapter, mostly on Sobolev spaces, are in general well known; when we do not provide a reference or a proof, these can be found in such fundamental books as Adams and Fournier [1], Aubin [7], Lieb and Loss [95], or Rudin [138, 139].

1.1. Basic Notation Unless otherwise specified, we consider real-valued functions, defined either on domains Ω ⊆ RN (i.e., Ω is open and connected) with a smooth boundary ∂Ω, or on cylinders Q := ]0, T [ × Ω, and which are at least Lebesgue measurable there. In this chapter, unless otherwise stated, we only consider the cases when Ω is bounded or Ω = RN ; we allow the possibility T = +∞. We denote by x := (x1 , . . . , xN ) and (t, x) the generic points of, respectively, Ω and Q, by | · | the norm in RN , and by (e1 , . . . , eN ) the standard orthonormal basis of RN . The abbreviations “a.e.” and “a.a.” stand, respectively, for “almost everywhere” and “almost all”, either in Q or in RN , with reference to the Lebesgue measure in these sets. 1. Intervals, Balls, Integer Part. Bounded intervals of R are denoted by [a, b] if closed, ]a, b[ if open, [a, b[ or ]a, b] otherwise. For unbounded intervals and a ∈ R, we occasionally adopt the notation R≥a := [a, +∞[, and similarly for R>a , R≤a , R0 denotes the set of the strictly positive 1

2

1. Functional Framework

integers. We also set (1.1.1)

m Rm | xi ≥ 0 , ≥0 := {x ∈ R

1 ≤ i ≤ N} .

For x ∈ RN and r > 0, B(x, r) := {y ∈ RN | |x − y| < r} denotes the open ball of center x and radius r, and B(x, r) its closure. Given x ∈ [0, +∞[, we denote by x its integer part; that is, x := max{n ∈ N | n ≤ x}. 2. Derivatives. If (t, x) → u(t, x) is a smooth function, we denote its first partial derivatives by ∂u ∂u (1.1.2) ∂t u := ∂j u := , 1≤j≤N. =: ut , ∂t ∂xj We also set ∇u := (∂1 u, . . . , ∂N u), Du := {ut , ∇u}, and (1.1.3)

∂2u utt := 2 , ∂t

Δ u :=

N  ∂2u j=0

∂x2j

.

More generally, given a multi-index α = (α1 , . . . , αN ) ∈ NN , we denote by |α| := α1 + · · · + αN its length, and set (1.1.4)

∂xα u :=

∂ |α| u αN = ∂1α1 · · · ∂N u. α α 1 N ∂ x1 · · · ∂ xN

Given a positive integer k, we denote by D k u (respectively, ∂xk u, ∂tk u) the set of all derivatives of u of order k (respectively, of the derivatives with respect to the space derivatives, or the time derivatives, only); that is, explicitly, (1.1.5)

D k u = {∂tr ∂xα u | r + |α| = k} ,

and so on. The notation ∂xα ϕ(u) means the partial derivative of order α of the composite function x → ϕ(u(x)); when necessary, we denote by (∂pα ϕ)(u), 1 ≤ p ≤ N , the image of u by the function ∂pα ϕ. Finally, the notation ∂xα f (x) ∂xβ g(x) means the product of the two functions ∂xα f and ∂xβ g, evaluated at x (as opposed to the α-th derivative ∂xα (f (x) ∂xβ g(x)) of the product f ∂xβ g). 3. Spaces of Continuously Differentiable Functions. Given m ∈ N, we introduce the following linear spaces: 1) C m (Ω), consisting of the functions f : Ω → R, which have continuous derivatives of order up to m; 2) Cbm (Ω), consisting of the functions in C m (Ω) which are bounded together with all their derivatives of order up to m; 3) C0m (Ω), consisting of the functions in C m (Ω), whose support is compact; 4) C m (Ω), consisting of functions which are restrictions to Ω of functions in C m (RN );

1.1. Basic Notation

3

5) Cbm (Ω), consisting of functions which are restrictions to Ω of functions in Cbm (RN ). Note that C0m (Ω) ⊂ Cbm (Ω), C m (RN ) = C m (RN ), and Cbm (RN ) = Cbm (RN ). When m = 0, we abbreviate C 0 (Ω) =: C(Ω), and similarly for the other spaces listed above. The spaces Cbm (Ω) and Cbm (Ω) are Banach spaces, with respect to the norm f Cbm (Ω  ) := max sup |∂xk f (x)| ,

(1.1.6)

0≤k≤m x∈Ω 

where Ω  denotes Ω or Ω. Analogous definitions hold for C ∞ (Ω), Cb∞ (Ω), C0∞ (Ω), C ∞ (Ω) and Cb∞ (Ω), as well as the corresponding spaces of continuously differentiable functions defined on a cylinder Q or its closure Q. 4. Integrals. When integration is over the whole of RN , we often abbreviate   (1.1.7) f (x) dx for f (x) dx . RN

Given an integrable function f from an interval [0, T ] into a Banach space X, and t ∈ [0, T ], we often write  t  t (1.1.8) uX dθ instead of u(θ)X dθ . 0

0

5. Convolution. The convolution of two functions f , g : RN → R is the function f ∗ g : RN → R defined by   (1.1.9) [f ∗ g](x) := f (x − y)g(y) dy = f (y)g(x − y) dy , whenever the integrals in (1.1.9) make sense. Because of Fubini’s theorem, the convolution product satisfies the associative property; that is, (1.1.10)

(f ∗ g) ∗ h = f ∗ (g ∗ h) ,

again, whenever the integrals in (1.1.10) make sense. 6. Conjugate Indices. Two numbers p, q ∈ R>1 are called conjugate indices if 1p + 1q = 1. This definition extends naturally to the cases p = 1, q = ∞, and p = ∞, q = 1. Unless otherwise stated, we will denote by p  the conjugate index of p ∈ [1, +∞]. Proposition 1.1.1. For all a, b ∈ R>0 and p ∈ R>1 , (1.1.11)

ab ≤

1 p

ap +

1 p



bp ,

which is known as Minkowski’s inequality. More generally, for all η > 0, 1  (1.1.12) ab ≤ η ap + Cη bp , Cη =  , p (pη)p  −1

4

1. Functional Framework

which is known as Minkowski’s inequality with weight η. Note that Cη → +∞ as η → 0. In particular, for p = p  = 2, (1.1.11) and (1.1.12) read (1.1.13)

ab ≤

1 2

a2 + 12 b2 ,

ab ≤ η a2 +

1 4η

b2 ,

which are known as Schwarz’s inequalities. 7. Constants. In the sequel, we will often establish estimates on the norm of solutions of differential equations such as (0.0.1), (0.0.2), or related ones, such as suitable approximations of them. We adopt the following convention on the use of constants appearing in these estimates. We denote by C a generic, universal constant, which may change from estimate to estimate or even within the same estimate. Thus, for example, we identify C, C 2 , eC , etc. Some of these constants depend only on N , either directly or via other quantities such as the norms of certain imbeddings, while others may depend on the coefficients of the equations; however, unless otherwise explicitly stated, these constants never depend on the data of the equation under consideration, nor on time t, nor on any of the functions involved in any of the formulas where such constants appear. When the specific value of a constant has to be fixed (for example, to define another), we number that constant, denoting it C0 , C1 , etc. When a constant depends on a parameter, such as ε, in a bounded way as the parameter grows or vanishes, we still denote the constant as C; when instead the constant grows unboundedly as the parameter grows or vanishes (e.g., as ε → 0), we denote it as Cε and likewise. An example of this use is the constant Cη in Minkowski’s inequality (1.1.12). Unless explicitly stated otherwise, we assume that all these constants are larger than 1. Finally, for m ∈ N≥1 we set (1.1.14) (1.1.15) (1.1.16)

K := {f ∈ C 1 (Rm ≥0 ; R≥0 ) | ∂j f ≥ 0 ,

1 ≤ j ≤ m} ,

K0 := {f ∈ K | f (0) = 0} , K∞ := {f ∈ C(R>0 ; R>0 ) | lim f (r) = +∞} . r→0

We use these spaces when we need to keep track of specific properties of the dependence of a constant on previously defined ones; in particular, we often introduce functions γ, κ, ψ ∈ K, while we reserve the letters ω and U for functions in K0 and K∞ , respectively. Note that functions in K are continuous, positive, and separately increasing with respect to each of their variables.

1.2. Functional Analysis Results We report some well-known definitions and results from functional analysis, that we use consistently in the sequel. All the vector spaces we consider are real.

1.2. Functional Analysis Results

5

1. Imbeddings. Let X1 and X2 be two Banach spaces with norms  · 1 and  · 2 , respectively. We say that: 1) X1 is continuously imbedded into X2 , with dense image, and we write X1 → X2 , if X1 is a dense subspace of X2 , and there is a constant K such that, for all x ∈ X1 , (1.2.1)

x2 ≤ K x1 .

2) X1 is compactly imbedded into X2 , and we write X1 →c X2 , if X1 → X2 and every bounded sequence of X1 contains a subsequence which converges in X2 . 2. Duality, Weak Convergence. If X and Y are Banach spaces, we denote by L(X, Y ) the linear space of all linear and continuous functions from X into Y . This space is itself a Banach space, with respect to the norm (1.2.2)

f L(X,Y ) := sup f (x)Y . x∈X xX =1

We denote by X  the topological dual of X, and by (1.2.3)

X  × X  (f, x) → f (x) =: f, xX  ×X ∈ R

the corresponding duality pairing. That is, X  := L(X, R), and the right side of (1.2.3) is the image, in R, of the element x ∈ X by the linear function f ∈ X  . X is norm reflexive if (X  )  =: X  is topologically isomorphic to X. If X is a Hilbert space, we identify f, xX  ×X = f, xX , the latter denoting the scalar product in X. A sequence (xn )n≥0 ⊂ X converges weakly to a limit x ∈ X if (1.2.4)

f, xn − xX  ×X → 0

in R

for all f ∈ X  . Likewise, a sequence (fn )n≥0 ⊂ X  converges weakly∗ to a limit f ∈ X  if (1.2.5)

fn − f, xX  ×X → 0

for all x ∈ X. In X  , weak∗ convergence is weaker than weak convergence. To see this, we note that, in accord with (1.2.4), fn → f weakly in X  if for all g ∈ X  , (1.2.6)

g, fn − f X  ×X  → 0 ;

thus, weak convergence in X  implies weak∗ convergence. Indeed, for each fixed x ∈ X, the map X   f → f, xX  ×X =: g(f ) is linear continuous; that is, g ∈ X  , so that (1.2.6) implies (1.2.5). On the other hand, (1.2.6) also shows that the converse is true if X is norm reflexive.

6

1. Functional Framework

Proposition 1.2.1. Let X and Y be Banach spaces, and f ∈ L(X, Y ). Let (xn )n≥1 ⊂ X and x ∈ X be such that xn → x weakly in X. Then, f (xn ) → f (x) weakly in Y . Weakly convergent sequences in X (respectively, weakly∗ convergent sequences in X  ) are bounded, while the converse is in general not true. Still, we have: Theorem 1.2.1. 1) Let (xn )n≥0 ⊂ X. If (xn )n≥0 converges weakly to some x ∈ X, then (1.2.7)

xX ≤ lim inf xn X . n→∞

That is, the norm is weakly lower-semicontinuous. If xn X ≤ M for some M independent of n, then xX ≤ M . 2) If X is norm reflexive, every bounded sequence (xn )n≥0 ⊂ X contains a weakly convergent subsequence (xnk )k≥0 . 3) Likewise, let (fn )n≥0 ⊂ X  . If (fn )n≥0 converges weakly∗ to some f ∈ X  , then (1.2.8)

f X  ≤ lim inf fn X  . n→∞

If fn X  ≤ M for some M independent of n, then f X  ≤ M . 4) (Alaoglu’s theorem [3]) If X is separable, every bounded sequence (fn )n≥0 ⊂ X  contains a weakly∗ convergent subsequence. 3. Bases. Let X be a Hilbert space, and let W = (wk )k≥1 be a sequence in X. 1) W is a total Fourier basis of X if W is a linearly independent set whose span is dense in X, and for all f ∈ X, the series (1.2.9)

∞  f, wk X wk k=1

converges to f in X (see (1.2.14) below). We call (1.2.9) the Fourier series of f with respect to W, and we write (1.2.10)

f =

∞ 

f, wk X wk .

k=1

If X is separable, it always admits a total Fourier basis. 2) W is an orthonormal basis of X if it is a basis and, in addition,  0 if i = j , (1.2.11) wi , wj X = δij := 1 if i = j .

1.3. H¨older Spaces

7

3) Let Y ⊂ X be a normed subspace of space X. W is a Y -regular basis if, whenever f ∈ Y , the series (1.2.9), which converges to f in X, also converges to f in Y (see (1.2.15) below). Let W be an orthonormal basis of X. For n ∈ N>0 , we set Wn := span{w1 , . . . , wn } ,

(1.2.12)

and we call Pn : X → Wn the projection of X into its finite-dimensional subspace Wn , defined by n  (1.2.13) Pn f := f, wk X wk , f ∈X. k=1

Then, each Pn is an orthogonal projection, and (1.2.10) means that lim f − Pn f X = 0 .

(1.2.14)

n→∞

In addition, if W is Y -regular, lim f − Pn f Y = 0 .

(1.2.15)

n→∞

1.3. H¨ older Spaces We briefly review the definitions of the H¨older spaces, for the main properties of which we refer, e.g., to Krylov [83, §3.1, §8.5]. 1. H¨ older Spaces in Ω. Let Ω ⊆ RN be a domain (not necessarily bounded). Given α ∈ ]0, 1[ and f : Ω → R, we set (1.3.1)

Hα (f ) := sup x,y∈Ω x=y

|f (x) − f (y)| . |x − y|α

For m ∈ N, we define the H¨older space on Ω of order α (1.3.2)

C m,α (Ω) := {f ∈ C m (Ω) | Hα (∂xm f ) < +∞} .

Functions in C 0,α (Ω) are called H¨older continuous in Ω and, in the limit case α = 1, Lipschitz continuous. Clearly, H¨ older continuous functions are uniformly continuous. The definition of the spaces C m,α (Ω) is somewhat more complicated, according to whether Ω is bounded or not; in either case, we say that f ∈ C m,α (Ω) if f is the extension to Ω of a function f˜ ∈ Cbm (Ω) ∩ C m,α (Ω). Clearly, C m,α (RN ) = Cbm (RN ) ∩ C m,α (RN ), while if Ω is bounded, C m,α (Ω) can be identified with C m,α (Ω) in the following sense. If f ∈ C m,α (Ω), we can evidently identify f with its restriction f˜ to Ω, which is in C m,α (Ω). Conversely, if f˜ ∈ C m,α (Ω), definition (1.3.2) implies that f˜ ∈ Cbm (Ω), because of the mean value theorem and the boundedness of Ω. Indeed, ∂xm f˜ is H¨older continuous, hence uniformly continuous, and therefore bounded, in Ω. Thus, f˜ can be uniquely extended to a function f on Ω, with f ∈ C m,α (Ω)

8

1. Functional Framework

by definition, and we identify f˜ with f . For example, the identity function on RN , i.e., the function x → i(x) := x, is such that i ∈ C 0,1 (RN ), but i∈ / C 0,α (RN ) if α ∈ ]0, 1[. In addition, i ∈ / C 0,α (RN ) if α ∈ ]0, 1], because it is not bounded. The space C m,α (Ω) is complete (thus, it is a Banach space), with respect to the norm (1.3.3)

|f |m,α := f Cbm (Ω) + Hα (∂xm f ) .

2. H¨ older Spaces in Q. Let Ω ⊆ RN be a domain, T ∈ ]0, +∞], and Q := ]0, T [×Ω. Given α ∈ ]0, 1[ and f : Q → R, we set (1.3.4)

˜ α (f ) := H

sup (t,x),(s,y)∈Q (t,x)=(s,y)

|f (t, x) − f (s, y)| . (|t − s| + |x − y|2 )α/2

For m ∈ N, we define the H¨older space on Q of order α (1.3.5)  C m+α/2,2m+α(Q) := f ∈ C(Q) | ∂tk ∂xλ f ∈ C(Q) , ˜ α (∂ k ∂ λ f ) < +∞ , H t x

2k + |λ| ≤ 2m ,

 2k + |λ| = 2m ,

and we say that f ∈ C m+α/2,2m+α (Q) if f is the extension to Q of a function f˜ ∈ C˜bm (Q) ∩ C m+α/2,2m+α(Q), where   (1.3.6) C˜bm (Q) := f ∈ Cb (Q) | ∂tk ∂xλ f ∈ Cb (Q) , 2k + |λ| ≤ 2m . Note that C˜bm (Q) ⊂ Cbm (Q). Again, if Ω = RN (1.3.7)

C m+α/2,2m+α (Q) = C˜bm (Q) ∩ C m+α/2,2m+α(Q) ,

while if both Ω and T are bounded we can identify C m+α/2,2m+α (Q) with C m+α/2,2m+α(Q). The latter is a Banach space, with respect to the norm   ˜ α (∂ k ∂ λ f ) . (1.3.8) |f |m,α := H ∂tk ∂xλ f Cb (Q) + t x 2k+|λ|≤2m

2k+|λ|=2m

In particular, we record explicitly that, for m = 1 and Q = ]0, T [×RN ,  C 1+α/2,2+α(Q) := f ∈ Cb1 (Q) | ∂x2 f ∈ C(Q) , (1.3.9)  ˜ α (ft ) + H ˜ α (∂x2 f ) < +∞ . H The asymmetry between the order of differentiation with respect to the time and space variables, measured respectively by 2k and |λ|, as per (1.3.5) and (1.3.6), is explicitly adapted to the study of parabolic equations such as (0.0.5). For this reason, the spaces C m+α/2,2m+α (Q) and C m+α/2,2m+α (Q) are sometimes called parabolic H¨older spaces.

1.4. Lebesgue Spaces

9

1.4. Lebesgue Spaces We briefly recall the definition and main properties of the Lebesgue spaces Lp (Ω) on domains Ω ⊆ RN , on which we consider the Lebesgue measure. For the main properties of these spaces, we refer, e.g., to Adams and Fournier [1]. When Ω = RN and there is no danger of confusion, we often abbreviate Lp (RN ) =: Lp . 1. Spaces Lp . Two measurable functions defined on Ω are said to be equivalent if they coincide except possibly on a subset of Ω with measure zero; this defines an equivalence relation. For 1 ≤ p ≤ ∞, the Lebesgue space Lp (Ω) is the vector space of all the (equivalence classes of) measurable functions f on Ω such that the quantities 

1/p p (1.4.1) |f |p := |f (x)| dx Ω

if 1 ≤ p < ∞, and |f |∞ := sup ess|f (x)|

(1.4.2)

x∈Ω

if p = ∞, are finite. The right sides of (1.4.1) and (1.4.2) define norms in Lp (Ω), with respect to which they are Banach spaces; for p = 2, L2 (Ω) is a Hilbert space, with inner product  (1.4.3) f, g := f (x)g(x) dx . Ω

If 1 ≤ p < ∞, is separable, while L∞ (Ω) is not, and C0∞ (Ω) is dense in Lp (Ω). If 1 < p < +∞, Lp (Ω) is reflexive; neither L1 (Ω) nor L∞ (Ω) is reflexive. Lp (Ω)

We can also give a definition of the spaces Lp (Γ), where Γ is a sufficiently smooth (N −1)-dimensional compact submanifold of RN (including Γ = ∂Ω); this is done in a natural way, by means of charts of local coordinates. More precisely, we say that f ∈ Lp (Γ) if for any local coordinate system, the corresponding expression of f is Lp -integrable in RN , with respect to the Riemannian volume element of Γ (see, e.g., Aubin [8, Ch. 4]). 2. Regularization. The density results mentioned above can be proven by means of a regularization process, one step of which usually consists of a convolution with the family of the Friedrichs’ mollifiers. The latter are functions in C ∞ (RN ), which can be constructed as follows. Let ρ ∈ C0∞ (RN ) be the nonnegative function, supported in {|x| ≤ 1}, defined by   −1 c0 exp 1−|x| if |x| < 1 , 2 (1.4.4) ρ(x) := 0 if |x| ≥ 1 ,

10

1. Functional Framework

with c0 chosen so that



(1.4.5)

ρ(x) dx = 1 .

For δ > 0, set ρδ (x) :=

(1.4.6)

1 x ρ . δN δ

Each of these functions, which are supported in the closed ball B(0, δ), is called a Friedrichs’ mollifier, and, if u is a locally integrable function on RN , the function

 1 x−y δ δ (1.4.7) x → u (x) := (ρ ∗ u)(x) = ρ u(y) dy δN δ is called a (Friedrichs) regularization of u. This terminology is motivated by the following well-known properties of the family (uδ )δ>0 . Theorem 1.4.1. Let u : RN → R be locally integrable, and for δ > 0 define uδ by (1.4.7). Then, uδ ∈ C ∞ (RN ). If u ∈ Lp , 1 ≤ p ≤ ∞, then uδ ∈ Lp as well, and |uδ |p ≤ |u|p .

(1.4.8)

If 1 ≤ p < ∞, uδ → u in Lp as δ → 0, and the map M : R>0 × Lp → Lp defined by M(δ, u) := uδ

(1.4.9)

is continuous. Finally, if u ∈ C(RN ), uδ → u uniformly on compact sets. Proof. Most results are proven, e.g., in Adams and Fournier [1, Sct. 2.28]. As for the continuity of M, note first that (1.4.8) implies that, for each fixed δ > 0, the map u → M(δ, u) is Lipschitz continuous on Lp , uniformly in δ, with Lipschitz constant L = 1. Fix now δ, δ0 > 0 and u, u0 ∈ Lp . Then, by (1.4.8), (1.4.10)

|uδ − uδ00 |p ≤ |uδ − uδ0 |p + | uδ0 − uδ00 |p ≤ |u − u0 |p + |wδ |p .

  =: wδ

By (1.4.7), wδ (x) = (1.4.11)

1 δN 



= |z|≤1

 x−y 1 u0 (y) dy ρ u0 (y) dy − N δ0 δ0  ρ(z) u0 (x − δz) dz − ρ(z) u0 (x − δ0 z) dz

ρ

x−y δ

|z|≤1



ρ(z)1/p

= |z|≤1



 +1/p

(u0 (x − δz) − u0 (x − δ0 z)) dz .

1.4. Lebesgue Spaces

11

From this, by H¨older’s inequality, (1.4.5), and Fubini’s theorem, we deduce that    δ p (1.4.12) |w (x)| dx ≤ ρ(z) |u0 (x − δz) − u0 (x − δ0 z)|p dx dz .

  =: ϕ(z)

Now,

 |u0 (x − δ0 z + (δ0 − δ)z) − u0 (x − δ0 z)|p dx

ϕ(z) =  (1.4.13)

|u0 (y + (δ0 − δ)z) − u0 (y)|p dy

= ≤

sup |u0 (· + (δ − δ0 )z) − u0 |pp =: (ω0 (δ − δ0 ))p .

|z|≤1

By the continuity of the translations in Lp , ω0 (δ − δ0 ) → 0 as δ → δ0 . Thus, because of (1.4.5), we deduce from (1.4.12) that |wδ |p ≤ ω0 (δ − δ0 ) ,

(1.4.14)

and conclude from (1.4.10) and (1.4.14) that M(δ, u) → M(δ0 , u0 ) as (δ, u) → (δ0 , u0 ) in R>0 × Lp .  3. Inequalities. The following inequalities are well known. Proposition 1.4.1. Let p and q be conjugate indices, f ∈ Lp (Ω), g ∈ Lq (Ω). For η > 0, let Cη be as in Proposition 1.1.1. Then, the pointwise product f g is in L1 (Ω), and (1.4.15)

|f g|1 ≤ |f |p |g|q ≤ η |f |pp + Cη |g|qq

(H¨ older’s inequalities). In particular for p = q = 2, (1.4.16)

|f g|1 ≤ |f |2 |g|2 ≤ η |f |22 +

1 4η

|g|22

(Schwarz’s inequalities). As a corollary, if Ω has finite measure μ(Ω) and 1 ≤ p ≤ q ≤ ∞, Lq (Ω) → Lp (Ω) ,

(1.4.17) and for all u ∈ Lq (Ω), (1.4.18)

|u|p ≤ KΩ |u|q ,

1

KΩ := (μ(Ω)) p

− q1

.

H¨older’s inequality also helps to characterize the dual of Lp (Ω), which, for  p ∈ [1, +∞[, is isometrically isomorphic to Lp (Ω). Consequently, if p ∈ ]1, +∞[ and (fm )m∈N ⊂ Lp (Ω), then, in accord with (1.2.4), fm → f weakly  in Lp (Ω) if and only if for all g ∈ Lp (Ω),   (1.4.19) fm (x) g(x) dx → f (x) g(x) dx . Ω

Ω

12

1. Functional Framework

Analogously, in accord with (1.2.5), fm → f weakly∗ in L∞ (Ω) if and only if for all g ∈ L1 (Ω),   (1.4.20) fm (x) g(x) dx → f (x) g(x) dx . Ω

Ω

H¨older’s inequality generalizes to more than two factors: Proposition 1.4.2. Let p1 , . . . , pm and q ∈ [1, ∞] be such that 1 1 1 (1.4.21) + ··· + = , p1 pm q and let fj ∈ Lpj (Ω) for 1 ≤ j ≤ m. Then, the pointwise product of the fj ’s is in Lq (Ω), and (1.4.22)

|f1 · · · fm |q ≤ |f1 |p1 · · · |fm |pm .

Proof. Proposition 1.4.2 is easily proven by induction, via H¨ older’s inequality (1.4.15). For simplicity, we consider only the case when all the indices are in ]1, ∞[. Thus, we first let m = 2, and assume fj ∈ Lpj (Ω), with j = 1, 2 and p11 + p12 = 1q . Then, α := pq1 and β := pq2 are conjugate indices, and, by (1.4.16),  q |f1 f2 |q = |f1 f2 |q dx Ω

 (1.4.23)

1/α 

≤

|f1 |

qα

1/β |f2 |

dx

Ω

qβ

dx

Ω

= |f1 |qp1 |f2 |qp2 . This proves (1.4.22) for m = 2. Assume then that (1.4.22) holds for some −1  m ≥ 2; that is, defining qm = p11 + · · · + p1m as per (1.4.21), that q m the product gm = f1 · · · fm is in L (Ω). Let q ∈ ]1, +∞[ be defined by 1 1 1 q q = qm + pm+1 . Then, the product gm fm+1 is in L (Ω), and we can conclude the proof of the proposition.  Finally, we recall the so-called Young’s inequality on convolutions in Lp spaces. Proposition 1.4.3. Let p ∈ [1, +∞[, f ∈ Lp , and g ∈ Lq , with q ∈ [1, p  ]. Define r ≥ 1 by 1r = 1q − p1 = 1q + 1p −1. Then, the convolutions f ∗g and g ∗f (recall definition (1.1.9)) are well defined in Lr . In addition, f ∗ g = g ∗ f a.e., and (1.4.24)

|f ∗ g|r = |g ∗ f |r ≤ |f |p |g|q .

In particular, if f ∈ Lp and g ∈ L1 , f ∗ g = g ∗ f ∈ Lp , and (1.4.25)

|f ∗ g|p ≤ |f |p |g|1 .

1.5. Sobolev Spaces

13

In fact, L1 is an algebra with respect to the convolution product. 4. Interpolation of Lebesgue Spaces. If p < q < r, Lq (Ω) can be characterized as an intermediate space between Lr (Ω) and Lp (Ω), in the following sense. Proposition 1.4.4. Assume 1 ≤ p ≤ q ≤ r ≤ ∞. Then, Lp (Ω) ∩ Lr (Ω) ⊆ Lq (Ω). Moreover, if θ ∈ [0, 1] is defined by 1 θ 1−θ (1.4.26) =: + , q p r then for all u ∈ Lp (Ω) ∩ Lr (Ω), |u|q ≤ |u|θp |u|r1−θ .

(1.4.27)

Proof. Let u ∈ Lp (Ω) ∩ Lr (Ω). Since q ∈ [p, r], it follows that, for a.a. x ∈ Ω, |u(x)|q ≤ |u(x)|p + |u(x)|r ;

(1.4.28)

hence, u ∈ Lq (Ω). Next, setting α = θq and β = (1 − θ)q, (1.4.26) implies that a = αp and b = βr are conjugate indices; thus, (1.4.27) follows by H¨older’s inequality, writing |u(x)|q = |u(x)|θq |u(x)|(1−θ)q .  Remarks. If Ω has finite measure, (1.4.17) implies that Lr (Ω) ∩ Lp (Ω) = Lr (Ω). (1.4.27) is an example of interpolation inequality. Indeed, the family (Lp (Ω))1≤p≤∞ is a family of interpolation spaces (see, e.g., Bergh and L¨ofstr¨om [14]). 

1.5. Sobolev Spaces We briefly H m (Ω) on Ω = RN . abbreviate

recall the definition and main properties of the Sobolev spaces domains Ω ⊆ RN ; again, we assume that Ω is bounded or that In the latter case, when there is no danger of confusion, we H m (RN ) =: H m .

1.5.1. Definitions and Main Properties. ˜ m (Ω). For m ∈ N, the Sobolev space H m (Ω) 1. Spaces H m (Ω) and H is the vector space of all Lebesgue-measurable functions f on Ω such that all distributional derivatives ∂xα f , |α| ≤ m, are in L2 (Ω). H m (Ω) is a Hilbert space with respect to the scalar product  (1.5.1) f, gm := ∂xα f, ∂xα g , |α|≤m

which induces the norm (1.5.2)

⎛

f m := ⎝



|α|≤m

⎞1/2 |∂xα f |22 ⎠

.

14

1. Functional Framework

In particular, L2 (Ω) = H 0 (Ω), and we abbreviate | · |2 =  · 0 =  · . More generally, for 1 ≤ p ≤ ∞ we define (1.5.3)

W m,p (Ω) := {f ∈ Lp (Ω) | ∂xα u ∈ Lp (Ω) ,

|α| ≤ m} ,

which is a Banach space with respect to the norm ⎞1/p ⎛  (1.5.4) f m,p := ⎝ |∂xα f |pp ⎠ ; |α|≤m

clearly, H m (Ω) = W m,2 (Ω). We abbreviate W m,p (RN ) =: W m,p , as well as H m (RN ) =: H m . Finally, for m ≥ 1 we define (1.5.5)

˜ m (Ω) := {u ∈ L∞ (Ω) | ∇u ∈ H m−1 (Ω)} , H

which is a Banach space with respect to the norm 1/2  2 2 (1.5.6) um . ˜ := |u|∞ + ∇um−1 2. Spaces H s (RN ). When Ω = RN , the spaces H m can be defined in an equivalent way, by means of the Fourier transform. We refer to Schwartz [141], Rudin [138, 139], or Yosida [165], for most of the definitions and properties of the Fourier transform that we recall and use in the sequel. The Fourier transform of a function f ∈ S (the Schwartz space of the so-called rapidly decreasing functions) is the function fˆ : RN → C l defined by  1 (1.5.7) fˆ(ξ) := e− ix·ξ f (x) dx , (2π)N/2 where i2 = − 1. The map f → fˆ =: F (f ) defined by (1.5.7) is the Fourier transform map. This map can immediately be extended to a map, still denoted by f → F f = fˆ, from L1 to L∞ , with (1.5.8)

|fˆ|∞ ≤ |f |1

(in fact, fˆ ∈ Cb (RN )). F can also be extended to a map from L2 to L2 , with (1.5.9)

fˆ = f 

(Parseval’s formula), as well as to a map from S  to S  (the space of the tempered distributions; i.e., the dual of the Schwartz space S). If f ∈ S  , its Fourier transform fˆ ∈ S  is defined by the identity (1.5.10)

fˆ, ϕS  ×S = f, ϕ ˆ S  ×S ,

which is a natural generalization of the identity (1.5.11)

fˆ, ϕ = f, ϕ ˆ ,

ϕ∈S,

1.5. Sobolev Spaces

15

which holds if f ∈ S, by Fubini’s theorem. For m ∈ N, we introduce the spaces F m consisting of all f ∈ S  such that  (1.5.12) (1 + |ξ|2 )m |fˆ(ξ)|2 dξ < +∞ . The left side of (1.5.12) is the square of a norm in F m , with associated scalar product  (1.5.13) f, gm := (1 + |ξ|2 )m fˆ(ξ) gˆ(ξ) dξ , where the bar denotes complex conjugation. We note that F is an isomorphism of H m onto F m , and that the definition of F m can be extended to any m ∈ R, simply by requiring that (1.5.12) hold for m ∈ R as well. Hence, for s ∈ R we define the Sobolev spaces (1.5.14)

H s := {f ∈ S  | (1 + | · |2 )s/2 fˆ ∈ L2 } ,

endowed with the norm defined by (1.5.12), i.e.,  (1.5.15) f 2s := (1 + |ξ|2 )s |fˆ(ξ)|2 dξ . It is clear that, for s ∈ N, the two definitions (1.5.14) and (1.5.3) (with p = 2 and Ω = RN ; recall that W m,2 = H m ) of H s are equivalent (in the sense of equivalent topological spaces). Moreover, it is also possible to prove that, for all s ∈ R, (H s )  is isomorphic to H −s . In addition, if f ∈ Lp , p ∈ [1, 2],  then fˆ ∈ Lp , and the Hausdorff-Young inequality (1.5.16)

|fˆ|p  ≤ C |f |p

holds, with C depending on N and p, but not on f (see, e.g., Lieb and Loss [95, ch. 5]). Finally, if Ω ⊂ RN is bounded and has a smooth boundary, for s ∈ R≥0 we define H s (Ω) as the space of restrictions to Ω of functions in H s (RN ), endowed with the norm (1.5.17)

us := inf{ U s | U |∂Ω = u} .

It can be seen (Lions and Magenes [101, ch. 1, §1.9] or Huet [61, ch. 6]) that, when s ∈ N, this definition of H s (Ω) is topologically equivalent to the one given in (1.5.3) (with p = 2). 3. Traces. We briefly recall the definition of the trace of a function in H 1 (Ω) at the boundary ∂Ω. The notion of a restriction, or trace, of a function u ∈ C 0 (Ω) to the boundary ∂Ω of a bounded set Ω cannot evidently be extended to functions in L2 (Ω), since these can be arbitrarily modified on subsets of measure zero. Nevertheless, a generalization of the notion of trace can be given for functions in H 1 (Ω), at least when ∂Ω is smooth, as a consequence of the density of C 1 (Ω) in H 1 (Ω) (in contrast, recall that

16

1. Functional Framework

C01 (Ω) is dense in L2 (Ω) but, in general, not in H 1 (Ω)). More precisely, the restriction map (1.5.18)

C 1 (Ω)  u → u |∂Ω ∈ C(∂Ω)

can be extended by continuity into a linear, continuous, and surjective map γ0 : H 1 (Ω) → H 1/2 (∂Ω) (see, e.g., Lions and Magenes [101, ch. 1]; the space H 1/2 (∂Ω) can be defined by means of the local coordinate charts defining the boundary, via the definition of the Sobolev spaces H s (RN ) with non integer s mentioned in (1.5.14)). We can then define the space (1.5.19)

H01 (Ω) := {u ∈ H 1 (Ω) | γ0 u = 0} = ker (γ0 ) ;

that is, functions in H01 (Ω) have a vanishing trace (in the above-generalized sense) at the boundary of Ω. More generally (i.e., without any requirement on the smoothness of ∂Ω), H01 (Ω) can be defined as the closure of C01 (Ω) in H 1 (Ω), with respect to the H 1 norm; when ∂Ω is smooth, the two definitions of H01 (Ω) are equivalent (see Kasuga [69]). In particular, since C01 (RN ) is dense in H 1 (RN ), H 1 (RN ) = H01 (RN ). We also define (1.5.20)

H − 1 (Ω) := (H01 (Ω))  ,

and note that, when Ω = RN , this definition is consistent with the one given in the previous section, that is, in accord with (1.5.14) with s = − 1,    (1.5.21) H − 1 = f ∈ S  | (1 + |ξ|2 )− 1 |fˆ(ξ)|2 dξ < +∞ . If m > 1, using the density of C m (Ω) in H m (Ω) (which requires ∂Ω to be of class at least C m−1,1 ), we can further define, for 0 ≤ j ≤ m − 1, the higher order traces γj (u) ∈ H m−j−1/2 (∂Ω), as generalization of the normal j derivatives ∂∂νuj , where ν is the outward unit normal to ∂Ω. Correspondingly, and in accord with Lions and Magenes [101, thm. 11.5], we can consider, as a generalization of (1.5.19) and (1.5.20), the spaces (1.5.22) (1.5.23)

H0m (Ω) := {u ∈ H m (Ω) | γj u = 0 ,

0 ≤ j ≤ m − 1} ,

H − m (Ω) := (H0m (Ω))  .

4. Extensions and Restrictions. If Ω ⊂ RN is bounded, the restrictions to Ω of functions in H m (RN ) are in H m (Ω). Indeed, if u ∈ H m (RN ), then its restriction R u to Ω is, evidently, in L2 (Ω); furthermore, recalling the definition of restriction of a distribution to an open set (see, e.g., Dautray and Lions [41, App., §2.4]) we can show that the identities (1.5.24)

∂xα (R u) = R(∂xα u) ,

|α| ≤ m ,

hold in D  (Ω). But since R(∂xα u) ∈ L2 (Ω), we conclude that R u ∈ H m (Ω), as claimed. In fact, the restriction operator u → R u is continuous from

1.5. Sobolev Spaces

17

H k (RN ) to H k (Ω), 0 ≤ k ≤ m, with (1.5.25)

R uH k (Ω) ≤ uH k (RN ) ,

u ∈ H m (RN ) .

Conversely, if the boundary ∂Ω is sufficiently smooth, functions in H m (Ω) can be extended, not necessarily in a unique way, to functions in H m (RN ). More precisely, as in Lions and Magenes [101, thm. 8.1]: Proposition 1.5.1. Let m ∈ N, and assume that ∂Ω is of class C m . There exists an extension operator E : H m (Ω) → H m (RN ), such that E is continuous from H k (Ω) into H k (RN ) for all k = 0, . . . , m, and Eu = u a.e. in Ω. In particular, there is C > 0, depending on N , m and Ω, such that, for all u ∈ H m (Ω), (1.5.26)

E uH k (RN ) ≤ C uH k (Ω) ,

0≤k ≤m.

Remarks. 1) If u ∈ H0m (Ω), one can take, for its extension, the function  u(x) if x∈Ω, (1.5.27) x → u ˜(x) := 0 if x∈ / Ω. Indeed, from Lions and Magenes [101, thm. 11.4], it follows that u ˜ ∈ m N H (R ) and (1.5.28)

˜ uH m (RN ) = uH m (Ω) .

2) Resorting to the density of C0m (Ω) in H0m (Ω), we can show that, if u ∈ H0m (Ω), the identities (1.5.29)

α ∂xα u ˜ = ∂ xu,

|α| ≤ m ,

hold, first in D  (RN ) and then, in fact, in L2 (RN ).



Proposition 1.5.2. Let ΩR = B(0, R) be a ball of radius R ≥ 1, and let E : H m (ΩR ) → H m (RN ) be the extension operator introduced in Proposition 1.5.1. Then, the constant C in (1.5.26) can be determined independently of R. More precisely, for all u ∈ H m (ΩR ), and 0 ≤ k ≤ m, (1.5.30)

E uH k (RN ) ≤ C1 uH k (B(0,R)) ,

where C1 is the value of C corresponding, in (1.5.26), to Ω = B(0, 1). Proof. The result follows from a homothetic-invariance argument, based on the explicit construction of the extension operator E in Lions and Magenes [101, thm. 8.1]. Indeed, given R > 1 and u ∈ H m (ΩR ), define v ∈ H m (Ω1 ) by v(y) := u(Ry). Then, by (1.5.26), (1.5.31)

E vH m (RN ) ≤ C1 vH m (Ω1 ) ,

18

1. Functional Framework

for some C1 independent of v. Let α ∈ NN , with |α| ≤ k ≤ m. By the x changes of variable x = Ry and y = R , we compute from (1.5.26) that ∂xα (Eu)2L2 (RN ) = RN −2|α| ∂yα (Ev)2L2 (RN ) ≤ C1 RN −2|α| v2H |α| (Ω ) 1  N −2|α| β ≤ C1 R ∂y v2L2 (Ω1 )

(1.5.32)

≤ C1



|β|≤|α|

R2(|β|−|α|) ∂xβ u2L2 (ΩR ) .

|β|≤|α|

Since R > 1 and |β| ≤ |α|, we deduce from (1.5.32) that   (1.5.33) Eu2H k (RN ) ≤ C1 ∂xβ u2L2 (ΩR ) , |α|≤k β≤α



from which (1.5.30) follows.

5. Regularization. Functions in Sobolev spaces can be regularized, by means of the Friedrichs’ mollifiers introduced in (1.4.7). In analogy to Theorem 1.4.1, we have (see, e.g., Adams and Fournier [1, Sct. 3.16]). Theorem 1.5.1. Let u : RN → R be locally integrable and, for δ > 0, define uδ ∈ C ∞ (RN ) by (1.4.7). If u ∈ Lp , 1 ≤ p ≤ ∞, then ∂xα uδ ∈ Lp for all multi-index α ∈ NN ; thus, uδ ∈ W m,p for every m ∈ N. In addition, if u ∈ W m,p , (1.5.34)

uδ m,p ≤ um,p .

If 1 ≤ p < ∞, uδ → u in W m,p as δ → 0, and the map M defined in (1.4.9) is continuous from R>0 × W m,p to W m,p . Note on Proof. The proof of the convergence uδ → u in W m,p is based on the inequality (1.5.35)

|uδ − u|p ≤ max |u(· − δz) − u|p |z|≤1

and the continuity of translations in Lp . To show (1.5.35), we note that, by (1.4.7) and (1.4.5), for a.a. x ∈ RN , (1.5.36)



  x−y x−y 1 1 u (x) − u(x) = N ρ ρ u(y) dy − u(x) N dy δ δ δ δ

 1 x−y = N ρ (u(y) − u(x)) dy δ δ  = ρ(z) (u(x − δ z) − u(x)) dz . δ

1.5. Sobolev Spaces

19

1

From this, writing ρ(z) = ρ(z) p  (1.5.37) |u (x) − u(x)| ≤ δ

+ p1

,

1/p  



1/p ρ(z) |u(x − δz) − u(x)| dz p

ρ(z) dz

.

Recalling then (1.4.5) again,    |uδ (x) − u(x)|p dx ≤ ρ(z) |u(x − δz) − u(x)|p dz dx   = ρ(z) |u(x − δz) − u(x)|p dx dz (1.5.38)  ≤ ρ(z) |u(· − δz) − u|pp dz , 

from which (1.5.35) follows.

Remarks. 1) It also holds that, if u ∈ s ∈ R, then → u in H s . If s < 0, this claim requires a proper interpretation of the convolution in (1.4.7) as the convolution of a test function (i.e., ρδ ) with a distribution (i.e., u); see, e.g., Schwartz [141, ch. 6], or Rudin [139, ch. 6]. H s,

uδ

2) In contrast to (1.5.34), if u ∈ W m,p and α ∈ NN is such that |α| > m, then ∂xα uδ ∈ Lp , but, in general, (1.5.39)

|∂xα uδ |p ≤

C δ |α|−m

um,p .

To see this, given such a multi-index α, we determine β and γ ∈ NN such that α = β + γ, |β| = m, |γ| > 0, and compute that  α δ γ ∂x (u (x)) = ∂x ρδ (x − y) ∂yβ u(y) dy   x−y  β 1 1 γ = (∂ ∂y u(y) dy (1.5.40) ρ) x δ N δ |γ| δ  1 = (∂ γ ρ)(z) ∂xβ u(x − δ z) dz , δ |γ| |z|≤1 z from which (1.5.39) follows, arguing as in (1.5.37) and (1.5.38).



6. Gagliardo-Nirenberg Inequalities. Let p, q and r ∈ [1, +∞]. The following result, due to Gagliardo and Nirenberg, describes conditions under which functions in Lr , which have distributional derivatives of some order in Lp , have intermediate derivatives in Lq . r Theorem 1.5.2. Let m ∈ N, p, r ∈ [1, +∞], and u ∈ Lp (Ω)  ∩ L (Ω). j Assume that ∂xm u ∈ Lp (Ω). For integer j, 0 ≤ j ≤ m, and θ ∈ m , 1 (with

20

1. Functional Framework

the exception θ = 1 if m − j − (1.5.41)

∈ N), define q by

1 j 1 m 1 = +θ − + (1 − θ) . q N p N r N 2

Then, for any γ ∈ NN , with |γ| = j, ∂xγ u ∈ Lq (Ω) and satisfies the Gagliardo-Nirenberg inequality (1.5.42)

|∂xγ u|q ≤ C |∂xm u|θp |u|r1−θ + C1 |u|s ,

with finite 1 ≤ s ≤ max{p, r}, and C > 0, C1 ≥ 0 are independent of u. The constant C is independent of Ω, while C1 → 0 as the volume of Ω grows to ∞. In particular, the choice C1 = 0 is admissible if Ω = RN . For a proof, see, e.g., Aubin [7, ch. 3], or Nirenberg’s own proof in [125]. j The choice of the lowest value θ = m in (1.5.41) yields the dimensionless version of estimate (1.5.42) (1.5.43)

1−j/m |∂xγ u|q ≤ C |∂xm u|j/m + C1 |u|s , p |u|r

with |γ| = j, and q ≥ 1 defined, independently of N , by

1 j 1 j (1.5.44) = + 1− . q pm r m Unless u satisfies some extra conditions, such as having vanishing trace at ∂Ω (in the sense of (1.5.19)), it is in general not possible to choose C1 = 0 in (1.5.42) if Ω = RN . To see this, it is sufficient to consider the example u(x) = x in Ω = ]0, 1[ ⊂ R, with m = 2, j = 1, θ = 12 and p = q = r = 2: if (1.5.42) held with C1 = 0, we would deduce the contradiction u  (x) ≡ 0. On the other hand, if Ω is bounded and u ∈ H01 (Ω), u satisfies the Poincar´e inequality (1.5.45)

|u|q ≤ CΩ,q ∇u

for all q such that 2 ≤ q ≤ q¯ := N2N −2 if N > 2, and all q ≥ 2 if N = 2, with CΩ,q independent of u. If Ω is bounded in at least one direction, CΩ,q can be chosen independently of Ω; otherwise, CΩ,q may grow unboundedly with the diameter of Ω. If instead Ω = RN , N > 2, Poincar´e’s inequality (1.5.45) holds, for all u ∈ H 1 (RN ), only in the limit case q = q¯ (> 2); that is, (1.5.46)

|u|q¯ ≤ Cq¯ ∇u .

This follows from the Gagliardo-Nirenberg inequality (1.5.42), with j = 0, θ = 1, p = 2, m = 1, and C1 = 0. If N = 2, however, inequality (1.5.46) does not hold, as one can see by a homogeneity argument (i.e., replacing u(x) by u(λ x), and letting λ → 0). 7. Interpolation of Sobolev Spaces. Clearly, H s2 (Ω) → H s (Ω) → if s2 ≥ s ≥ s1 . In fact, (H s (Ω))s≥0 is a family of interpolation spaces. H s1 (Ω)

1.5. Sobolev Spaces

21

Proposition 1.5.3. Let s2 ≥ s ≥ s1 ≥ 0, and θ ∈ [0, 1] be such that s = (1 − θ)s1 + θ s2 . Then, for all u ∈ H s2 , the inequality us ≤ C us1−θ uθs2 1

(1.5.47)

holds, with C independent of u (compare to (1.4.27) of Proposition 1.4.4). Proof. We prove (1.5.47) only when Ω = RN , or when s1 , s and s2 are integers. For the general case, we refer again to Huet [61], or to Lions and Magenes [101, sct. 1.9], where in fact the spaces H s (Ω), for non integer s, are defined by interpolation between integers. Note that (1.5.47) is another example of interpolation inequality. 1) When Ω = RN , (1.5.47) is an immediate consequence of H¨older’s inequality. Indeed, let u ∈ H s2 . Then, (1.5.48)



u2s = =



(1 + |ξ|2 )s |fˆ(ξ)|2 dξ (1 + |ξ|2 )(1−θ)s1 |fˆ(ξ)|2(1−θ) (1 + |ξ|2 )θ s2 |fˆ(ξ)|2 θ dξ

 ≤

1−θ  (1 + |ξ|2 )s1 |fˆ(ξ)|2 dξ

θ (1 + |ξ|2 )s2 |fˆ(ξ)|2 dξ

,

from which (1.5.47) follows, with C = 1. 2) When s1 , s and s2 are integers, (1.5.47) is a consequence of the Gagliardo-Nirenberg inequality. Indeed, let u ∈ H s (Ω), and α ∈ NN be such that |α| =: a ≤ s. If a ≤ s1 , (1.5.49)

∂xα u ≤ us1 = us1−θ uθs1 ≤ us1−θ uθs2 . 1 1

If s1 < a ≤ s2 , there are multi-indices β and γ such that |β| = s1 and α = β + γ. Then, (1.5.42) implies that (1.5.50)

∂xα u ≤ C ∂xs2 uθa ∂xβ u1−θa + C1 ∂xβ u ,

1 with θa = sa−s , in accord with (1.5.41). From (1.5.50), estimating its last 2 −s1 term as in (1.5.49), we deduce that

(1.5.51)

a ∂xα u ≤ (C + C1 ) u1−θ uθs2a ; s1

then using the inequality (1.5.52)

x1−λ y λ ≤ x1−μ y μ ,

x ≤ y,

λ≤μ,

with x = us1 , y = us2 , λ = θa and μ = θ := (1.5.47)), we obtain from (1.5.51) that (1.5.53)

s−s1 s2 −s1

(as required in

∂xα u ≤ (C + C1 ) us1−θ uθs2 . 1

Together with (1.5.49), this implies (1.5.47).



22

1. Functional Framework

Remark. Using interpolation, it is possible to define the spaces H s (Ω) also for s ∈ R 2 (i.e., with higher order integrability), or even H¨ older continuous (recall the definition of the spaces C r,α (Ω) in part 1 of section 1.3). Theorem 1.5.3. Let Ω be a bounded domain with smooth boundary, or Ω = RN , and m ∈ R≥0 , p ∈ R≥1 . Then: 1) If mp < N and p ≤ q ≤ (1.5.54) N p

Np N −mp ,

or if mp = N and p ≤ q < +∞,

W m,p (Ω) → Lq (Ω) .

2) If mp > N , r = m −  Np  − 1 ∈ N, and 0 < α ≤  Np  + 1 − ∈ / N, or 0 < α < 1 if Np ∈ N,

(1.5.55)

if

W m,p (Ω) → C r,α (Ω) .

3) In particular, if p = 2 and m > (1.5.56)

N p

N 2,

H m (Ω) → Cb (Ω) → L∞ (Ω) ;

hence, recalling definition (1.5.5), (1.5.57)

˜ m (Ω) . H m (Ω) → H

In fact, also (1.5.58)

˜ m (Ω) → Cb (Ω) . H

4) If m ≥ 1, (1.5.59)

W m,∞ (Ω) → C m−1,1 (Ω) .

5) Finally, if Ω is bounded and m > 0, the imbedding W m,p (Ω) → is compact for all ε ∈ ]0, m].

W m−ε,p (Ω)

Notes on Proof. For a proof of most of the results of Theorem 1.5.3, except for (1.5.58), see, e.g., Adams and Fournier [1], Aubin [7, ch. 2], or Lions and Magenes [101, ch. 1]. For the case when s is not an integer, see also Triebel [160, scts. 2.3, 2.6]. ˜ m (Ω) and assume first that m − 1 ≤ N . 1) To prove (1.5.58), fix u ∈ H 2 Then, by (1.5.54), there is p > N such that ∇u ∈ Lp (Ω); thus, for any ball B ⊂ Ω, (1.5.55) implies that u ∈ W 1,p (B) → C(B). Since B is arbitrary, we conclude that u ∈ C(Ω); since also u ∈ L∞ (Ω), (1.5.58) follows. If

1.5. Sobolev Spaces

23

instead m − 1 > N2 , (1.5.55) implies that ∇u ∈ Cb (Ω), so that u is Lipschitz continuous in Ω. 2) If p = 2, imbeddings (1.5.54) and (1.5.55) read explicitly as H m (Ω) → Lq (Ω)

(1.5.60) if m


N 2,

H m (Ω) → C r,α (Ω) ,

(1.5.61)

with r = m − N 2+1  and 0 < α <  N2  + 1 − N2 . Imbedding (1.5.60) follows from the Gagliardo-Nirenberg inequality; if N > 2m, the choice j = 0, θ = 1 and p = 2 in (1.5.41) yields the original Sobolev imbedding H m (Ω) → Lq (Ω), with q = N2N −2m , as per (1.5.60). The imbedding N m N ∞ N H (R ) → L (R ) if m > 2 follows from the estimate 

1/2    −1 ˆ  2 −m (1.5.62) |f (x)| = [F (f )](x) ≤ CN (1 + |ξ| ) dξ f m , with f ∈ C0∞ (RN ) and CN = (2π)− N/2 , via a density argument. 3) To prove (1.5.59), it is sufficient to consider the case m = 1; that is, to show that W 1,∞ (Ω) → C 0,1 (Ω) .

(1.5.63)

In fact, we show that, if u ∈ W 1,∞ (Ω), then for all x, y ∈ Ω, |u(x) − u(y)| ≤ |∇u|∞ |x − y| .

(1.5.64)

To see this, suppose first that there is a ball B ⊂ Ω containing x and y. Then, u ∈ W 1,p (B) for all p ∈ [1, +∞[. Taking p > N , imbedding (1.5.55) implies that u ∈ C(B). Let (uδ )δ>0 be the family of the Friedrichs’ regularizations of u. Then, by Theorem 1.4.1, uδ → u in C(B) and, by (1.4.8), (1.5.65)

|∇uδ |L∞ (B) ≤ |∇uδ |L∞ (Ω) ≤ |∇u|L∞ (Ω) .

Consequently, for any x, y ∈ Ω, (1.5.66)

|uδ (x) −

uδ (y)|



1

≤

|∇uδ (λ x + (1 − λ)y) · (x − y)| dλ

0

≤ |∇u|L∞ (Ω) |x − y| , from which (1.5.64) follows. In the general case, we use a connectedness argument, replacing |x − y| by the geodesic distance between x and y, which is equivalent to the Euclidean distance (see, e.g., Aubin [7, ch. 1]). Thus, u is Lipschitz continuous. Recalling our remarks on the definition of the spaces C m,α (Ω) in section 1.3, we conclude that, if Ω is bounded, u can be extended to a function u ˜ ∈ C 0,1 (Ω), which we identify with u; if instead N N ∞ Ω = R , u ∈ C(R )∩L (RN )∩C 0,1 (RN ) = Cb (Ω)∩C 0,1 (RN ) = C 0,1 (RN ). In either case, (1.5.59) follows, for m = 1. 

24

1. Functional Framework

Corollary  1 1  1.5.1. Let r ∈N[2, +∞], and s1 , s2 ∈ N be such that s1 ≥ N 2 − r and s1 + s2 > 2 . There exist p, q ∈ [2, +∞] such that, for all f ∈ H s1 and g ∈ H s2 , (1.5.67)

|f g|r  ≤ |f |p |g|q ≤ C f s1 gs2 ,

where r  is the conjugate index of r, and C is independent of f and g. Proof. The given conditions on r, s1 , and s2 allow us to fix q > 0 such that     (1.5.68) max 12 − sN2 , 12 − 1r ≤ 1q ≤ min 12 , sN1 ; note that 2 ≤ q ≤ +∞. We define p ∈ [2, +∞] by p1 = 1 − 1q − 1r . Then, (1.5.67) follows by H¨older’s inequality, and the injections H s1 → Lp , H s2 → Lq . To verify these injections, it is sufficient to proceed case by case, depending on whether s1 and s2 are less than, equal to, or larger than N2 . For example, if both s1 , s2 > N2 , the imbedding (1.5.56) implies that f and g ∈ L∞ ; since also f , g ∈ L2 , by interpolation (Proposition 1.4.4) it follows that f ∈ Lp and g ∈ Lq . The latter holds also if s2 ≤ N2 , by (1.5.60). The other cases are similar.  Remark. The imbeddings (1.5.56) imply, in particular, that if m > there is a constant C such that, for all u ∈ H m (Ω),

N 2,

|u|∞ ≤ C um .

(1.5.69)

When N ≥ 3 and Ω = RN , (1.5.69) can be improved into |u|∞ ≤ C ∇um−1 ,

(1.5.70)

as follows from the estimate

(1.5.71)



|u|∞ ≤ CN |ˆ u(ξ)| dξ u|1 = CN |ˆ   = CN |ˆ u(ξ)| |ξ| (1 + |ξ|2 )(m−1)/2  · |ξ|−1 (1 + |ξ|2 )−(m−1)/2 dξ 

1/2 ≤ CN |ξ u ˆ(ξ)|2 (1 + |ξ|2 )m−1 dξ  ·

|ξ|

−2

2 −(m−1)

(1 + |ξ| )

noting that the last integral is finite because m >

N 2

1/2 dξ

,

and N ≥ 3.



9. Sobolev Product Estimates. Many of the standard rules of differentiation, such as the product and the chain rules, can be extended to functions in Sobolev spaces. For the product rule, this is illustrated by the fact that, in some cases, the product of functions in Sobolev spaces is again

1.5. Sobolev Spaces

25

in a Sobolev space. This result is based on an extension to functions in Sobolev spaces of Leibniz’ formula   α β α−β (1.5.72) ∂xα (f g) = g, β ∂x f ∂x β≤α

where β ≤ α means that βj ≤ αj for all j = 1, . . . , N , and





α α1 αN (1.5.73) := · ...· . β β1 βN Theorem 1.5.4. Let m, n and s ∈ N be such that m ≥ s, n ≥ s, and m + n − s > N2 . Let f ∈ H m (Ω) and g ∈ H n (Ω). Then, the product f g ∈ H s (Ω), and (1.5.74)

f gs ≤ C f m gn ,

with C independent of f and g. Moreover, if m > s, for all η > 0 there is Cη ∈ K∞ such that (1.5.75)

f gs ≤ (η f m + Cη f 0 ) gn .

Proof. Since f, g ∈ L2 (Ω), the product f g ∈ L1 (Ω); thus, for all multiindex α ∈ NN , ∂xα (f g) can be defined in D  (Ω). We show that if |α| ≤ s, ∂xα (f g) ∈ L2 (Ω), that Leibniz’ formula (1.5.72) holds, as an identity in L2 (Ω), and that ∂xα (f g) satisfies the estimate (1.5.76)

∂xα (f g) ≤ C f m gn ,

with C independent of f and g. Let first α > 0, with |α| ≤ s, and, for δ > 0, let f δ and g δ be the Friedrichs’ regularizations of f and g. Then, since Leibniz’ formula (1.5.72) does hold for smooth functions,   α β δ α−β δ (1.5.77) ∂xα (f δ g δ ) = g =: Σ(f δ , g δ ) . β ∂x f ∂x β≤α

We now show that each of the terms of Σ(f δ , g δ ) can be estimated in L2 (Ω), independently of δ, by means of H¨older’s inequality, the Sobolev imbeddings of Theorem 1.5.3, and Theorem 1.5.1. In fact, if we consider ∂xβ f δ ∈ H m−|β| (Ω) and ∂xα−β g δ ∈ H n−|α|+|β| (Ω), we can apply Corollary 1.5.1 with f = ∂xβ f δ , g = ∂xα−β g δ , s1 = m − |β|, s2 = n − |α| + |β|, and r = r  = 2, because s1 ≥ m − |α| ≥ m − s ≥ 0, and s1 + s2 = m + n − |α| ≥ m + n − s > N2 . Thus, from (1.5.67) it follows that there are p, q ∈ [2, +∞] such that 1p + 1q = 12 , and |∂xβ f δ ∂xα−β g δ |2 ≤ C |∂xβ f δ |p |∂xα−β g δ |q (1.5.78)

≤ C ∂xβ f δ m−|β| ∂xα−β g δ n−|α|+|β| ≤ C f δ m g δ n ≤ C f m gn ,

26

1. Functional Framework

having used (1.5.34) in the last step. When α = 0, we use the GagliardoNirenberg inequality |f δ g δ |2 ≤ C |∇(f δ g δ )|θ2 |f δ g δ |11−θ ,

(1.5.79)

θ=

N N +2

,

to deduce from (1.5.78), with |α| = 1, that (1.5.80)

|f δ g δ |2 ≤ C f δ θm g δ θn f δ 01−θ g δ 01−θ ≤ C f m gn .

Now, for all ϕ ∈ D(Ω), by the definition of differentiation in D  (Ω),  ∂xα (f g), ϕD  ×D = (− 1)|α| f g ∂xα ϕ dx Ω (1.5.81)  |α| = (− 1) lim f δ g δ ∂xα ϕ dx , δ→0 Ω

the convergence being a consequence of Theorem 1.4.1 and the obvious fact that f δ g δ → f g in L1 (Ω), because f δ → f and g δ → g in L2 (Ω). In fact, Theorem 1.5.1 also implies that ∂xβ f δ → ∂xβ f in H m−|β| (Ω) and ∂xα−β g δ → ∂xα−β g in H n−|α|+|β| (Ω); thus, each term in the sum (1.5.77) converges to the corresponding term of the sum Σ(f, g) in L2 (Ω). Indeed, by (1.5.78), ∂xβ f δ ∂xα−β g δ − ∂xβ f ∂xα−β g0 ≤ ∂xβ (f δ − f ) ∂xα−β g δ 0 + ∂xβ f (∂xα−β g δ − ∂xα−β g)0

(1.5.82)

≤ C f δ − f m gn + C f m g δ − gn Consequently, (− 1)

|α|



 δ δ

f g

∂xα ϕ dx

∂xα (f δ g δ ) ϕ dx

= 

Ω

(1.5.83)

→0.

Ω

Σ(f δ g δ ) ϕ dx

= Ω →

Ω

Σ(f g) ϕ dx = Σ(f g), ϕD  ×D .

From (1.5.81) and (1.5.83) we deduce the identity   α β α−β (1.5.84) ∂xα (f g) = g, β ∂x f ∂x β≤α

that is, Leibniz’ formula holds in D  . In fact, since Σ(f, g) ∈ L2 , (1.5.84) holds in L2 (Ω). Moreover, (1.5.74) follows from (1.5.78) and (1.5.80), which imply (1.5.76). As for (1.5.75), choose first a > 0 such that m − a ≥ s and (m − a) + r − s > N2 . Then, by (1.5.74), (1.5.85)

f gs ≤ C f m−a gr .

1.5. Sobolev Spaces

27

By the interpolation inequality (1.5.47), and the weighted Minkowski’s inequality (1.1.12), for all β > 0 and corresponding Cβ ∈ K∞ , (1.5.86)

f m−a ≤ β f m + Cβ f 0 ,

so that (1.5.75) follows by choosing β =

η C.



We shall occasionally abbreviate the content of Theorem 1.5.4 by writing (1.5.87)

H m · H n → H s .

Theorem 1.5.4 allows us to prove Corollary 1.5.2. Let s, m ∈ N, with s > N2 and 0 ≤ m ≤ s. Then, H s · H m → H m , and for all f ∈ H s (Ω) and g ∈ H m (Ω), (1.5.88)

f gm ≤ C f s gm ,

with C independent of f and g. In particular, H s (Ω) is an algebra under pointwise multiplication (which makes sense, because, by (1.5.56), functions in H s (Ω) can be assumed to be continuous if s > N2 ), and for all f, g ∈ H s (Ω), (1.5.89)

f gs ≤ C f s gs ,

in accord with (1.5.88) for m = s. Proof. Corollary 1.5.2 follows from Theorem 1.5.4 with m, n and s replaced, respectively, by s, m, and n.  Remark. When N ≥ 3 and Ω = RN , (1.5.89) can be improved into either of (1.5.90)

f gs ≤ C ∇f s−1 gs

or

f gs ≤ C f s ∇gs−1 .

To see this, recalling Leibniz’ formula (1.5.72) we need to estimate finitely many terms of the form ∂xβ f ∂xα−β g, with 0 ≤ |α| ≤ s and β ≤ α. If |β| = 0, by (1.5.70), (1.5.91)

|f ∂xα g|2 ≤ C |f |∞ |∂xα g|2 ≤ C ∇f s−1 gs .

If |β| ≥ 1, there is a multi-index γ, and k ∈ {1, . . . , N }, such that ∂xβ = ∂xγ ∂k . Then, by (1.5.74), (1.5.92)

|∂xβ f ∂xα−β g|2 ≤ C ∂xγ ∂k f s−1−|γ| ∂xα−β gs−|α|+|β| ≤ C ∇f s−1 gs .

Thus, the first of (1.5.90) follows from (1.5.91) and (1.5.92). The proof of the second is analogous.  In applications to quasi-linear equations, we need to generalize Theorem 1.5.4 as follows.

28

1. Functional Framework

˜  (Ω) is an algebra under pointwise Theorem 1.5.5. If  > N2 , the space H multiplication (which is defined, because of (1.5.58)). That is, if f and ˜  (Ω), their product f g is in H ˜  (Ω), and g∈H (1.5.93)

f g˜ ≤ C f ˜ g˜ ,

˜  (Ω) and g ∈ with C independent of f and g. More generally, if f ∈ H r r H (Ω), 0 ≤ r ≤ , then f g ∈ H (Ω), and (1.5.94)

f gr ≤ C f ˜ gr ,

with C independent of f and g. ˜  (Ω). Then, f and g ∈ L∞ (Ω); thus, Sketch of Proof. Let f and g ∈ H f g ∈ L∞ (Ω), and |f g|∞ ≤ |f |∞ |g|∞ .

(1.5.95)

The fact that ∇(f g) ∈ H −1 (Ω) is a consequence of Leibniz’ formula (1.5.77), ˜  (Ω) (the only modification in its proof in which also holds if f and g ∈ H Theorem 1.5.4 being in the justification of (1.5.81), which is now a consequence of the fact that f δ → f and g δ → g uniformly on K = supp(ϕ)). Finally, estimates (1.5.93) and (1.5.94) follow from the analogous of estimate (1.5.76), which now reads (1.5.96)

∂xα (f g)0 ≤ C f ˜ gr ,

|α| ≤ r. This estimate is proven exactly as estimate (1.5.76); the only modification being in the estimate (1.5.78) when β = 0, which we replace with (1.5.97)

|f δ ∂xα g δ |2 ≤ |f δ |∞ |∂xα g δ |2 ≤ f ˜ gr . 

We can then complete the proof of Theorem 1.5.5.

10. Spaces V For integer m ≥ 1, we define V = V as the completion of H m with respect to the norm u → ∇um−1 . The space C0∞ (RN ) is dense in V m , because it is dense in H m , and the operator ∇ : H m → H m−1 extends continuously to an operator in V m , which we still denote by ∇. Likewise, the Laplace operator −Δ is a positive-definite operator from H m+1 into H m−1 , with square root (−Δ)1/2 : (H m → H m−1 ) → H m−1 . In fact, (−Δ)1/2 is an isometry from V m into H m−1 , with m (RN ).

(1.5.98)

m

m (RN )

(−Δ)1/2 um−1 = ∇um−1

for all u ∈ V m . Also, note that, if m > N2 and N ≥ 3, by (1.5.70) V m is continuously imbedded into Cb (RN ); thus, V m can be identified with a linear subspace of Cb (RN ).

1.5. Sobolev Spaces

29

Proposition 1.5.4. Let m ≥ 1 and h ∈ H m−1 . The function h is in the range space (−Δ)1/2 (H m ) if and only if  2 ˆ |h(ξ)| 2 (1.5.99) κ := dξ < +∞ . |ξ|2 RN In particular, (1.5.99) holds if h ∈ H m−1 ∩ Lp , with 1 ≤ p < N2N +2 (which m−1 p 1/2 requires N ≥ 3). If h ∈ H ∩ L and h = (− Δ) h0 for some h0 ∈ H m , there is C > 0, depending only on N and m, such that (1.5.100)

hm−1 ≤ h0 m ≤ C (hm−1 + |h|p ) .

Proof. If h ∈ H m−1 and h = (− Δ)1/2 h0 for some h0 ∈ H m , then κ = ˆ 0  = h0  is finite. Conversely, assume h ∈ H m−1 satisfies (1.5.99), and h ˆ Then, (1.5.99) implies that h0 ∈ L2 ; an induction let h0 := F −1 (| · |−1 h). procedure based on the identity  2 ˆ 0 (ξ)|2 (1 + |ξ|2 ) dξ h0 k = (1 + |ξ|2 )k−1 |h (1.5.101) = h0 2k−1 + h2k−1 , k ≥1, shows that, in fact, h0 ∈ H m . Assume now that N ≥ 3. If h ∈ Lp and q ˆ 1 ≤ p < N2N +2 < 2, the Hausdorff-Young theorem implies that h ∈ L , q 1 1 N p + q = 1. Let r be the conjugate index of 2 . We compute that r < 2 , so ˆ that the function ξ → 1/|ξ| is in L2r (RN ). Thus, the function ξ → h(ξ)/|ξ| loc

is in L2loc (RN ), which implies (1.5.99). The inequalities of (1.5.100) follow from (1.5.101): the first directly, with k = m; for the second, we first sum inequalities (1.5.101) from k = 1 to k = m, which yields (1.5.102)

h0 2m = h0 20 +

m 

h2k−1 ≤ h0 2 + C h2m−1 .

k=1

ˆ ∈ Lq , and | · |−1 ∈ L2r (RN ), Now, by (1.5.99), and recalling that h ∈ Lp , h loc   1 ˆ 1 ˆ h0 2 = |h(ξ)|2 dξ + |h(ξ)|2 dξ 2 2 |ξ| |ξ| |ξ|≥1 |ξ|≤1  1/r  2/q 1 q ˆ2+ ˆ ≤ |h| dξ |h(ξ)| dξ 2 (1.5.103) 2r |ξ|≤1 |ξ| |ξ|≤1 ˆ2 ≤ |h|2 + C |h| 2

q

≤ h2m−1 + C |h|2p . Inserting (1.5.103) into (1.5.102) yields the second inequality of (1.5.100). 

30

1. Functional Framework

1.5.2. The Laplace Operator. 1. Given a bounded domain Ω ⊂ RN , with smooth boundary ∂Ω and outward unit normal ν, the Laplace operator with homogeneous Dirichlet boundary conditions on Ω is the operator − ΔΩ , defined as the realization in H01 (Ω) of the standard second order differential operator defined in the second of (1.1.3); that is, − Δ := −

(1.5.104)

N  ∂2 = − div∇ . ∂x2i i=1

More precisely, for u ∈ H01 (Ω), − ΔΩ u is the distribution in H − 1 (Ω) defined by the identities − ΔΩ u, vH −1 ×H01 := ∇u, ∇v ,

(1.5.105)

v ∈ H01 (Ω) .

Clearly, − ΔΩ ∈ L(H01 (Ω); H − 1 (Ω)); as an unbounded operator in L2 (Ω), − ΔΩ has domain D(− ΔΩ ) = H 2 (Ω) ∩ H01 (Ω) (see, e.g., Lions [98, ch. 2]). If u ∈ D(− ΔΩ ), both ∇u and div(∇u) are in L2 (Ω), so that the normal component γ1 (u) := ν · ∇u can be defined as an element of H − 1/2 (∂Ω) (see, e.g., Milani and Koksch [119, thm. A84]). Hence, the integration by parts formula − ΔΩ u, v = −γ1 (u), γ0 (v)∂Ω + ∇u, ∇v

(1.5.106)

holds for u ∈ D(− ΔΩ ) and v ∈ H 1 (Ω), where  · , · ∂Ω denotes the duality pairing between H − 1/2 (∂Ω) and H 1/2 (∂Ω). Equation (1.5.106) is the natural generalization of the well-known Green’s formula    (1.5.107) (− Δu) v dx = − (ν · ∇u) v dS + ∇u · ∇v dx Ω

∂Ω

Ω

for u ∈ and v ∈ In particular, (1.5.106) implies that − ΔΩ is a self-adjoint and strictly positive operator of L(H01 (Ω); H − 1 (Ω)); this follows, via a density argument, from the identities C 2 (Ω)

C 1 (Ω).

(1.5.108)

− ΔΩ u, vH −1 ×H01 = ∇u, ∇v = − ΔΩ v, uH −1 ×H01 ,

(1.5.109)

− ΔΩ u, uH −1 ×H01 = ∇u, ∇u = ∇u2 .

For future reference, we note that a repeated application of (1.5.106) shows that a sufficient condition for the validity of the integration by parts formulas u, (− ΔΩ )m v = (− ΔΩ )m/2 u, (− ΔΩ )m/2 v ,

(1.5.110) if m is even, and (1.5.111)

u, (− ΔΩ )m v = ∇(− ΔΩ )(m−1)/2 u, ∇(− ΔΩ )(m−1)/2 v ,

if m is odd, is that u ∈ H m (Ω), v ∈ H 2m (Ω), and (1.5.112)

(− ΔΩ )j u , (− ΔΩ )k v ∈ H01 (Ω)

1.5. Sobolev Spaces

for 0 ≤ j ≤

 m−1 2

31

and 0 ≤ k ≤ m − 1. Likewise, the identity u, (− ΔΩ )m v = (− ΔΩ )m u, v

(1.5.113)

holds, if u, v ∈ H 2m (Ω), and (− Δ)k u, (− Δ)k v ∈ H01 (Ω), for 0 ≤ k ≤ m−1. 2. We recall the following elliptic regularity result. Proposition 1.5.5. Let m ≥ 0 and u ∈ H01 (Ω) be such that the distribution − Δu (defined by (1.5.105)) is in H m (Ω). Then, u ∈ H m+2 (Ω), and the estimate um+2 ≤ C ( − Δum + u0 )

(1.5.114)

holds, with C depending on m, N and Ω, but independent of u. If Ω is a ball of radius R ≥ 1, C can be determined independently of R. Proof. For a proof that u ∈ H m+2 (Ω) and satisfies (1.5.114), see, e.g., Gilbarg and Trudinger [52, thm. 8.13]. That proof is by induction on m; (1.5.114) is first established when Ω is a half-plane (in which case C depends only on m, and the term u0 can be omitted), and is then reported to general Ω by means of a partition of unity argument. The last claim of Proposition 1.5.5 follows by a homothetic-invariance argument, as in the proof of Proposition 1.5.2. Indeed, let ΩR = B(0, R), R ≥ 1. Given u ∈ H m+1 (ΩR ) ∩ H01 (ΩR ), define v ∈ H m+1 (Ω1 ) ∩ H01 (Ω1 ) by v(y) := u(Ry). Then, since (∂yα v)(y) = R|α| (∂xα u)(Ry), letting C1 denote the value of C in (1.5.114) when Ω = Ω1 , for |α| = m + 2 we estimate (1.5.115) ∂xα u2L2 (ΩR ) = RN −2(m+2) ∂yα v2L2 (Ω1 )  ≤ RN −2(m+2) C12 Δy v2H m (Ω1 ) + v2L2 (Ω1 ) . Now, since R ≥ 1, Δy v2H m (Ω1 ) = (1.5.116)

! |β|≤m

R2(|β|+2)−N ∂xβ Δx u2L2 (Ωr )

≤ R2(m+2)−N Δx u2H m (ΩR ) ; hence, we proceed from (1.5.115) with (1.5.117)

∂xα u2L2 (ΩR ) ≤ C12 Δx u2H m (ΩR ) + C12 R− 2(m+2) u2L2 (ΩR )  ≤ C12 Δx u2H m (ΩR ) + u2L2 (ΩR ) .

The right side of (1.5.117) is in accord with (1.5.114), with C1 independent of R, as claimed. When |α| ≤ m+1, we use instead the Gagliardo-Nirenberg

32

1. Functional Framework

inequality (1.5.118) |α|/(m+2) 1−|α|/(m+2) uL2 (Ω ) R) R

∂xα uL2 (ΩR ) ≤ C ∂xm+2 uL2 (Ω

+ CR uL2 (ΩR ) ,

noting that, as stated in Theorem 1.5.2, C is independent of R, and CR → 0 as R → +∞. This concludes the proof of Proposition 1.5.5.  For m ∈ N≥1 , we introduce the space (1.5.119) m HΔ (Ω) := {u ∈ H m (Ω) | (− Δ)k u ∈ H01 (Ω) ,

0 ≤ k ≤  m−1 2 } ,

1 (Ω) = H 1 (Ω). endowed with the norm induced by H m (Ω). Note that HΔ 0 m (Ω), and the space Proposition 1.5.6. For all m ≥ 1, H0m (Ω) ⊂ HΔ m (Ω) is dense in H m (Ω). C ∞ (Ω) ∩ HΔ Δ m (Ω) in H m (Ω), Sketch of Proof. 1) To prove the density of C ∞ (Ω) ∩ HΔ Δ m assume first that m = 2μ is even. Let u ∈ HΔ (Ω). Then, (− Δ)μ u ∈ L2 (Ω), and since C0∞ (Ω) is dense in L2 (Ω), given any ε > 0 there is ϕ ∈ C0∞ (Ω) such that

(1.5.120)

ϕ − (− Δ)μ u0 ≤ ε .

For 0 ≤ r ≤ μ − 1, we define ϕr recursively, starting from ϕ0 := ϕ, as the solution of the Dirichlet boundary value problem  − Δϕr+1 = ϕr , (1.5.121) ϕr+1 | = 0. ∂Ω

With the help of Proposition 1.5.5, we verify that each ϕr is in C ∞ (Ω) ∩ 2r (Ω); in particular, by (1.5.114), HΔ (1.5.122)

ϕμ − um ≤ C ( − Δ (ϕμ − u)m−2 + ϕμ − u0 ) .

Since ϕμ and u ∈ H01 (Ω), by the Poincar´e inequality (1.5.45) (1.5.123)

ϕμ − u20 ≤ C ∇ϕμ − ∇u20 = C− Δ(ϕμ − u), ϕμ − u ≤ C  − Δ(ϕμ − u)0 ϕμ − u0 ,

from which we deduce that (1.5.124)

ϕμ − u0 ≤ C  − Δ(ϕμ − u)0 .

Replacing (1.5.124) into (1.5.122) yields, recalling (1.5.121), (1.5.125)

ϕμ − um ≤ C  − Δ(ϕμ − u)m−2 = C ϕμ−1 − (− Δu)m−2 .

1.5. Sobolev Spaces

33

Iterating this procedure μ times and recalling (1.5.120), we obtain that (1.5.126)

ϕμ − um ≤ C μ ϕ0 − (− Δ)μ u0 ≤ C ε .

m (Ω), (1.5.126) proves the density claim when m is Since ϕμ ∈ C ∞ (Ω) ∩ HΔ even. When m = 2μ + 1 is odd, the procedure is the same, except that now (− Δ)μ u ∈ H01 (Ω). Since C0∞ (Ω) is dense in H01 (Ω), given any ε > 0 there is ϕ ∈ C0∞ (Ω) such that

ϕ − (− Δ)μ u1 ≤ ε

(1.5.127)

(compare to (1.5.120)). The rest of the proof proceeds in the same way. m (Ω); we only consider 2) We now turn to the proof that H0m (Ω) ⊂ HΔ the case m = 2p + 1 odd, the case of even m being analogous. We proceed by induction on p. The case p = 0 is obvious, because, as we have already 1 (Ω). Assume then that H 2p−1 (Ω) ⊂ H 2p−1 (Ω), for remarked, H01 (Ω) = HΔ 0 Δ p ≥ 1, and let f ∈ H02p+1 (Ω). Then, f ∈ H 2p+1 (Ω), and (− Δ)k f ∈ H 1 (Ω) for all k, 0 ≤ k ≤ p. We wish to prove that (− Δ)k f ∈ H01 (Ω), that is, that the traces γ0 ((− Δ)k f ) vanish on ∂Ω. To this end, we resort to the following identity on the values of the Laplace operator on the boundary ∂ Ω, valid, e.g., for all h ∈ C ∞ (Ω):

(1.5.128)

¯ h − (N − 1) H ∇ν h + ∇νν h , Δh = Δ

¯ denotes the Laplace operator on the manifold ∂ Ω, with respect to where Δ the metric induced by RN , H is the mean curvature of ∂ Ω, and ∇ν h, ∇νν h are, respectively, the first and second covariant derivatives of h with respect to the normal direction ν to ∂ Ω (see, e.g., Aubin [8, Ch. 3, 4]). We apply (1.5.128) to the function h = (− Δ)p−1 f , and obtain that, on ∂ Ω, − (− Δ)p f (1.5.129)

¯ ((− Δ)p−1 f ) − (N − 1) H ∇ν ((− Δ)p−1 f ) = Δ + ∇νν ((− Δ)p−1 f ) =: F1 + F2 + F3 .

¯ is a tangential differential operator, Of these terms, F1 = 0, because Δ and, by the induction assumption, (− Δ)p−1 f = 0 on ∂ Ω. To see that also F2 = 0 and F3 = 0, we use an orthonormal moving frame adapted to ∂ Ω, and expand F2 and F3 as sums of covariant derivatives of f of partial order with respect to ν at most 1 + 2(p − 1) = 2p − 1 for F2 , and at most 2 + 2(p − 1) = 2p for F3 . Since f ∈ H02p+1 (Ω), all these derivatives vanish on ∂ Ω. Thus, we conclude from (1.5.129) that (− Δ)p f ∈ H01 (Ω), as desired.  A repeated application of Proposition 1.5.5 yields m+1 Corollary 1.5.3. Let m ≥ 0 and u ∈ HΔ (Ω). Assume that (− Δ)(m+2)/2 u ∈ L2 (Ω) if m is even, or that (− Δ)(m+1)/2 u ∈ H 1 (Ω) if m is odd. Then

34

1. Functional Framework

u ∈ H m+2 (Ω) and satisfies the estimates  (1.5.130) um+2 ≤ C (− Δ)(m+2)/2 u0 + u0 , if m is even, and (1.5.131)

 um+2 ≤ C ∇(− Δ)(m+1)/2u0 + u0 ,

if m is odd, with C independent of u. If Ω is a ball of radius R ≥ 1, C can be determined independently of R. Sketch of Proof. Let first m be even. Then, (− Δ)r u |∂Ω = 0 for 0 ≤ r ≤ m 2 (in fact, this is true for all r ≥ 0, but we need to be careful to use only the m+1 conditions allowed by the assumption that u ∈ HΔ (Ω)). Hence, we can m apply (1.5.114) 2 + 1 times, and obtain Rm 

um+2 ≤ C (− Δ) u + C (− Δ)r u r=0  (m+2)/2 ≤ C (− Δ) u + u , (m+2)/2

(1.5.132)

where Rm :=  m+1 2 , and the last step is obtained by means of a repeated application of the inequalities (1.5.133) (− Δ)r u2 = (− Δ)r u, (− Δ)r u = (− Δ)r+1 u, (− Δ)r−1 u 1 1 (− Δ)r+1 u2 + (− Δ)r−1 u2 , 2 2 again justified by the fact that (− Δ)r u |∂Ω = 0 for 0 ≤ r ≤  m 2= m+1 if u ∈ HΔ (Ω). Note that (1.5.133) implies that   (1.5.134) (− Δ)r u ≤ max u, (− Δ)Rm +1 u . ≤

m 2

= Rm ,

Thus, (1.5.130) follows. If m is odd, the procedure is similar; the only change is in (1.5.132), which now would read 

Rm−1

um+2 ≤ C (− Δ)(m+1)/2 u1 + C

(− Δ)r u

r=0

(1.5.135)

≤ C ∇(− Δ)(m+1)/2u + C

Rm 

(− Δ)r u

r=0

 ≤ C ∇(− Δ)(m+1)/2 u + u , which is (1.5.131). Finally, the fact that C does not depend on R if Ω = B(0, R), R ≥ 1, follows from the analogous statement in Proposition 1.5.5. 

1.5. Sobolev Spaces

35

3. Since the boundedness of Ω implies that H01 →c L2 (Ω) (as per the last claim of Theorem 1.5.3), − ΔΩ has compact inverse; consequently, it admits an unbounded sequence of positive eigenvalues (λj )j≥1 , which can be ordered so that (1.5.136)

0 < λ1 ≤ λ2 ≤ · · · ≤ λj ≤ · · · ,

λj → +∞ .

m (Ω) ∩ For each j ≥ 1, the corresponding eigenfunction wj is in C ∞ (Ω) ∩ HΔ C(Ω) for all m ≥ 1; if Ω has a C ∞ boundary (in particular, if Ω is a ball), wj ∈ C ∞ (Ω). The sequence W = (wj )j≥1 is a complete system, orthonormal in L2 (Ω) and orthogonal in H01 (Ω) (recall part 3 of section 1.2). Together with Corollary 1.5.3, this allows us to consider different norms in the space m (Ω), defined in (1.5.119). HΔ m (Ω), the maps Proposition 1.5.7. Let m ≥ 1, and define, on HΔ ⎧  1/2 ⎪ ⎨ u2 + (− ΔΩ )m/2 u2 if m even , (1.5.137) N1 (u) := ⎪ ⎩ u2 + ∇(− Δ )(m−1)/2 u2 1/2 if m odd , Ω

and  (1.5.138)

N2 (u) =

∞ 

1/2 2 λm k |u, wk |

.

k=1 m (Ω), equivalent to the one induced by Then, N1 and N2 are norms in HΔ m H (Ω). More specifically, there are constants C1 and C2 , depending on Ω, m (Ω), such that, for all u ∈ HΔ

(1.5.139)

N1 (u) ≤ um ≤ C1 N1 (u) ,

(1.5.140)

N2 (u) ≤ N1 (u) ≤ C2 N2 (u) .

m (Ω). Clearly, if m is even, Proof. Let u ∈ HΔ

(1.5.141)

N1 (u) ≤ u0 + (− ΔΩ )m/2 u0 ≤ u0 + ∂xm u0 ≤ um ,

and analogously if m is odd. This proves the first of (1.5.139); the second follows from the elliptic estimates of Corollary 1.5.3. Thus, N1 is an equivam (Ω). To prove that N and N are equivalent, we first show lent norm in HΔ 1 2 that ⎧ ⎨ (−ΔΩ )m/2 u if m even , (1.5.142) N2 (u) = ⎩ ∇(−ΔΩ )(m−1)/2 u if m odd .

36

1. Functional Framework

Indeed, e.g., let m be even. Since (−Δ)m/2 u ∈ L2 (Ω), and (−Δ)r u |∂Ω = 0 m−1 for 0 ≤ r ≤ m 2 − 1 =  2 , by (1.5.113) we deduce that ∞ 

(−ΔΩ )m/2 u2 =

|(−ΔΩ )m/2 u, wk |2

k=1

(1.5.143)

∞ 

=

|u, (−ΔΩ )m/2 wk |2

k=1 ∞ 

=

m/2

|u, λk

wk |2 = (N2 (u))2 .

k=1

If instead m is odd, from the identity ∇(−ΔΩ )(m−1)/2 u =

(1.5.144)

∞ 

(−ΔΩ )(m−1)/2 u, wk  ∇wk ∈ L2 (Ω)

k=1

we deduce, again by (1.5.113), that (1.5.145) ∇(− ΔΩ )(m−1)/2 u2 =

=

=

=

∞ 

(−ΔΩ )(m−1)/2 u, wi  (−ΔΩ )(m−1)/2 u, wj  ∇wi , ∇wj 

i, j=1 ∞ 

u, (−ΔΩ )(m−1)/2 wi  u, (−ΔΩ )(m−1)/2 wj  ΔΩ wi , wj 

i, j=1 ∞ 

(m−1)/2

λi

(m−1)/2

λj

u, wi  u, wj  λi wi , wj 

i, j=1 ∞ 

2 2 λm i |u, wi | = (N2 (u)) .

i=1

Together with (1.5.143), (1.5.145) implies (1.5.142); in turn, this implies that N2 (u) ≤ N1 (u); that is, the first of (1.5.140). To show the second, recalling (1.5.136) we see that (1.5.146)

u = 2

∞  j=1

∞ 1  m 1 |u, wj | ≤ m λj |u, wj |2 = m (N2 (u))2 . λ1 λ1 2

j=1

Together with (1.5.145), (1.5.146) implies the second of (1.5.140), with C2 = 1  λm + 1. This ends the proof of Proposition 1.5.7. 1

1.5. Sobolev Spaces

37

m (Ω)-regular, for all m ≥ As a consequence of Proposition 1.5.7, W is HΔ 1. This follows from ∞ ∞ &2 &   & & (1.5.147) u − Pn u2m = & u, wj  wj & ≤ C |u, wj |2 λm j , m

j=n+1

j=n+1

which vanishes as n → +∞ because it is the tail of the series in (1.5.138), m (Ω). In addition, W is also a total orthogwhich converges because u ∈ HΔ m onal Fourier basis in HΔ (Ω), with respect to either of the norms N1 and N2 defined, respectively, in (1.5.137) and (1.5.138). For example, denoting by m (Ω) by the norm N , recalling  · , · N2 the scalar product induced on HΔ 2 2 that W is orthonormal in L (Ω) we deduce from (1.5.142) that, e.g., if m is even and i = j, wi , wj N2 (1.5.148)

= (− ΔΩ )m/2 wi , (− ΔΩ )m/2 wj  m/2

= λi

m/2

wi , λ j

wj  = 0 .

An analogous argument holds if m is odd, and for the scalar product induced m (Ω) by N . Furthermore, W is H m+1 (Ω)-regular. Indeed, by the on HΔ 1 Δ elliptic regularity estimate (1.5.114), we can replace (1.5.147) with (1.5.149) f − Pn f m+1 ≤ C (f − Pn f 0 + (− Δ)(f − Pn f )m−1 )  ≤ C f − Pn f 0 +  − Δf − Pn (− Δf )m−1  + Pn (− Δf ) − (− Δ)Pn f m−1 . In (1.5.149), the term − Δf −Pn (− Δf )m−1 vanishes, because of (1.5.147) m−1 with u = − Δf ∈ HΔ (Ω), and the last term equals zero, because Pn (− Δf ) =

N 

− Δf, wj  wj =

j=1

(1.5.150)

=

N 

=

f, − Δwj  wj

j=1

f, λ1j wj  wj =

j=1 N 

N 

N 

f, wj  λ1j wj

j=1

f, wj (− Δwj ) = − Δ

j=1

N  f, wj  wj j=1

= − Δ(Pn f ) . 1.5.3. Chain Rules and Commutator Estimates. In this section we first present an extension of the standard chain rule of differentiation to functions in Sobolev spaces, and then apply this result

38

1. Functional Framework

to the estimate of the commutator between differentiation and composition of such functions. 1. For m ∈ N and 1 ≤ p ≤ +∞, we define Wm,p := W m,p ∩ Cb (RN ) ,

(1.5.151)

which is a Banach space with respect to the norm (1.5.152)

uWm,p := um,p + sup |u(x)| . x∈RN

Lemma 1.5.1. Given m, b ∈ N, with 0 ≤ b ≤ m, and p ∈ [1, +∞], set q := m b ∈ [1, +∞], with the understanding that q = 1 even if b = m = 0, and q = +∞ if b = 0 < m. Then, Wm,p → Wb,pq . Proof. The case b = m is trivial. If b = 0, then q = ∞, and Wm,p → Cb (RN ) = W0,∞ . Assume then 1 ≤ b ≤ m − 1. Then, q ∈ ]1, +∞[, and pq > p, so that by Proposition 1.4.4 Wm,p → Lp ∩ L∞ → Lp q . Next, let m β ∈ NN , with 1 ≤ |β| ≤ b, and set r := |β| ≥ m b = q. Let w ∈ Wm,p . Then by Theorem 1.5.2 ∂xβ w ∈ Lp r , and, by Proposition 1.4.4 again, ∂xβ w ∈ Lp ∩ Lp r → Lp q . In addition, ∂xβ w satisfies the Gagliardo-Nirenberg and interpolation inequalities (see (1.5.41) and (1.4.27)) (1.5.153)

1−1/r |∂xβ w|p r ≤ C |∂xm w|1/r ≤ C wWm,p , p |w|∞

(1.5.154)

|∂xβ w|p q ≤ C |∂xβ w|θp |∂xβ w|p1−θ r ≤ C wWm,p ,

r−q with θ = q(r−1) . Estimates (1.5.153) and (1.5.154) show that the imbedding Wm,p → Wb, pq is continuous. 

2. Given ϕ ∈ C m (RM ), with m ≥ 0 and M ≥ 1, we set (1.5.155)

hm,ϕ (λ) := max sup |∂xα ϕ(z)| , |α|≤m |z|≤λ

λ ∈ R≥0 ;

note that, for all m ≥ 0, (1.5.156)

hm,ϕ  ≤ C hm+1,ϕ .

Proposition 1.5.8 (Chain Rule). Let m ∈ N≥1 , p ∈ [1, +∞], u = (u1 , . . . , uM ) ∈ (Wm,p )M , and ϕ ∈ C m (RM ), with ϕ(0) = 0. Then, ϕ(u) ∈ Wm,p , with (1.5.157) (1.5.158)

|ϕ(u)|∞ ≤ h0,ϕ (|u|∞ ) , |ϕ(u)|p ≤ h1,ϕ (|u|∞ ) |u|p .

In addition, if p ∈ ]1, +∞], then for α ∈ NN , with 1 ≤ |α| ≤ m, (1.5.159)

|α| |∂xα ϕ(u)|p ≤ C0 h|α|,ϕ (|u|∞ ) (1 + |u|m−1 ∞ ) |∂x u|p .

The constant C0 ≥ 1 depends only on m, N , M , and p.

1.5. Sobolev Spaces

39

Proof. First, note that the boundedness condition on u is indeed required, as the example ϕ(r) = r2 shows: if u ∈ H 1 but u ∈ / L∞ , then u2 need not 1 be in H . 1) (1.5.157) is clearly true, because the continuity of ϕ implies its boundedness over the compact ball {z ∈ RN | |z| ≤ |u|∞ }. Thus, for all x ∈ RN , (1.5.160)

|ϕ(u(x))| ≤

sup |ϕ(z)| = h0,ϕ (|u|∞ ) ,

|z|≤|u|∞

from which (1.5.157) follows. Also, writing  1 (1.5.161) ϕ(u) = ϕ  (λ u) · u dλ , 0

we see that ϕ(u) ∈ Lp , 1 ≤ p ≤ ∞, and satisfies (1.5.158). 2) Assume now that p ∈ ]1, ∞[, and that u is also in (C ∞ )M . We prove (1.5.159) by induction on m, via repeated use of the H¨older and the Gagliardo-Nirenberg inequalities. Indeed, (1.5.159) is clearly true for m = 1; assuming then that it is true for 1 ≤ m ≤ s, we show that it is also true for m = s + 1. Let α ∈ NN be such that |α| = s, and k ∈ {1, 2, . . . , N }. Then,    α β  α−β (1.5.162) ∂xα ∂k (ϕ(u)) = ∂xα ϕ  (u) · ∂k u = ∂k u . β ∂x ϕ (u) · ∂x β≤α

If β = 0, by (1.5.157) applied to to ϕ  , and (1.5.156) with m = 0, (1.5.163)

|ϕ  (u) · ∂xα ∂k u|p ≤ |ϕ  (u)|∞ |∂xα ∂k u|p ≤ h1,ϕ (|u|∞ ) |∂xs+1 u|p .

b If β > 0, we set b := |β| and define q ∈ ]1, +∞[ by 1q := s+1 . By H¨older’s inequality and the induction assumption applied to u (considered as a function in Wb,pq , as allowed by Lemma 1.5.1), and since b ≤ |α| = s,

(1.5.164)

|∂xβ ϕ  (u) · ∂xα−β ∂k u|p ≤ C |∂xβ ϕ  (u)|p q |∂xα−β ∂k u|p q  b s−b ≤ C hb,ϕ  (|u|∞ ) (1 + |u|s−1 ∞ ) |∂x u|p q |∂x ∂k u|p q  .

Because of our choice of q, the Gagliardo-Nirenberg inequalities (1.5.42) imply that (1.5.165) (1.5.166)

1−1/q |∂xb u|p q ≤ C |∂xs+1 u|1/q , p |u|∞

|∂xs−b ∂k u|p q 

≤ C |∂xs+1 u|p1−1/q |u|1/q ∞ .

By (1.5.165) and (1.5.166), and recalling that b + 1 ≤ |α| + 1 = s + 1, as well as (1.5.156) with m = b, we deduce from (1.5.164) that (1.5.167)

|∂xβ ϕ  (u) · ∂xα−β ∂k u|p s+1 ≤ C hs+1,ϕ (|u|∞ ) (1 + |u|s−1 ∞ ) |u|∞ |∂x u|p .

Together with (1.5.163), (1.5.167) shows that (1.5.159) also holds for m = s + 1 (and p = 1 as well, if u ∈ C ∞ ).

40

1. Functional Framework

3) Let now u ∈ (Wm,p )M . By Theorems 1.4.1 and 1.5.1, the Friedrichs’ regularizations un := u1/n , defined in (1.4.7), are in (Wm,p ∩ C ∞ )M , with |un |∞ ≤ |u|∞

(1.5.168) for all n > 0 and, as n → ∞, un → u

(1.5.169)

in (W m,p )M and pointwise in RN .

Since ϕ is continuous, (1.5.168) implies that there is R > 0 such that |ϕ(un (x))| ≤ R

(1.5.170)

for all x ∈ RN . By (1.5.159), and (1.5.168), a n |∂xα ϕ(un )|p ≤ C0 ha,ϕ (|u|∞ ) (1 + |u|m−1 ∞ ) |∂x u |p ;

(1.5.171)

thus, by (1.5.169), the sequence (∂xα ϕ(un ))n≥1 is bounded in Lp . Since Lp is reflexive, there are a subsequence, still denoted (∂xα ϕ(un ))n≥1 , and a function χα ∈ Lp , such that ∂xα ϕ(un ) → χα

(1.5.172)

weakly in Lp ; that is, recalling (1.4.19),   α n (1.5.173) ∂x ϕ(u ) g dx → χα g dx 

for all g ∈ Lp . Let ζ ∈ C0∞ . Then, by (1.5.169) and Lebesgue’s dominated convergence theorem (via (1.5.170)), (1.5.174) 

 ∂xα ϕ(un ) ζ

dx = (− 1)

a

ϕ(un ) ∂xα ζ dx 

→ (− 1)

a

 ϕ(u) ∂xα ζ

dx =

∂xα ϕ(u) ζ dx .

Comparing this to (1.5.173), we deduce that χα = ∂xα ϕ(u). Hence, by (1.2.7), (1.5.171), and (1.5.169), |∂xα ϕ(u)|p ≤ lim inf |∂xα ϕ(un )|p n→∞

(1.5.175)

a n ≤ C0 ha,ϕ (|u|∞ ) (1 + |u|m−1 ∞ ) lim |∂x u |p

= C0 ha,ϕ (|u|∞ ) (1 +

n→∞ m−1 |u|∞ ) |∂xa u|p

.

That is, (1.5.159) holds. 4) If p = ∞, the proof of (1.5.159) is based on the weak∗ sequential compactness of balls in L∞ . We refer to Racke [136, lemma 4.7], where one can find an alternative proof of (1.5.159), based on the explicit representation of composite derivatives provided by the Fa` a di Bruno formula. 

1.5. Sobolev Spaces

41

Remarks. If ϕ(0) = 0, estimate (1.5.159) still holds for |α| ≥ 1; however, it is not necessarily true that ϕ(u) ∈ Lp , as we can see, for example, if ϕ is constant. On the other hand, the conclusion of Proposition 1.5.8 can be rephrased, in the sense that the function x → ϕ(u(x)) − ϕ(0) is in Wm,p , and, if 1 ≤ p < +∞ and α ∈ NN , with |α| ≤ m, (1.5.176)

|α| |∂xα (ϕ(u) − ϕ(0))|p ≤ C0 hm,ϕ (|u|∞ ) (1 + |u|m−1 ∞ ) |∂x u|p .

In fact, (1.5.176) follows from (1.5.159) if α > 0, while if α = 0, it follows from (1.5.177)

|ϕ(u) − ϕ(0)|p ≤ C0 h1,ϕ (|u|∞ ) |u|p ,

which in turn is a consequence of (1.5.161), with ϕ(u) replaced by ϕ(u) − ϕ(0).  Proposition 1.5.9. Let m ∈ N≥1 , p ∈ ]1, +∞[, and ϕ ∈ C m (RM ), with ϕ(0) = 0. Then, the map u → ϕ(u) is locally Lipschitz continuous from (Wm−1,p )M into Wm−1,p , and continuous from (Wm,p )M into Wm,p . Proof. First, note that Proposition 1.5.8 implies that the map u → ϕ(u) can be properly defined both from (Wm,p )M into Wm,p , and from (Wm−1,p )M into Wm−1,p . We also note that, in order to estimate the difference ϕ(u) − ϕ(v), we would not need the assumption ϕ(0) = 0, since we can replace ϕ(·) with ϕ(·) − ϕ(0). 1) Fix u, v ∈ (Wm−1,p )M with (e.g.) uWm−1,p , vWm−1,p ≤ R. Since ϕ is at least of class C 1 , we can write  1 (1.5.178) ϕ(u) − ϕ(v) = ϕ  (λu + (1 − λ)v) · (u − v) d λ . 0

Letting (1.5.179)

C1 (R) := max |ϕ  (w)| , |w|≤R

we deduce from (1.5.178) that, for 1 ≤ p ≤ ∞, (1.5.180)

|ϕ(u) − ϕ(v)|p ≤ C1 (R) |u − v|p ,

which shows that ϕ is locally Lipschitz continuous from (W0,p )M into W0,p . Next, fix α ∈ NN , with 1 ≤ |α| ≤ m−1. From (1.5.178), by Leibniz’ formula (Theorem 1.5.4),

(1.5.181)

∂xα (ϕ(u) − ϕ(v))    α = β β≤α

1 0

∂xβ ϕ  (λu + (1 − λ)v) ∂xα−β (u − v) d λ

42

1. Functional Framework

(this identity can be justified by regularization, as in part 3 of the proof of Proposition 1.5.8). We proceed as in the estimate of (1.5.162). If β = 0, as in (1.5.180), |ϕ  (λu + (1 − λ)v)∂xα (u − v)|p

(1.5.182)

≤ C1 (R) |∂xα (u − v)|p ≤ C1 (R) u − vm−1,p . |α| |β| ∈ [1, +∞[. Lp q , with

If β > 0, we set q := so that

∂xβ u,

∂xβ v

∈

Then, by Lemma 1.5.1, W|α|,p → W|β|,pq , 1/q

(1.5.183)

1−1/q

|∂xβ u|p q ≤ C |∂xα u|p |u|∞

≤ C (|∂xα u|p + |u|∞ ) ≤ C uWm−1,p , 

and analogously for v. In addition, ∂xα−β (u − v) ∈ Lp q , as follows from the estimate (1.5.184)

|∂xα−β (u − v)|p q 

1−1/q

≤ C |∂xα (u − v)|p

1/q

|u − v|∞

≤ C u − vWm−1,p .

Since ϕ  ∈ C m−1 (RM ) and 1 ≤ |β| ≤ m − 1, Proposition 1.5.8 and (1.5.183) imply that ∂xβ ϕ  (λu + (1 − λ)v) ∈ Lpq , with  |∂xβ ϕ  (λu + (1 − λ)v)|pq ≤ Ψ(U, V ) |∂xb u|pq + |∂xb v|pq (1.5.185)

≤ C(uWm−1,p , vWm−1,p ) ≤ C2 (R) ,

where C2 ∈ K, b = |β|, U := |u|∞ , V := |v|∞ , and  (1.5.186) Ψ(U, V ) := C0 hb,ϕ  (U + V ) 1 + U b−1 + V b−1 . In conclusion, from (1.5.181), (1.5.182), (1.5.185), and (1.5.184), we deduce the estimate (1.5.187)

|∂xα (ϕ(u) − ϕ(v))|p ≤ C2 (R) u − vWm−1,p .

Together with (1.5.180) for p = ∞, (1.5.187) shows the asserted local Lipschitz continuity of ϕ from (Wm−1,p )M into Wm−1,p . 2) To prove the continuity of the map u → ϕ(u) from (Wm,p )M into Wm,p , it is sufficient to estimate the difference ∂xα (ϕ(u) − ϕ(v)), u, v ∈ (Wm,p )M , for |α| = m. We proceed by induction on m, considering p ∈ ]1, ∞[ and ϕ ∈ C m (RM ) as arbitrary. As in the previous part of this proof, we assume that uWm,p , vWm,p ≤ R.

1.5. Sobolev Spaces

43

2a) Let first m = 1. For k ∈ {1, . . . , N } and u, v ∈ (W1,p )M , we decompose (1.5.188) ∂k (ϕ(u) − ϕ(v)) = ϕ  (u) · (∂k u − ∂k v) + (ϕ  (u) − ϕ  (v)) · ∂k v . At first, recalling (1.5.179), (1.5.189)

|ϕ  (u) · (∂k u − ∂k v)|p ≤ |ϕ  (u)|∞ |∂k u − ∂k v|p ≤ C1 (R) u − v1,p .

Similarly, (1.5.190)

|(ϕ  (u) − ϕ  (v)) · ∂k v|p ≤ |ϕ  (u) − ϕ  (v)|∞ |∂k v|p ≤ |ϕ  (u) − ϕ  (v)|∞ v1,p .

Since ϕ  is uniformly continuous on the compact ball B(0, R), given ε > 0 there is δ ∈ ]0, 1] such that |ϕ  (a) − ϕ  (b)| ≤ ε if |a|, |b| ≤ R, and |a − b| ≤ δ. Thus, |ϕ  (u) − ϕ  (v)|∞ ≤ ε if u − vCb (RN ) ≤ δ, and in this case we deduce from (1.5.190) that (1.5.191)

|(ϕ  (u) − ϕ  (v)) ∂k v|p ≤ ε v1,p .

Thus, from (1.5.189) and (1.5.191) we conclude that, if u − vW1,p ≤ δ, (1.5.192)

|∂k (ϕ(u) − ϕ(v))|p ≤ C3 (R) (δ + ε) .

Clearly, (1.5.192) allows us to show the asserted continuity of ϕ on (W1,p )M . 2b) Assume then that ϕ is continuous from (Wm−1,p )M into Wm−1,p ; that is, the induction step holds up to m − 1, m ≥ 2, for any p ∈ ]1, ∞[. Fix α ∈ NN with |α| = m, and let k ∈ {1, . . . , N }, γ ∈ NN , be such that ∂xα = ∂xγ ∂k . As in (1.5.188), we decompose (1.5.193) ∂xα (ϕ(u) − ϕ(v)) = ∂xγ (ϕ  (u) · ∂k u − ϕ  (v) · ∂k v) = ∂xγ (ϕ  (u) · ∂k (u − v)) + ∂xγ ((ϕ  (u) − ϕ  (v)) · ∂k v) =: Φ1 (u, v) + Φ2 (u, v) . The estimate of |Φ1 (u, v)|p is analogous to that of ∂xα (ϕ(u) − ϕ(v)) of (1.5.181), via Leibniz’ formula   γ β  γ−β (1.5.194) Φ1 (u, v) = ∂k (u − v) . β ∂x ϕ (u) ∂x β≤γ

Proceeding as in the first part of this proof, we arrive at (1.5.195)

|Φ1 (u, v)|p ≤ C4 (R) u − vWm,p .

44

1. Functional Framework

The estimate of Φ2 (u, v) is only slightly different. Again by Leibniz’ formula,   γ β   γ−β (1.5.196) Φ2 (u, v) = ∂k v . β ∂x (ϕ (u) − ϕ (v)) ∂x β≤γ

Set q := (1.5.197)

m b

∈ ]1, +∞]. Then by Lemma 1.5.1 Wm,p → W|β|,pq , and |∂xβ (ϕ  (u) − ϕ  (v)) ∂xγ−β ∂k v|p ≤ C |∂xβ (ϕ  (u) − ϕ  (v))|p q |∂xγ−β ∂k v|p q  .

By the Gagliardo-Nirenberg inequalities, |∂xγ−β ∂k v|p q  (1.5.198)

= |∂xα−β v|p q  1−1/q

≤ C |∂xm v|p

1/q

|v|∞ ≤ C vWm,p ;

thus, since vWm,p ≤ R, we obtain from (1.5.197) that (1.5.199) |∂xβ (ϕ  (u) − ϕ  (v)) ∂xγ−β ∂k v|p ≤ C5 (R) |∂xβ (ϕ  (u) − ϕ  (v))|p q . Since |β| ≤ |γ| = m − 1 and ϕ  ∈ C m−1 (RM ), the induction assumption up to m − 1, with ϕ and p replaced by ϕ  and p q, implies that ϕ  is continuous from (W|β|, pq )M into W|β|,p q . Hence, given ε > 0 there is δβ ∈ ]0, 1] such that, if u − vW|β|, pq ≤ δβ , (1.5.200)

|∂xβ (ϕ  (u) − ϕ  (v))|p q ≤ ϕ  (u) − ϕ  (v)W|β|,p q ≤ ε .

Let δ := min0≤|β|≤m−1 Cβ−1 δβ , where Cβ is the norm of the imbedding Wm,p → W|β|,p q , and assume that u − vWm,p ≤ δ. Then, (1.5.201)

u − vW|β|,p q ≤ Cβ u − vWm,p ≤ Cβ δ ≤ δβ ;

thus, (1.5.200) holds. By (1.5.199), we deduce from (1.5.196) that (1.5.202)

|Φ2 (u, v)|p ≤ C6 (R) ε .

Putting (1.5.195) and (1.5.202) into (1.5.193), we conclude that (1.5.203)

|∂xα (ϕ(u) − ϕ(v))|p ≤ C7 (u) (δ + ε)

if u − vWm,p ≤ δ. As in (1.5.192), (1.5.203) allows us to show the asserted continuity of ϕ on (Wm,p )M ; in fact, ϕ is locally uniformly continuous on (Wm,p )M . This ends the proof of Proposition 1.5.8. We remark explicitly that, when b = 0, the right side of (1.5.199) contains the factor |ϕ  (u) −ϕ  (v)|∞ , which can be estimated as we did to deduce (1.5.191) from (1.5.190). 

1.5. Sobolev Spaces

45

Proposition 1.5.10 (Linear Commutator Estimate). Let m, s ∈ N be such ˜ s and w ∈ H m . Then, for all that s > N2 + 1 and 1 ≤ m ≤ s. Let ζ ∈ H N α ∈ N with |α| ≤ m, the commutator Gα (ζ, w) := ∂xα (ζw) − ζ ∂xα w

(1.5.204) is in H m−|α| , and (1.5.205)

Gα (ζ, w)m−|α| ≤ C ∇ζs−1 wm−1 .

Remark. The importance of this proposition lies in that estimate (1.5.205) only involves the lower order norm of w in H m−1 .  Proof. As in the proof of Theorem 1.5.4, let ζ δ and wδ be the Friedrichs’ regularizations of ζ and w. Then, by Leibniz’ formula,    α β δ α−β δ (1.5.206) Gα (ζ δ , wδ ) = w =: Σ(ζ δ , wδ ) . β ∂x ζ ∂x 0 0, define (1.5.215)

C δ (h, u) := ρδ ∗ (h u) − h (ρδ ∗ u) .

Then, C δ (h, u) ∈ H 1 for all δ > 0, and, if δ ∈ ]0, 1], (1.5.216)

C δ (h, u)1 ≤ C |∇h|∞ u0 ,

with C independent of h, u, and δ. In fact, C δ (h, u) → 0 in H 1 as δ → 0. Remark. The importance of this result lies in that the assumption u ∈ H 1 is not required for the convergence of C δ (h, u) to 0 to hold in H 1 (compare to the corollary after lemma 6.1 of Mizohata [122, ch. 6]).  Proof. Since h u ∈ L2 , Theorem 1.5.1 implies that C δ (h, u) ∈ H 1 , and C δ (h, u) → 0 in L2 as δ → 0. Thus, we need to prove that, for 1 ≤ k ≤ N , ∂k [C δ (h, u)] → 0 in L2 as well. We can write  (1.5.217) C δ (h, u)(x) = ρδ (x − y) (h(y) − h(x)) u(y) dy ,

1.5. Sobolev Spaces

47

so that (1.5.218) ∂xk C δ (h, u)(x)  = By (1.5.214), (1.5.219) in addition,

∂xk (ρδ (x − y)) (h(y) − h(x)) u(y) dy − ∂xk h(x) uδ (x) .   ∂xk ρδ (x − y) = − ∂yk ρδ (x − y) ; 

  ∂yk ρδ (x − y) (h(x) − h(y)) u(x) dy = 0 .

(1.5.220)

Consequently, we obtain from (1.5.218) that (1.5.221)

 δ

∂xk C (h, u)(x) =

∂yk [ρδ (x − y) (h(x) − h(y))] (u(y) − u(x)) dy + (ρδ ∗ (u ∂xk h))(x) − ∂xk h(x) uδ (x)

=: I1,δ (x) + I2,δ (x) − I3,δ (x) . Since ∂k h ∈ L∞ , Theorem 1.4.1 yields that I2,δ − I3,δ → 0 in L2 . We decompose then I1,δ = I4,δ + I5,δ , where  I4,δ (x) := ∂yk [ρδ (x − y)] (h(y) − h(x)) (u(x) − u(y)) dy , (1.5.222)  I5,δ (x) := ρδ (x − y) (∂yk h(y)) (u(x) − u(y)) dy . (1.5.223) To estimate I4,δ , we note that, by (1.5.63), h is Lipschitz continuous; thus, setting L := |∇h|∞ and  (1.5.224) M := |∂k ρ(z)| dz , |z|≤1

by H¨older’s inequality (1.5.225)

 '    |x − y| |u(x) − u(y)| dy |I4,δ (x)| ≤ L δN1+1 |x−y|≤δ ∂k ρ x−y δ  ≤L |∂k ρ(z)| |u(x) − u(x − δz)| dz |z|≤1

√ ≤L M



1/2 |∂k ρ(z)| |u(x) − u(x − δz)| dz 2

.

48

1. Functional Framework

From this, it follows that (1.5.226) 

 |I4,δ (x)| dx ≤ L M 2

2

 |∂k ρ(z)|

|u(x) − u(x − δz)|2 dx dz

(LM )2 sup |u(· − δz) − u|22 , |z|≤1

so that I4,δ → 0 in L2 as δ → 0. Acting likewise, we obtain that |I5,δ |2 ≤ L sup |u(· − δz) − u|2 ,

(1.5.227)

|z|≤1

which shows that also I5,δ → 0 in L2 as δ → 0. In conclusion, we deduce from (1.5.221) that ∂k [C δ (h, u)] → 0 in L2 , as desired. This proves the convergence claim of Lemma 1.5.2. To show (1.5.216), we first note that proceeding as in (1.5.225), we can deduce from (1.5.217) that  (1.5.228)

|C (h, u)(x)| ≤ L δ δ

1/2 ρ(z) |u(x − δz)| dz 2

,

from which, as in (1.5.226), if δ ∈ ]0, 1], (1.5.229)

|C δ (h, u)|2 ≤ L δ |u|2 ≤ |∇h|∞ |u|2 .

Next, by (1.4.8) of Theorem 1.4.1, we deduce from (1.5.221) that (1.5.230)

|I2,δ − I3,δ |2 ≤ |u (∂k h)|2 + |(∂k h) uδ |2 ≤ 2 |∇h|∞ |u|2 ,

and, from (1.5.226) and (1.5.227), that (1.5.231)

|I4,δ + I5,δ |2 ≤ 2 L(M + 1) |u|2 = 2(M + 1)|∇h|∞ |u|2 .

Thus, (1.5.216) follows from (1.5.229), (1.5.230), and (1.5.231), and the proof of Lemma 1.5.2 is complete.  ˜ s, Theorem 1.5.6. Let s, m ∈ N, with s > N2 +1 and 1 ≤ m ≤ s. Let h ∈ H u ∈ H m−1 , and define C δ (h, u) as in (1.5.215). Then, C δ (h, u) ∈ H m for all δ > 0, and (1.5.232)

ρδ ∗ (h u) − h (ρδ ∗ u)m ≤ C ∇hs−1 um−1 .

In fact, C δ (h, u) → 0 in H m as δ → 0. Proof. We first note that, by Theorem 1.5.5, h u ∈ H m−1 . We fix a multiindex α, with |α| ≤ m − 1, and show that, as δ → 0, ∂xα C δ (h, u) → 0 in H 1 .

1.5. Sobolev Spaces

49

We compute that (1.5.233) ∂xα C δ (h, u) =

   α δ β α−β β δ α−β ρ ∗ (∂ h ∂ u) − ∂ h (ρ ∗ ∂ u) x x x x β β≤α

=

  α β

C δ (∂xβ h, ∂xα−β u) .

β≤α

In this sum, the term corresponding to β = 0 converges to 0 in H 1 by ˜ s → W 1,∞ (by (1.5.56)), and ∂xα u ∈ L2 . If Lemma 1.5.2, because h ∈ H β > 0, Theorem 1.5.4 implies that ∂xβ h ∂xα−β u ∈ H s−|β| ·H m−1−|α|+|β| → H 1 , because s − |β| ≥ 1, m − 1 − |α| + |β| ≥ 1, and (s − |β|) + (m − 1 − |α| + |β|) > 1 + N2 . Hence, by the last part of Theorem 1.5.1 (first with s = 1, and then with s = m − 1 − |α| + |β|), (1.5.234)

ρδ ∗ (∂xβ h ∂xα−β u) → ∂xβ h ∂xα−β u

(1.5.235)

ρδ ∗ (∂xα−β u) → ∂xα−β u

in H 1 , in H m−1−|α|+|β| ,

so that (1.5.236)

∂xβ h (ρδ ∗ ∂xα−β u) → ∂xβ h ∂xα−β u

in H 1 as well. In conclusion, ∂xα C δ (h, u) → 0 in H 1 , and the convergence claim of Theorem 1.5.6 is proven. To show (1.5.232), we first note that, by (1.5.216), the term corresponding to β = 0 in the sum (1.5.233) can be estimated by (1.5.237)

C δ (h, ∂xα u)1 ≤ C |∇h|∞ ∂xα u ≤ C ∇hs−1 um−1 .

The other terms of the sum can be estimated by means of (1.5.34) and Theorem 1.5.4; recalling that |β| ≥ 1, we find C δ (∂xβ h, ∂xα−β u)1 ≤ C ∇hs−|β|−1 ∂xα−β um−|α|+|β|−1 (1.5.238) ≤ C ∇hs−1 um−1 . Thus, (1.5.232) follows, and the proof of Theorem 1.5.6 is complete.



2. In our applications to second order PDEs, we will often resort to the following version of Theorem 1.5.6. Corollary 1.5.4. Let s, m ∈ N, with s > N2 + 1 and 1 ≤ m ≤ s. For ˜ s , and u ∈ H m+1 . Then, C δ (aij , ∂i ∂j u) ∈ H m i, j = 1, . . . , N , let aij ∈ H for all δ > 0, and (1.5.239)

ρδ ∗ (aij ∂i ∂j u) − aij ∂i ∂j (ρδ ∗ u)m ≤ C ∇aij s−1 ∇um .

In fact, C δ (aij , ∂i ∂j u) → 0 in H m as δ → 0.

50

1. Functional Framework

Proof. This result follows from Theorem 1.5.6, with h = aij , and u replaced by ∂i ∂j u ∈ H m−1 , noting that ∂i ∂j (ρδ ∗ u) = ρδ ∗ (∂i ∂j u).  Corollary 1.5.4 can be extended to the case m = 0, as follows: Theorem 1.5.7. Let s ∈ N, s > let

N 2

˜ s , and u ∈ H 1 . For δ > 0, + 1, aij ∈ H

 Fijδ := ρδ ∗ (∂j (aij ∂i u)) − ∂j aij ∂i (ρδ ∗ u) .

(1.5.240)

Then, Fijδ ∈ L2 for all δ > 0, and Fijδ  ≤ C ∇aij s−1 ∇u .

(1.5.241)

In fact, Fijδ → 0 in L2 as δ → 0. The proof of this theorem follows the same lines of that of Theorem 1.5.6, so we omit it. We only need to note that, in (1.5.240), the first convolution has to be understood in the sense of the convolution of the distribution ∂j (aij ∂i u), which is in H −1 , with the test functions y → ρδ (x − y) (see, e.g., Yosida [165, ch. VI, §3]). More precisely, (1.5.242)

[ρδ ∗ (∂j (aij ∂i u))](x)

 1 ∂ x−y ∂ := − ρ u(y) dy , aij (y) δ N ∂yj δ ∂yi

so that



1 ∂ x−y ∂ ρ u(y) dy aij (y) δ N ∂yj δ ∂yi

 ∂2 1 x−y − aij (x) ρ u(y) dy δ N ∂xi ∂xj δ

 1 ∂ x−y − ∂j aij (x) ρ u(y) dy . δ N ∂xi δ 

Fijδ (x) = − (1.5.243)

Since, by (1.5.214), (1.5.244)

∂2 ρ ∂xi ∂xj



x−y δ



∂2 ρ = ∂yi ∂yj



x−y δ

,

integrating by parts once in the middle term of the right side of (1.5.243), we deduce that

 1 ∂ x−y ∂ Fijδ (x) = ρ u(y) dy (aij (x) − aij (y)) δ N ∂yj δ ∂yi (1.5.245)

 1 ∂ x−y − ∂j aij (x) ρ u(y) dy . δ N ∂xi δ 

1.6. Orthogonal Bases in H m (RN )

51

1.6. Orthogonal Bases in H m (RN ) In this section we construct orthonormal Fourier bases for the spaces H m = H m (RN ), m ∈ N, by means of the Hermite functions. As usual, we denote by u ˆ = F u and u ˇ = F −1 u, respectively, the Fourier transform of a function 2 u ∈ L , and its inverse. In accord with (1.5.7), these are defined by  u ˆ(ξ) := CN e− i x·ξ u(x) dx , (1.6.1)  u ˇ(ξ) := CN ei x·ξ u(x) dx , (1.6.2) with CN := (2π)− N/2 . 1. Hermite Functions. The Hermite polynomials (Hn )n≥0 on R can be defined in various equivalent ways; for example, by means of the so-called Rodrigues’ formula (1.6.3)

2

Hn (x) = (−1)n ex

dn − x2 e , dxn

or by the recursive relations (1.6.4) (1.6.5)

Hn+1 (x) = 2(x Hn (x) − n Hn−1 (x)) , Hn  (x) = 2n Hn−1 (x) ,

starting from (1.6.6)

H− 1 (x) := 0 ,

H0 (x) := 1 .

One then defines the normalized Hermite functions (1.6.7)

hn (x) := cn e− x

2 /2

Hn (x) ,

√ cn := ( π 2n n!)−1/2 .

It is then known (see, e.g., Szeg¨ o [152]) that the sequence (hn )n≥0 forms a complete orthonormal system in L2 ; recalling (1.2.11), this means that for all j, k ∈ N,  (1.6.8) hj (x) hk (x) dx = δjk , and that for each f ∈ L2 , the identity (1.6.9)

f=

∞ 

f, hj  hj

j=0

holds in L2 and, in fact, a.e. in R. Explicitly, this means that n & &  & & (1.6.10) lim &f − f, hj  hj & = 0 , n→∞

j=0

0

52

1. Functional Framework

and for a.a. x ∈ R, (1.6.11)

f (x) = lim

n→+∞

n  f, hj  hj (x) . j=0

(The a.e. convergence of the series (1.6.9) follows from an equiconvergence result proved by Muckenhoupt [123], together with the Carleson-Hunt theorem (Carleson [20] and Hunt [62])1 ). The Hermite functions hn are eigenvectors of the Fourier transform, corresponding to the eigenvalues (− i)n ; that is, for all n ≥ 0, (F hn )(ξ) = (−i)n hn (ξ) .

(1.6.12)

Each hn is also an eigenvector of the second-order differential operator u → − u  + x2 u ,

(1.6.13)

corresponding to the eigenvalue λn = 2n + 1 (see, e.g., Dautray and Lions [42, ch. 8, §2.7]). Since for all n ≥ 0, cn (1.6.14) cn+1 = ( , 2(n + 1) we deduce from (1.6.4) and (1.6.5) the recursive relations √ √ √ (1.6.15) n + 1 hn+1 (x) = 2 x hn (x) − n hn−1 (x) , (1.6.16)

√ √ √ 2 hn  (x) = n hn−1 (x) − n + 1 hn+1 (x) ,

n ≥ 0, starting from (1.6.17)

h0 (x) =

√ 2 − 1/2 π ex ,

h−1 (x) = 0 .

2. A Basis of H m . We now define a sequence of functions (wj )j∈NN ⊂ S → H m by  ˆ j = am ∗ h j , (1.6.18) wj := F −1 (1 + | · |2 )− m/2 h where, for j = (j1 , . . . , jN ) ∈ NN , hj is defined by the product hj (x) = hj1 (x1 ) · · · hjN (xN ) , x = (x1 , . . . , xN ) ∈ RN .   The functions am := F −1 (2π)− N/2 (1 + | · |2 )−m/2 can be found in closed form (a2 is a scalar multiple of e−| · | , and for the other values of m, am is the product of a polynomial and a modified Bessel function; see, e.g., Erd´ely et al. [46]). Note that each hj is an eigenvector of the second-order differential operator (1.6.19)

(1.6.20) 1 We

u → − Δ u + |x|2 u , are grateful to Professor K. Stempak for this information.

1.6. Orthogonal Bases in H m (RN )

53

corresponding to the eigenvalue λj = 2|j| + N ; in addition, (1.6.19) implies that, for ξ = (ξ1 , . . . , ξN ) ∈ RN , ˆ j (ξ) = h ˆ j (ξ1 ) · · · h ˆ j (ξN ) . h 1 N

(1.6.21)

In (1.6.21), with some abuse of notation we use the same symbolˆto denote the Fourier transforms both in RN , at the left side of (1.6.21), and in R, at its right side. Finally, for 1 ≤ r ≤ N we define the projections x → μr (x) := er · x = xr , and, for m ≥ 1, introduce the space H∗m := {f ∈ H m | μr f ∈ H m−1 ,

(1.6.22)

1 ≤ r ≤ N} ,

in which we consider the norm induced by H m . Note that, if Ω is bounded, H m (Ω) ⊆ H∗m (Ω). In the sequel, when the context is clear and there is no risk of confusion, we keep the same notation  ,  to denote the scalar product in L2 (RN ) and L2 (R). Recalling the definition in part 3 of section 1.2, we claim: Theorem 1.6.1. For all m ∈ N, the sequence W := (wj )j∈NN is a total, orthonormal Fourier basis in H m , which is H∗m+1 -regular. Proof. 1) Recalling (1.6.18) and (1.6.21), we compute that (1.6.23)



wj , wk m =



(1 + |ξ|2 )m w ˆj (ξ) w ˆk (ξ) dξ

ˆ j (ξ) h ˆ k (ξ) dξ h 



ˆ j (ξ1 ) h ˆ j (ξN ) h ˆ k (ξ1 ) dξ1 · · · ˆ k (ξN ) dξN . = h h 1 1 1 1 =

If j = k, there is r ∈ {1, . . . , N } such that jr = kr . Then, by (1.6.8) and Parseval’s identity,  ˆ j (ξr ) h ˆ k (ξr ) dξr = hj , hk  = 0 , (1.6.24) h r r r r and (1.6.23) implies that wj , wk m = 0. If instead j = k, then jr = kr for all r ∈ {1, . . . , N }, and (1.6.23) implies that wj , wk m = 1. This shows that W is an orthonormal system in H m ; thus, W is linearly independent. 2) To show that W is a total basis of H m , let f ∈ H m and, for n ∈ N, define, in accord with (1.2.13), (1.6.25)

Pn f :=

n 

f, wj m wj ,

|j|=0

54

1. Functional Framework

where |j| := j1 + · · · + jN . Clearly, each Pn is an orthogonal projection of H m onto the subspace Wn := span{wj | |j| ≤ n}; hence, f, Pn f m = Pn f 2m

(1.6.26)

(which also implies that Pn f m ≤ f m ). Let ζ := (1 + | · |2 )m/2 fˆ ,

(1.6.27)

and note that ζ ∈ L2 , because f ∈ H m . Recalling (1.6.12), we compute that  f, wj m = (1 + |ξ|2 )m fˆ(ξ) w ˆj (ξ) dξ   ˆ j (ξ) dξ = ζ(ξ) (−i)|j| hj (ξ) dξ = ζ(ξ) h (1.6.28) = i|j| ζ, hj  . Since ζ ∈ L2 , it can be approximated by functions ϕ ∈ L2 , of the form ϕ(x) = ϕ1 (x1 ) · · · ϕN (xN ) ,

(1.6.29) for which

N )

ϕ2 =

(1.6.30)

ϕr 2 .

r=1

Let now P˜n : L2 → span{hj | |j| ≤ n} be the orthogonal projection in L2 , defined, as in (1.6.25), by n 

P˜n ζ :=

(1.6.31)

ζ, wj  wj ,

ζ ∈ L2 .

|j|=0

We compute lim P˜n ϕ2 =

n→∞

(1.6.32) =

lim

n→∞

n ) N 

|ϕr , hjr |2

|j|=0 r=1

∞ N  )

|ϕr , hjr |2 =

r=1 jr =0

N )

ϕr 2 = ϕ2 .

r=1

From this, we can deduce that P˜n ζ → ζ in L2 ; therefore, by (1.6.28), f

− Pn f 2m

=

f 2m

−

(1.6.33) = ζ2 −

Pn f 2m n 

=

f 2m

−

n 

|f, wj m |2

|j|=0

|ζ, hj |2 = ζ − P˜n ζ20 → 0 ,

|j|=0

which shows that Pn f → f in claimed.

H m.

Hence, W is a total basis of H m , as

1.6. Orthogonal Bases in H m (RN )

55

3) To show that W is H∗m+1 -regular, assume first that f ∈ H m+1 . Then, for all n ≥ 0, f − Pn f m+1 (1.6.34)

≤

f − Pn f m + ∇f − ∇(Pn f )m

≤

f − Pn f m + ∇f − Pn (∇f )m + Pn (∇f ) − ∇(Pn f )m

=: R1n + R2n + R3n . Since f and ∇f ∈ H m , R1n and R2n vanish as n → +∞. This shows that if the projections Pn commuted with differentiation (e.g., as in (1.5.150) of part 3 of section 1.5.2), W would be H m+1 -regular. In the present situation, we need to show that, for each r ∈ {1, . . . , N }, (1.6.35)

lim Pn (∂r f ) − ∂r (Pn f ) = 0 .

n→∞

To this end, we fix r ∈ {1, . . . , N } and, given a multi-index j = (j1 , . . . , jN ), we denote by j  ∈ NN −1 the multi-index obtained from j by suppressing the index jr ; thus, |j  | = |j| − jr . Calling ek , 1 ≤ k ≤ N , the multiindices corresponding to the vectors of the standard basis of RN , we also set j(r±1) := j ±er ; that is, in the multi-index j(r±1) the index at the rth position is either increased or decreased by 1, with the exception that if jr = 0, the index at the rth position of the multi-index j(r−1) remains 0 (this amounts to the formal agreement that jr − 1 = 0). With this, we proceed to show that

(1.6.36)

Dn,r := Pn (∂r f ) − ∂r (Pn f )  1 ( = √ jr + 1 βj(r+1) wj + βj wj(r+1) , 2 |j|=n

where βj := f, wj m ; note that, in (1.6.36) the summation is only carried over multi-indices j in which jr = n − |j  |. By the definition (1.6.25) of Pn ,

(1.6.37)

Dn,r =

n  |j|=0

(∂r f, wj m wj − f, wj m ∂r wj ) .

56

1. Functional Framework

Recalling (1.6.18), we start by computing  ∂r wj , wk m =



(1 + |ξ|2 )m [F (∂r wj )](ξ) wˆk (ξ) dξ

(1 + |ξ|2 )m (iξr )w ˆj (ξ) w ˆk (ξ) dξ  = − (1 + |ξ|2 )m w ˆj (ξr ) (iξr )w ˆk (ξ) dξ  ˆ j (ξ) (i ξr )h ˆ k (ξ) dξ ; = − h =

(1.6.38)

thus, by (1.6.19),  ˆ j (ξ) [F (∂r hk )](ξ) dξ h 

ˆ j (ξ1 ) h ˆ k (ξ1 ) dξ1 = − h 1 1 

ˆ j (ξr ) [F (hk  )](ξr ) dξr ··· h r r 

ˆ ˆ ··· hjN (ξN ) hkN (ξN ) dξN .

∂r wj , wk m = −

(1.6.39)

In this product, all factors with j = k , 1 ≤  = r ≤ N vanish; when instead j = k for all  ∈ {1, . . . , N } \ {r}, recalling (1.6.16) we can proceed with  ˆ j (ξr ) [F (hk  )](ξr ) dξr ∂r wj , wk m = − h r r  ( ( 1 ˆ k −1 (ξr ) dξ ˆ k +1 (ξr ) − kr h ˆ j (ξr ) (1.6.40) = √ kr + 1 h h r r r 2 ( 1 ( = √ kr + 1 δjr ,kr +1 − kr δjr ,kr −1 . 2 Thus, setting j˜ := (j1 , . . . , jr−1 , kr , jr+1 , . . . , jN ), ∂ r wj

=

∞ 

∂r wj , wk m wk

|k|=0

(1.6.41)

= =

∞ ( 1  ( √ kr + 1 δjr ,kr +1 − kr δjr ,kr −1 wj˜ 2 k =0 r ( 1 ( √ jr wj(r−1) − jr + 1 wj(r+1) . 2

1.6. Orthogonal Bases in H m (RN )

57

From this, integrating by parts in the first term of the sum in (1.6.37), it follows that n  Dn,r = (− f, ∂r wj m wj − βj ∂r wj ) |j|=0

(1.6.42)

=

n ( 1  ( √ jr + 1 βj(r+1) wj − jr βj(r−1) wj 2 |j|=0 ( ( − jr βj wj(r−1) + jr + 1 βj wj(r+1) ,

from which (1.6.43) 

Dn,r

n n−|j | ( 1    ( =√ jr + 1 βj(r+1) wj − jr βj wj(r−1) 2 |j  |=0 jr =0

+

(

jr + 1 βj wj(r+1) −

(

jr βj(r−1) wj

.

Changing the index jr into jr +1 in the second and fourth terms of this sum, we see that all terms cancel, except those in which jr assumes its maximum value n − |j  |. Thus, this sum reduces precisely to the right side of (1.6.36). Consequently, 1   ( (1.6.44) Dn,r 2m = (jr + 1)(kr + 1) Qjk , 2 |j|=n |k|=n

where (1.6.45)

Qjk := βj(r+1) wj + βj wj(r+1) , βk(r+1) wk + βk wk(r+1) m .

We now note that Qjk = 0 if j = k, because (1.6.46)

wj , wk m = 0 ,

wj(r+1) , wk(r+1) m = 0 ,

and wj(r+1) , wk m = 0 only if j  = k  and jr + 1 = kr . But in this case, |j  | = |k  |, so that jr = n − |j  | = n − |k  | = kr , in contradiction with jr = kr − 1. Thus, wj(r+1) , wk m = 0. A similar argument shows that wj , wk(r+1) m = 0 as well. In conclusion, we deduce from (1.6.44) that   (1.6.47) Dn,r 2m = 12 (jr + 1) βj2(r+1) + βj2 . |j|=n

Set now (1.6.48)

Sn,r :=



(jr + 1) βj2 .

|j|=n

58

1. Functional Framework

Then, by (1.6.47), Dn,r 2m ≤

(1.6.49)

1 2

(Sn+1,r + Sn,r ) ,

and we conclude that Pn f → f in H m+1 if Sn,r → 0. We now proceed to show that the latter does hold, if f ∈ H∗m+1 . With ζ defined in (1.6.27), (1.6.50) (

jr + 1 βj =

( jr + 1 f, wj m 

=

ˆ j (ξ1 ) · · · [ ζ(ξ) h 1

(

ˆ j (ξr )] · · · h ˆ j (ξN ) dξ . jr + 1 h r N

From (1.6.7) and (1.6.5), we deduce that, for each  ∈ N, √   (1.6.51)  h−1 (x) = √12 h  (x) + x h (x) ; thus, recalling that the hj ’s are real-valued, we continue from (1.6.50) with (1.6.52) ( 2(jr + 1) βj  ˆ j (ξ1 ) · · · [(hj +1  )ˆ(ξr ) + (μr hj +1 )ˆ(ξr )] · · · h ˆ j (ξN ) dξ = ζ(ξ) h r r 1 N  * + ˇ hj (x1 ) · · · hj +1  (xr ) + xr hj +1 (xr ) · · · hj (xN ) dx = ζ(x) r r 1 N  ˇ hj (x1 ) · · · hj +1 (xr ) · · · hj (xN ) dx = − ∂r ζ(x) r 1 N  ˇ hj (x1 ) · · · hj +1 (xr ) · · · hj (xN ) dx ; + xr ζ(x) r 1 N that is, (1.6.53)

( jr + 1 βj =

√1 2

ˇ hj μr ζˇ − ∂r ζ, . (r+1)

From this, since |j(r+1) | = |j| + 1,  ˇ hj Sn := 12 |μr ζˇ − ∂r ζ, |2 (r+1) (1.6.54)

≤

|j|=n 

ˇ hk |2 . |μr ζˇ − ∂r ζ,

|k|=n+1

Thus, it is sufficient to show that μr ζˇ − ∂r ζˇ ∈ L2 , because in this case the right side of (1.6.54), being the (n + 1)-th Fourier coefficient of μr ζˇ − ∂r ζˇ in L2 , will vanish as n → ∞. Thus, since μr ζˇ = i (∂r ζ)ˇ, we wish to show that both ∂r ζˇ and (∂r ζ)ˇ ∈ L2 . Of these, the first is an immediate consequence of the fact that f ∈ H m+1 , which implies that ζˇ ∈ H 1 . For the other,

1.6. Orthogonal Bases in H m (RN )

59

by Parseval’s formula it is sufficient to show that ∂r ζ ∈ L2 . Recalling the definition (1.6.27) of ζ, (1.6.55)

∂r ζ(ξ) = m (1 + |ξ|2 )(m/2)−1 ξr fˆ(ξ) + (1 + |ξ|2 )m/2 ∂r fˆ(ξ) .

Of these terms, the first is in L2 , because, obviously, f ∈ H m−1 ; as for the second, we note that, since ∂r fˆ = − i (μr f )ˆ, (1.6.56)

(1 + | · |2 )m/2 ∂r fˆ 0 = (∂r fˆ)ˇm = μr f 2m ,

which is finite because f ∈ H∗m+1 . Thus, ∂r ζ ∈ L2 as desired, and we conclude that μr ζˇ − ∂r ζˇ ∈ L2 . Hence, Sn → 0, and this ends the proof of Theorem 1.6.1.  3. Although we will not need this in the sequel, we report a property of the space H∗m , which is not easily found in the literature. Proposition 1.6.1. For all m ∈ N≥1 , the space H∗m is compactly imbedded into H m−1 . Proof. 1) We first show that H∗m →c L2 . Let (fn )n≥1 be a bounded sequence in H∗m . Then, (fn )n≥1 is bounded in H m ; thus, it contains a subsequence, still denoted (fn )n≥1 , which converges weakly to a function f ∈ H m . We show that fn → f strongly in L2 . Indeed, the sequence of functions (x → |x|(fn (x) − f (x)))n≥1 is also bounded in L2 ; thus, given arbitrary R > 0,   2 2 1 fn − f  ≤ |fn − f | dx + R2 |x|2 |fn − f |2 dx |x|≤R |x|≥R  (1.6.57) ≤ |fn − f |2 dx + RC2 , |x|≤R

with C > 0 independent of n. By the compactness of the imbedding of H m (B(0, R)) into L2 (B(0, R)) (as per the last claim of Theorem 1.5.3), and resorting to a diagonalization argument, we can choose a subsequence (fnk )k≥1 , such that, for all R > 0, the restrictions of fnk to B(0, R) converge to the restriction of f to B(0, R) strongly in L2 (B(0, R)). Thus, given ε > 0, taking R such that RC2 ≤ ε we conclude from (1.6.57) that fnk → f in L2 as k → ∞. 2) The general case follows by interpolation. In fact, let (fn )n≥1 and f be as above. Then, by the interpolation inequality (1.5.47), 1−1/m

(1.6.58)

fnk − f m−1 ≤ C fnk − f m

1/m

≤ C1 fnk − f 0

1/m

fnk − f 0

,

from which it follows that fnk → f strongly in H m−1 .



60

1. Functional Framework

1.7. Sobolev Spaces Involving Time We briefly review the definition and the main properties of the spaces H m (a, b; X) and C m ([a, b]; X), where a ∈ [−∞, +∞[, b ∈ ] − ∞, +∞], a < b, and X is a Banach space. For the main properties of these spaces, we refer, e.g., to Edwards [45, ch. 8], and Diestel and Uhl [43, ch. 4]. 1. Bochner Spaces Lp (a, b; X). Let ]a, b[ ⊆ R be an interval, and X a Banach space. For 1 ≤ p ≤ +∞, Lp (a, b; X) is the space of the (equivalence classes of) functions u : ]a, b[→ X which are strongly measurable, and such that the quantities  b (1.7.1) u(t)pX dt a

if 1 ≤ p < ∞, and sup ess u(t)X

(1.7.2)

a 0 and a sequence (tn )n≥0 ⊂ ]a, +∞[ such that tn → +∞ and f (tn ) ≥ 2ε0 for all n. Then, for each n there is θn > tn such that f (θn ) ≤ ε0 ; indeed, if there were n0 such that f (θ) > ε0 for all θ ≥ tn0 , f would not be integrable on ]a, +∞[. Taking subsequences if necessary, we can assume that tn < θn < tn+1 < θn+1 for all n. Consequently, there is a sequence of intervals [an , bn ] ⊂ ]a, +∞[, with θn ≤ an < bn ≤ tn+1 , and such that, for all t ∈ [an , bn ], (1.7.24)

ε0 = f (an ) ≤ f (t) ≤ 2 ε0 = f (bn ) .

64

1. Functional Framework

Recalling that f ∈ L1 (a, +∞; R≥0 ), and that f  is a.e. bounded above, we can set  ∞ (1.7.25) Λ0 := f (t) dt , Λ1 := sup ess f  (t) . t≥0

0

Recalling that f (t) ≥ 0 for a.a. t, from the estimates ∞  bn ∞   (1.7.26) Λ0 ≥ f (t) dt ≥ ε0 (bn − an ) , n=0 an

we see that the series

∞ !

n=0

(bn −an ) converges; hence, (bn −an ) → 0. However,

n=0

since f ∈ AC([an , bn ]), this contradicts the inequality  bn (1.7.27) ε0 = f (bn ) − f (an ) = f  (t) dt ≤ Λ1 (bn − an ) , an

which shows that the sequence (bn − an )n≥0 cannot vanish.



2.4. The classical derivative of a function f ∈ C([a, b]; X) is defined in the usual way; that is, (1.7.28)

f  (t) := lim

θ→t

f (θ) − f (t) , θ−t

whenever this limit exists in X. We denote by ∂tk f the classical derivative of order k of f , and set (1.7.29) C m ([a, b]; X) := {u ∈ C([a, b]; X) | ∂tj u ∈ C([a, b]; X) , 0 ≤ j ≤ m} . This is a Banach space, with respect to the norm (1.7.30)

uC m ([a,b];X) := max

a≤t≤b

m 

∂tj u(t)X .

j=0

An analogous definition holds for the space Cbm ([a, +∞[; X) of the space of continuously differentiable functions, with bounded derivatives, of order up to m. More generally, if X and Y are Banach spaces, with X → Y , we set (1.7.31)

C m ([a, b]; X, Y ) := {u ∈ C([a, b]; X) | ∂tm u ∈ C([a, b]; Y )} .

Again, this is a Banach space, with respect to the norm (1.7.32)

uC m ([a,b];X,Y ) := max (u(t)X + ∂tm u(t)Y ) , a≤t≤b

and an analogous definition holds for the space Cbm ([a, +∞[; X, Y ). 3. Regularization. Many approximation results for functions in Lp (0, T ; H m ) can be obtained by regularization, e.g., via Friedrichs’ mollifiers; note that a function can be mollified either by convolution with respect to t only, or x only, or both.

1.7. Sobolev Spaces Involving Time

65

Theorem 1.7.1. Let p ∈ [1, +∞[, m ≥ 0, and u ∈ Lp (0, T ; H m ). For δ > 0, define uδ (t, ·) := ρδ ∗ u(t, ·). Then, uδ ∈ Lp (0, T ; H m ), and uδ → u ˜ from R>0 × Lp (0, T ; H m ) into in Lp (0, T ; H m ), as δ → 0. The map M p m L (0, T ; H ), defined by (1.7.33)

˜ u)](t) := uδ (t) , [M(δ,

t ∈ [0, T ] ,

is continuous. Analogous claims hold if u ∈ C([0, T ]; H m ); in particular, uδ → u in C([0, T ]; H m ). In addition, the map (δ, t) → uδ (t) is continuous from R>0 × [0, T ] into H m . ˜ defined in (1.7.33) is formally the same as the Note that the map M map M defined in (1.4.9), in the sense that both map the pair (δ, u) into the function uδ . However, this identity is only up to the identification (1.7.34)

˜ u)](t) = M(δ, u(t)) , [M(δ,

for a.a. t ∈ [0, T ], which holds because the regularization involves the space variables only. Proof. The first claim is a consequence of (1.5.34), which implies that for almost all t ∈ ]0, T [, (1.7.35)

uδ (t)m ≤ u(t)m .

Moreover, as δ → 0, uδ (t) → u(t) in H m , again for a.a. t; hence, (1.7.35) yields the second claim, via Lebesgue’s dominated convergence theorem. ˜ can be proven with an argument similar to that of The continuity of M the proof of the continuity of M in Theorem 1.4.1, again via Lebesgue’s dominated convergence theorem. To prove that uδ → u in C([0, T ]; H m ) if u ∈ C([0, T ]; H m ), we argue by contradiction. If the claim were false, there would exist ε0 > 0, an infinitesimal sequence (δn )n≥1 , and a sequence (tn )n≥1 ⊂ [0, T ], such that (1.7.36)

uδn (tn ) − u(tn )m ≥ ε0 .

Recalling (1.5.35), we deduce that, for all n ≥ 1, (1.7.37)

ε0 ≤ sup u(tn , · − δn z) − u(tn )m . |z|≤1

Thus, for all n ≥ 1 there is zn ∈ B(0, 1) such that (1.7.38)

u(tn , · − δn zn ) − u(tn )m ≥

1 2

ε0 .

1+N , it contains a subsequence Since (tn , zn ) varies in a compact subset of R (tnk , znk ) k∈N , converging to some (t∗ , z∗ ) ∈ [0, T ] × B(0, 1). Then, from

66

1. Functional Framework

(1.7.38) it follows that (1.7.39) 1 2

ε0 ≤ u(tnk , · − δnk znk ) − u(tnk )m ≤ u(tnk , · − δnk znk ) − u(t∗ , · − δnk znk )m + u(t∗ , · − δnk znk ) − u(t∗ )m + u(t∗ ) − u(tnk )m =: Ak + Bk + Ck .

The change of variables x → x − δnk znk shows that Ck = Ak ; since tnk → t∗ , and u is continuous from [0, T ] into H m , it follows that Ck and Ak → 0 as k → ∞. Since t∗ is fixed in the term Bk , and |δnk znk | = δnk |znk | ≤ δnk → 0, the continuity of the translations in H m implies that Bk → 0 as well. For k sufficiently large, this implies a contradiction with (1.7.39); hence, uδ → u in C([0, T ]; H m ), as claimed. Finally, assume that u ∈ C([0, T ]; H m ), and fix (δ0 , t0 ). Then, (1.7.40)

uδ (t) − uδ0 (t0 )m ≤ uδ (t) − uδ (t0 )m + uδ (t0 ) − uδ0 (t0 )m ≤ u(t) − u(t0 )m + ω0 (δ − δ0 ) ,

where ω0 is analogous to the one introduced in the proof of Theorem 1.4.1. Since ω0 (δ − δ0 ) → 0 as δ → δ0 , we can conclude the proof of Theorem 1.7.1.  Corollary 1.7.1. For T > 0 and m ∈ N, let f , g ∈ C([0, T ]; H m ). For η > 0, there is ωg ∈ K0 such that (1.7.41)

f − (ρη ∗ g)C([0,T ];H m ) ≤ f − gC([0,T ];H m ) + ωg (η) .

Proof. This is a consequence of the estimate (1.7.42)

f − (ρη ∗ g)m ≤ f − gm + g − (ρη ∗ g)m ,

and the fact that, by Theorem 1.7.1, ρη ∗ g → g in C([0, T ]; H m ) as η → 0.  Theorem 1.7.2. Let Q := ]0, T [ × RN , and u ∈ Cb (Q). Define uδ as in Theorem 1.7.1. Then, for all compact set K ⊂ RN , uδ → u uniformly on [0, T ] × K. Proof. Proceeding by contradiction, assume there are a compact set K0 ⊂ RN , ε0 > 0, and an infinitesimal sequence (δn )n≥1 , such that (1.7.43)

max |uδn (t, x) − u(t, x)| ≥ ε0 ,

(t,x)∈Q0

1.7. Sobolev Spaces Involving Time

67

where Q0 := [0, T ] × K0 . Then, for all n ≥ 1 there is (tn , xn ) ∈ Q0 such that |uδn (tn , xn ) − u(tn , xn )| ≥

(1.7.44)

1 2

ε0 .

Since Q0 is compact, there is a subsequence, still denoted ((tn , xn ))n≥1 , converging to some (t∗ , x∗ ) ∈ Q0 . From    1 (1.7.45) uδ (t, x) − u(t, x) = N ρ x−y (u(t, y) − u(t, x)) dy , δ δ recalling (1.4.5) we obtain that (1.7.46)

|uδ (t, x) − u(t, x)| ≤ sup |u(t, y) − u(t, x)| . |y−x|≤δ

Hence, from (1.7.44), (1.7.47)

1 2

ε0 ≤

sup |x−xn |≤δn

|u(tn , x) − u(tn , xn )| ,

and this leads to a contradiction. Indeed, as n → ∞, δn → 0, so that xn → x; therefore, (tn , x) → (t∗ , x∗ ), and, by the uniform continuity of u on Q0 , the right side of (1.7.47) vanishes.  Theorem 1.7.3. Let X and Y be Banach spaces, with X → Y . For all m ∈ N≥1 , the space D([a, b]; X) is dense in C m ([a, b]; X, Y ). Sketch of Proof. We only consider the case m = 1. Acting as in Lions and Magenes [101, ch. 1, sct. 2.2], it is sufficient to prove the density of D(R; X) in Cbm (R; X, Y ). Given u ∈ Cbm (R; X, Y ), for δ > 0 let uδ denote its Friedrichs’ regularization in the time variable, defined as in (1.7.12). Then, as in Theorem 1.7.1, uδ (t) → u(t) in X, uniformly in t. Next, we compute that

 1 d t−θ δ ut (t) = ρ u(θ) dθ δ dt δ

 1 d t−θ = − (1.7.48) ρ u(θ) dθ δ dθ δ

 1 t−θ = ρ ut (θ) dθ , δ δ from which we conclude that uδt (t) → ut (t) in Y , uniformly in t. W m (a, b; X, Y ).

Lp



4. Spaces As was the case for spaces, functions f ∈ Lp (a, b; X) can be differentiated in distributional sense, with (1.7.49)

f  ∈ D  (]a, b[; X) := L(D(]a, b[); X)

defined by (1.7.50)

f  (ϕ) := −



b a

ϕ  (t)f (t) dt ,

ϕ ∈ D(]a, b[)

68

1. Functional Framework

(the integral in (1.7.50) being an element of X). For k ∈ N and u ∈ Lp (a, b; X), we denote by u(k) its distributional derivative of order k. More generally, if Y is a Banach space, with X → Y , and f ∈ Lp (a, b; X), then f ∈ Lp (a, b; Y ) as well, and can be differentiated in distributional sense, with f  ∈ D  (]a, b[; Y ); note that the condition X → Y implies that D  (]a, b[; X) → D  (]a, b[; Y ). For m ∈ N and p ∈ [1, +∞], we define (1.7.51)

W m,p (a, b; X, Y ) := {u ∈ Lp (a, b; X) | u(m) ∈ Lp (a, b; Y )} ;

this is a Banach space, with respect to the norm  b  1/p p p (m) (1.7.52) f W m,p (a,b;X,Y ) := f (t)X + f (t)Y dt . a

We abbreviate (1.7.53)

W m,p (a, b; X, X) =: W m,p (a, b; X) ,

(1.7.54)

W m,2 (a, b; X, Y ) =: H m (a, b; X, Y ) ;

in particular, (1.7.55)

W m,2 (a, b; X) =: H m (a, b; X) .

When X is a Hilbert space, the latter is also a Hilbert space, with scalar product  b (1.7.56) u, vH m (a,b;X) := u(t), v(t) + u(m) (t), v (m) (t)X dt . a

When X and Y are Hilbert spaces, H m (a, b; X, Y ) is a generalization of the space called W (a, b) in Lions and Magenes [101, ch. 1, sct. 2.2]). 5. Intermediate Derivatives and Traces. Since functions in the space L2 (a, b; X) can be arbitrarily modified on subsets of [a, b] of measure zero, one cannot evidently define their value, or trace, on any point of [a, b]. However, just as in the case of traces of functions in H 1 (Ω) (see part 3 of section 1.5.1), a generalization of the trace can be given for functions f ∈ W 1,p (a, b; X, Y ). For example, functions in H 1 (a, b; X, Y ) can be modified on a set of measure zero, so that the modified functions are in C([a, b]; [X, Y ]1/2 ), where [X, Y ]1/2 denotes an intermediate space between X and Y , in the sense of interpolation theory (see, e.g., Bergh and L¨ofstr¨om [14]). From now on, we restrict ourselves to the case when X = H s (Ω) and Y = H m (Ω), with s, m ∈ R≥0 , s ≥ m, and, as usual, either Ω = RN , or Ω a bounded set of RN with smooth boundary. In either case, we abbreviate H m (Ω) =: H m . In particular, we consider cylinders Q := ]0, T [ ×Ω, with the induced product Lebesgue measure, and we allow the case T = +∞.

1.7. Sobolev Spaces Involving Time

69

For , m ∈ R>0 , with m ≥  ≥ 1, and integer k, 0 ≤ k ≤  m  , we set W m,,k (a, b) := W k,2 (a, b; H m , H m− k ) .

(1.7.57)

Theorem 1.7.4. Let m,  and k as above. The space D([a, b]; H m ) is dense in W m,,k (a, b). If u ∈ W m,,k (a, b) and 0 ≤ j ≤ k, u(j) ∈ L2 (a, b; H m− j ) .

(1.7.58)

Furthermore, u can be modified on a set of measure 0 in [a, b], so that, if 0 ≤ j ≤ k − 1, u(j) ∈ C([a, b]; H m−(j+1/2)) .

(1.7.59)

The linear maps u → u(j) of W m,,k (a, b) into L2 (a, b; H m− j ) if 0 ≤ j ≤ k, and into C([a, b]; H m−(j+1/2)) if 0 ≤ j ≤ k − 1, are continuous. In particular, there are positive constants C0 , C1 , and C2 , such that for all u ∈ W m,,k (a, b) and 0 ≤ j ≤ k, 1−j/k

j/k

u(j) L2 (a,b;H m−j ) ≤ C0 uL2 (a,b;H m ) u(k) L2 (a,b;H m−k ) ,

(1.7.60)

and for 0 ≤ j ≤ k − 1, (1.7.61) max u(j) (t)m−(j+1/2)

a≤t≤b

1−(j+1/2)/k

(j+1/2)/k

≤ C1 uL2 (a,b;H m ) u(k) L2 (a,b;H m− k ) + C2 uL2 (a,b;H m ) . The constants C0 and C1 are independent of a and b, while C2 → 0 if |b − a| → +∞. In particular, the choice C2 = 0 is admissible if b = +∞. For a proof, we refer to Lions and Magenes [101, ch. 1], where the claims of Theorem 1.7.4 can be deduced as particular cases of the results established for general Banach spaces X → Y .  6. Imbeddings. Part (1.7.59) of Theorem 1.7.4 is known as the trace theorem. In particular, the choice  = k = 1 and j = 0 in (1.7.59) yields the imbedding (1.7.62)

W 1,2 (a, b; H m+1 , H m ) → C([a, b]; H m ) .

6.1. We first show that the content of the imbedding (1.7.62) can be made more precise.

70

1. Functional Framework

Proposition 1.7.5. Let m ∈ N. The map t → u(t)2m is absolutely continuous on [a, b], and for a.a. t ∈ [a, b], (1.7.63)

d u(t)2m = 2u(t), u  (t)m . dt

Sketch of Proof. By the trace theorem (1.7.59) (replacing m by m + 1 and taking  = k = 1 and j = 0), u ∈ C([a, b]; H m+1/2) → C([a, b]; H m ); in fact, u is H¨older continuous from [a, b] into H m , as follows from   t1     u(t1 ) − u(t2 )m ≤  u (θ)m dθ t2 (1.7.64)  b

1/2 ≤ u  (θ)2m dθ |t1 − t2 |1/2 , a

for any t1 , t2 ∈ [a, b]. Thus, u ∈ AC([a, b]; H m ), and u(·)m is differentiable a.e. in [a, b]. Finally, (1.7.63) is first proven for u ∈ C 1 ([a, b]; H m ), and then extended by a density argument, noting that the right side of (1.7.63) is in L1 (a, b).  Remark. The result of Proposition 1.7.5 is similar to a more general one, concerning functions u ∈ L2 (a, b; V ) with u  ∈ L2 (a, b; V  ), where V → H → V  is a so-called Gel’fand triple of Hilbert spaces; see e.g., Zeidler [166, ch. 23].  6.2. We next show that the assumptions of Proposition 1.7.5 can be somewhat relaxed. Proposition 1.7.6. Let m ∈ N≥1 . Then, (1.7.65) (1.7.66)

W 1,2 (a, b; H m , L2 ) → C([a, b]; H m/2 ) → AC([a, b]; L2 ) , W 1,1 (a, b; H m , H m−1 ) → AC([a, b]; H m−1 ) .

If u ∈ W 1,2 (a, b; H m , L2 ), the identity d u(t)2 = 2u(t), ut (t) dt holds for a.a. t ∈ [a, b]. More generally, (1.7.67)

(1.7.68)

W 1,1 (a, b; X) → AC([a, b]; X) ,

even if the Banach space X is not reflexive. Sketch of Proof. 1) The first inclusion of (1.7.65) follows again from (1.7.59), with  = m, k = 1, and j = 0. The proof of the second inclusion and of (1.7.67) is similar to that of the analogous parts of Proposition 1.7.5. 2) Let u ∈ W 1,1 (a, b; H m , H m−1 ). The continuity of u is proven by showing that u coincides, up to a constant, with the absolutely continuous

1.7. Sobolev Spaces Involving Time

71

't function U (t) := a u  (θ) dθ in D  (a, b; H m−1 ). Note also that, for t, t0 ∈ [a, b], as in (1.7.64),  t      (1.7.69) u(t) − u(t0 )m−1 ≤  u (θ)m−1 dθ , t0

and the right side of (1.7.69) vanishes as t → t0 , because u  ∈ L1 (a, b; H m−1 ). (1.7.69) also shows that u ∈ AC([a, b]; H m−1 ); that is, (1.7.66) holds. 3) The last claim follows from the results quoted in problems 23.5 and 23.8 of Zeidler [166, ch. 23] (with u replaced by u  in problem 23.5). At 't first, the function v(t) := a u  (θ) dθ is in AC([a, b]; X), and v  = u  in D  (]0, T [; X) (and also a.e. on ]0, T [). Thus, since the difference w := u − v is in L1 (a, b; X), and w  = 0 in D  (]0, T [; X), w(t) is constant in X; that is, there is c ∈ X such that w(t) = c for a.a. t ∈ ]0, T [. Then, u(·) = v(·) + c ∈ AC([a, b]; X) since, as in (1.7.69),  t      (1.7.70) u(t) − u(t0 )X = v(t) − v(t0 )X ≤  u (θ)X dθ → 0 t0

as t → t0 .



6.3. Conversely, if m is sufficiently large, the conclusions of Proposition 1.7.5 can be significantly strengthened. Proposition 1.7.7. Let m ∈ N, with m >

N 2,

and Q = ]a, b[ ×Ω. Then,

(1.7.71)

W 1,2 (a, b; H m+1 , H m−1 ) → Cb (Q) ,

(1.7.72)

C([a, b]; H m+1 ) ∩ C 1 ([a, b]; H m ) → Cb1 (Q) .

Proof. We only consider the most significant case N2 < m < N2 + 1. Let f ∈ W 1,2 (a, b; H m+1 , H m−1 ). By the trace theorem (1.7.59) and the Sobolev imbedding (1.5.61), (1.7.73)

f ∈ C([a, b]; H m ) → C([a, b]; C 0,α (Ω)) ,

0 < α <  N2  + 1 − N2 . Let (t, x) and (s, y) ∈ Q, with t = s and x = y. Then, (1.7.74)

|f (t, x) − f (s, y)| ≤ |f (t, x) − f (t, y)| + |f (t, y) − f (s, y)| .

Using (1.5.61) for f (t, ·), we first estimate |f (t, x) − f (t, y)| (1.7.75)

≤

C f (t)m |x − y|α

≤

C max f (t)m |x − y|α a≤t≤b

=: C1 (f ) |x − y|α . Next, we note that H m−1 → Lq , with Nirenberg inequality (1.5.42), with θ =

1 q N 2

= 12 − m−1 N . By the Gagliardo− (m − 1) ∈ ]0, 1[ (and different

72

1. Functional Framework

C), |f (t, y) − f (s, y)| ≤ |f (t) − f (s)|∞ ≤ C ∂xm (f (t) − f (s))θ |f (t) − f (s)|q1−θ

(1.7.76)

+ C1 |f (t) − f (s)|q , with C1 = 0 if Ω = RN . In the right side of (1.7.76), we first estimate (1.7.77)

∂xm (f (t) − f (s)) ≤ f (t) − f (s)m ≤ 2 C1 (f ) ,

and, in the last term, |f (t) − f (s)|q ≤ C f (t) − f (s)m−1 (1.7.78)

θ+(1−θ)

= C f (t) − f (s)m−1

1−θ ≤ C f (t) − f (s)θm f (t) − f (s)m−1 .

Consequently, from (1.7.76), (1.7.79)

1−θ |f (t, y) − f (s, y)| ≤ C (2 C1 (f ))θ f (t) − f (s)m−1 .

Since f  ∈ L2 (a, b; H m−1 ), f (t) − f (s)m−1

(1.7.80)

≤ ≤

 t      f (θ)m−1 dθ   s  1/2  t   1/2   2  |t − s|  f (θ)m−1 dθ 

≤

|t − s|

s

b

1/2

f



1/2 (θ)2m−1 dθ

a

=: |t − s|1/2 C2 (f ) . Thus, from (1.7.75), (1.7.79), and (1.7.80),  (1.7.81) |f (t, x) − f (s, y)| ≤ C(f ) |t − s|(1−θ)/2 + |x − y|α , with C(f ) := C1 (f ) + C (2C1 (f ))θ (C2 (f ))1−θ . An analogous estimate holds if either t = s or x = y. In conclusion, f is H¨older continuous in both the time and space variables; hence, f ∈ Cb (Q), as claimed in (1.7.71). The proof of (1.7.72) is similar.  Remark. The proof of Proposition 1.7.7 shows that, if m > N2 , the assumptions f ∈ C([a, b]; H m ) and f  ∈ L1 (a, b; H m−1 ), are sufficient to deduce that f ∈ Cb (Q). Indeed, (1.7.73) still holds, and in the first line of (1.7.80), that is,  t      (1.7.82) f (t) − f (s)m−1 ≤  f (θ)m−1 dθ , s

1.7. Sobolev Spaces Involving Time

73

the right side vanishes as |t − s| → 0, if f  ∈ L1 (a, b; H m−1 ).



6.4. The following result improves and generalizes that of Proposition 1.7.7. , Theorem 1.7.5. Let m ≥ N2 + 1 + 2r, r ≥ 0. Then, W r+1,2 (0, T ; H m+1 , H m−1−2r ) → C r+α/2,2r+α(Q) , for all α < α0 := N2 + 1 − N2 .

(1.7.83)

Proof. 1) Let first r = 0, and let f ∈ W 1,2 (0, T ; H m+1 , H m−1 ). Recalling ˜ α (f ) < ∞, since Proposition 1.7.7 (1.3.5), it is sufficient to show that H implies that f ∈ Cb (Q). Assume first that t = s and x = y. Then, (1.7.84)

|f (t, x) − f (s, y)| |f (t, x) − f (t, y)| |f (t, y) − f (s, y)| ≤ + . α/2 α |x − y|α |t − s| + |x − y| |t − s|α/2

From (1.7.75) we deduce the estimate (1.7.85)

|f (t, x) − f (t, y)| ≤ C1 (f ) . |x − y|α

Likewise, from (1.7.79) and (1.7.80), |f (t, y) − f (s, y)| |t − s|α/2 (1.7.86)

≤

C (2C1 (f ))θ 1−θ f (t) − f (s)m−1 |t − s|α/2

≤

C (2C1 (f ))θ (C2 (f ))1−θ |t − s|(1−θ)/2 |t − s|α/2

≤ C3 (f ) |t − s|(1−θ−α)/2 ≤ C3 (f ) T (1−θ−α)/2 =: C4 (f, T ) , where θ = N2 − (m − 1) and C3 (f ) = C (2C1 (f ))θ (C2 (f ))1−θ ; note that 1 − θ − α ≥ 1 − θ − α0 = 0. From this and (1.7.85) we conclude that (1.7.87)

Hα (f ) ≤ C1 (f ) + C4 (f, T ) ,

as desired. The proof of (1.7.87) when t = s or x = y is analogous. 2) Let now r ≥ 1, f ∈ W r+1,2 (0, T ; H m+1 , H m−1−2r ), and set gr := 2k + |λ| = 2r. By (1.7.58) of Theorem 1.7.4,

∂tk ∂xλ f ,

(1.7.88)

gr ∈ L2 (0, T ; H m+1−2r ) ,

(gr )  ∈ L2 (0, T ; H m−1−2r ) ;

hence, by the trace theorem, gr ∈ C([0, T ]; H m−2r ). Since m − 2r =  N2  + 1 > N2 , by Proposition 1.7.7 it follows that gr ∈ Cb (Q). In turn, this implies that f ∈ C˜br (Q). Moreover, again because m − 2r =  N2  + 1, we can repeat the same procedure of part 1 above, with f and m replaced by gr and m−2r, ˜ α (∂tk ∂xλ f ) < ∞. We and deduce that gr = ∂tk ∂xλ f ∈ C α/2,α (Q); that is, H can then conclude that f ∈ C r+α/2,2r+α (Q), as claimed. 

74

1. Functional Framework

6.5. The condition on f  in Theorem 1.7.5 may be somewhat relaxed, at the cost of decreasing the value of α. , Proposition 1.7.8. Let m ≥ N2 + 1. Then, (1.7.89)



f ∈ C([0, T ]; H m ) | f  ∈ L2 (Q)

for all α ≤ α1 := 1 −



→ C α/2,α (Q) ,

N 2m .

Proof. The proof is very similar to that of Proposition 1.7.7. Indeed, (1.7.75) still holds, and in (1.7.76) we just need to replace the Lq norm of f (t) − f (s) N with its L2 norm. Then, in (1.7.76), the value of θ becomes θ = 2m , and in (1.7.86) the condition 1 − θ − α ≥ 0 translates into α ≤ α1 , as assumed. Note that α1 < α0 .  6.6. Finally, we present a compact imbedding result, analogous to some of the results of Theorem 1.5.3. Theorem 1.7.6. Let X, Y and Z be reflexive Banach spaces, with X →c Y → Z. Let p, q ∈ ]1, +∞[. Then the injection (1.7.90)

{u ∈ Lp (a, b; X) | u  ∈ Lq (a, b; Z)} → Lp (a, b; Y ) ,

which is continuous, is also compact. For a proof, see, e.g., Lions [99, sct. 1.5]. The following consequence of Theorem 1.7.6 is fundamental. Proposition 1.7.9. Let m ∈ N, T > 0, and let Ω ⊂ RN be bounded. Let (un )n≥1 be a bounded sequence in W 1,2 (0, T ; H m+1 , H m ). There is a subsequence (unk )k≥1 ⊆ (un )n≥1 , and a function u ∈ W 1,2 (0, T ; H m+1 , H m ), such that (1.7.91)

unk → u nk 



in

L2 (0, T ; H m+1 )

weakly ,

(1.7.92)

(u ) → u

(1.7.93)

nk

u

→u

in

L (0, T ; H )

(1.7.94)

unk → u

in

C([0, T ]; H m−1/2)

in

2

m

weakly ,

2

m

strongly ,

L (0, T ; H )

strongly .

Proof. (1.7.91) follows from the second part of Theorem 1.2.1, which also implies that a subsequence of the subsequence ((unk )  )k≥1 converges weakly to some element v ∈ L2 (0, T ; H m ). A standard argument shows that v = u  in D  (0, T ; L2 ); thus, (1.7.92) holds. By the last statement of Theorem 1.5.3, H m+1 →c H m , because Ω is bounded; hence, (1.7.93) follows from Theorem 1.7.6. To prove (1.7.94), we recall the trace inequality (1.7.61),

1.7. Sobolev Spaces Involving Time

75

with  = k = 1 and j = 0: setting wk := unk − u, we estimate  T k 2 max w (t)m−1/2 ≤ C1 wk 2m dt 0≤t≤T

(1.7.95)

 +C 0

0

T

1/2 

T

wk 2m dt

(wk )  2m−1 dt

1/2 .

0

Consequently, (1.7.94) follows, since (1.7.93) implies that the first two integrals at the right side of (1.7.95) vanish as k → +∞, and the last is bounded, by (1.7.92).  6.7. We conclude with an approximation result for functions in the space W 1,2 (0, T ; H m , L2 ). When m ≥ 1, (1.7.65) allows us to consider the subspace (1.7.96)

WTm (Q) := {u ∈ W 1,2 (0, T ; H m , L2 ) | u(T, ·) = 0} ,

because u(T, ·) ∈ L2 . Theorem 1.7.7. The set of the restrictions to [0, T [ × RN of the product functions (t, x) → ϕ(t) ψ(x), with ϕ ∈ D(] − T, T [) and ψ ∈ D(RN ), is total in WTm (Q). Sketch of Proof. Let f ∈ WTm (Q). Adapting the result of the extension theorem 2.2 of Lions and Magenes [101, ch. 1, sct. 2], we can extend f to ˜ m (Q), where a function f˜ ∈ W T (1.7.97) ˜ m (Q) := {u ∈ W 1,2 (− T, T ; H m , L2 ) | u(− T, ·) = u(T, ·) = 0} , W T the map f → f˜ being continuous. As mentioned in Lions [98, ch. V, sct. ˜ m (Q) (in contrast to the first 4], the space D(] − T, T [; H m ) is dense in W T claim of Theorem 1.7.4, we can take the open interval ] − T, T [, because of the conditions u(± T ) = 0). In turn, functions ψ ∈ D(] − T, T [; H m ) can ˜ m (Q), by functions in D(] − T, T [×RN ). be approximated, in the norm of W T One way of doing this is by Friedrichs’ mollification in space and truncation; that is, by functions of the type  (1.7.98) ψ δ (t, x) := ζ δ (x) ρδ (x − y) ψ(t, y) dy , δ >0, where ρδ is as in (1.4.6) of section 1.4, and ζ δ ∈ C0∞ (RN ), with 0 ≤ ζ δ (x) ≤ 1 for all x ∈ RN , ζ δ (x) ≡ 1 for |x| ≤ 1δ , and ζ δ (x) ≡ 0 for |x| ≥ 2δ . Finally, we use Schwartz’s result of [141, ch. 4, §3] (see also Friedlander and Joshi [50, Thm. 4.3.1]), by which the tensor product set D(] − T, T [) ⊗ D(RN ) is dense in D(] − T, T [ × RN ) with respect to the Schwartz topology, and, hence, ˜ m (Q). The continuity of the restriction operator from for the topology of W T

76

1. Functional Framework

˜ m (Q) to W m (Q) allows us then to conclude the proof of Theorem 1.7.7. W T T  Remark on Notational Convention. From now on, in the light of Theorems 1.7.4 and 1.7.5, with abuse of notation we identify classical and distributional derivatives with respect to t; that is, we write ∂tk u to also denote u(k) . When k = 1 or k = 2, we also write ut and utt , instead of ∂t1 u and ∂t2 u, to denote u  and u  . 

Chapter 2

Linear Equations

2.1. Introduction 1. In this chapter we consider the Cauchy problem for the linear hyperbolic evolution equation (2.1.1)

utt − aij (t, x) ∂i ∂j u = f (t, x) + bi (t, x) ∂i u + c(t, x) u ,

where summation for i, j from 1 to N is understood. This means that we take (t, x) in the cylinder Q = ]0, T [ ×RN , where T > 0 is fixed but arbitrary, and seek solutions of (2.1.1) which satisfy the initial conditions (or Cauchy data) (2.1.2)

u(0, x) = u0 (x) ,

ut (0, x) = u1 (x) ,

where u0 and u1 are given functions on RN . Our goal is to show that the Cauchy problem (2.1.1)+(2.1.2) is well-posed in a suitable class of Sobolev spaces; we call the corresponding solutions strong. In section 2.4, we will also briefly consider linear equations in the divergence form (2.1.3)

utt − ∂j (aij ∂i u) = f + bi ∂i u + c u ,

and show that the Cauchy problem (2.1.3)+(2.1.2) is well-posed in a suitable class of weak solutions. Note that (2.1.1) can formally be rewritten in the divergence form (2.1.4)

utt − ∂j (aij ∂i u) = f + (bi − ∂j aij )∂i u + c u .

As we mentioned in the Preface, the results we establish for (2.1.1) are not specifically dependent on the fact that the equation is hyperbolic; in fact, our solution theory, based on the Faedo-Galerkin method, can be 77

78

2. Linear Equations

readily adapted to obtain strong solutions of linear parabolic equations in non divergence form (2.1.5)

ut − aij (t, x) ∂i ∂j u = f (t, x) + bi (t, x) ∂i u + c(t, x)u ;

indeed, in section 2.5 we will present corresponding well-posedness results for the Cauchy problem relative to (2.1.5). 2. Throughout this chapter, we will adopt the following notations and conventions, some of which we have already introduced in Chapter 1. We denote first-order derivatives by D := {∂t , ∇}. The abbreviations “a.e.” and “a.a.” stand, respectively, for “almost everywhere” and “almost all”, either in Q or in RN , with reference to the Lebesgue measure in these sets. For 1 ≤ p ≤ ∞, we set Lp := Lp (RN ), and denote by | · |p its norm. For m ∈ N, we set H m := H m (RN ), and denote by  · m and  ·, · m its norm and scalar product. We identify L2 = H 0 , and abbreviate  · 0 = | · |2 =  · ,  ·, · 0 =  ·, · . When there is no risk of confusion, we often write u(t)m instead of u(t, ·)m , or even um , especially under integration over time intervals. Finally, when we say that a constant “depends on the data” (respectively, “on the coefficients”), we assume that we are in a context where the data u0 , u1 , and f (respectively, the coefficients aij , bi , c) have been specified in some function spaces, and we mean that the constant can be estimated by a continuous function of the norm of the data (respectively, the coefficients) in these spaces. Typically, we denote such functions by γ, κ, or ψ, with γ, κ, ψ ∈ K. These functions can be explicitly determined and, without loss of generality, we may assume them to have range in R≥1 . Finally, we reserve the letter s to always denote an integer strictly larger than N2 + 1. This condition plays a crucial role in the sequel, because it implies that H s−1 is an algebra (see Corollary 1.5.2), and also, by the imbeddings (1.5.61) and (1.5.56), that (2.1.6)

H s−1 → C 0,α (RN ) → L∞ .

2.2. The Hyperbolic Cauchy Problem 1. We consider the Cauchy problem for the hyperbolic equation (2.1.1). We assume that the coefficients aij in (2.1.1) are bounded, symmetric, and uniformly strongly elliptic in some cylinder Q; that is, aij ∈ L∞ (Q), aij = aji a.e. in Q, and there are numbers α1 > α0 > 0 such that, for a.a (t, x) ∈ Q and all q ∈ RN , (2.2.1)

α0 |q|2 ≤ aij (t, x) q i q j ≤ α1 |q|2 .

2.2. The Hyperbolic Cauchy Problem

79

In addition, we assume that, for some integer s >

N 2

+ 1,

(2.2.2) D aij ∈ L1 (0, T ; H s−1 ) ,

bi , c ∈ L1 (0, T ; H s ) ∩ L2 (0, T ; H s−1 ) .

Correspondingly, we set μ1 (t) := Daij (t)s−1 + bi (t)s + c(t)s ,  T M1 := μ1 (t) dt ;

(2.2.3) (2.2.4)

0

in addition, noting that, by the second claim of Proposition 1.7.6 (with m = s − 1), ∇ aij ∈ W 1,1 (0, T ; H s−1 , H s−2 ) → AC([0, T ]; H s−2 ), we set   A := max |aij |L∞ (Q) + ∇aij C([0,T ];H s−2 ) , (2.2.5) i,j

(2.2.6)

M22



T

2

:= A T + max

1≤i≤N

 bi 2s−1 dt

0

T

+

c2s−1 dt .

0

We also note that aij ∈ L∞ (Q) → L1 (0, T ; L∞ ), and, by (2.1.6), the first of (2.2.2) implies that ∂t aij ∈ L1 (0, T ; H s−1 ) → L1 (0, T ; L∞ ); hence, the third claim of Proposition 1.7.6, with X = L∞ (which is not reflexive), implies that aij ∈ AC([0, T ]; L∞ ) .

(2.2.7)

Likewise, (2.2.2) implies that (2.2.8)

D aij , bi , c ∈ L1 (0, T ; H s−1 ) → L1 (0, T ; L∞ ) .

This observation will be essential in the application of Theorem 2.2.1 below to the quasi-linear equations we study in Chapter 3. 2. For m ∈ N and T > 0, we define (2.2.9)

Xm (T ) := C([0, T ]; H m+1 ) ∩ C 1 ([0, T ]; H m ) ,

(2.2.10)

Ym (T ) := {u ∈ Xm (T ) | utt ∈ L2 (0, T ; H m−1 )} ;

these are Banach spaces, with respect to their natural norms  1/2 uXm (T ) = max u(t)2m+1 + ut (t)2m (2.2.11) , 0≤t≤T

(2.2.12)

uYm (T ) =

 u2Xm (T ) +

T

1/2 utt 2m−1 dt

.

0

Note that, with the notation of (1.7.31), Xm (T ) = C 1 ([0, T ]; H m+1 , H m ). We are ready to state the main result on strong solutions of linear hyperbolic equations.

80

2. Linear Equations

Theorem 2.2.1. Let s and m ∈ N, with s > N2 + 1 and 1 ≤ m ≤ s. Let u0 ∈ H m+1 , u1 ∈ H m , and f ∈ L2 (0, T ; H m ). Under the above stated assumptions on the coefficients of (2.1.1), the following holds: (1) Existence: There exists a unique u ∈ Ym (T ), which is a strong solution of the Cauchy problem (2.1.1)+(2.1.2), in the sense that equation (2.1.1) holds in H m−1 for a.a. t ∈ ]0, T [, as well as a.e. in Q. (2) A Priori Estimates: u satisfies the estimates (2.2.13)

uXm (T ) ≤

(2.2.14)

utt L2 (0,T ;H m−1 ) ≤ I0 ψ1 ,

2 I0 ψ0 ,

where I0 depends on the data u0 , u1 , f , and ψ0 depends exponentially on T and the coefficients (via α0 , a bound a1 on aij (0), defined in (2.3.18) below, and on M1 ), and ψ1 depends on ψ0 , T , and M2 . (The constants I0 , ψ0 and ψ1 are defined in (2.3.21), (2.3.24) and (2.3.28) below). (3) Well-Posedness: As a consequence of (2.2.13), the Cauchy problem (2.1.1)+(2.1.2) is well-posed; that is, the map (u0 , u1 , f ) → u =: Φ(u0 , u1 , f )

(2.2.15)

is continuous from H m+1 × H m × L2 (0, T ; H m ) into Ym (T ). (4) Regularity: If, for some k ∈ N≤m−1 , and all r ∈ N≤k , (2.2.16)

∂tr f ∈ L2 (0, T ; H m−r ) ,

(2.2.17)

∂tr bi ,

∂tr aij ∈ L2 (0, T ; H s−r ) ,

∂tr c ∈ L2 (0, T ; H s−r ) ,

then (2.2.18)

u ∈

k+1 .

C  ([0, T ]; H m+1− ) ,

∂tk+2 u ∈ L2 (0, T ; H m−k−1 ) .

=0

Remarks. 1) Estimates (2.2.13) and (2.2.14) are a priori, in the sense that they are automatically satisfied by any solution u ∈ Ym (T ) that problem (2.1.1)+(2.1.2) may have. 2) As a consequence of the Sobolev product estimates, we can see that the right side of (2.1.1) is in L2 (0, T ; H m ). Thus, while each of the two terms at the left side of (2.1.1) is in L2 (0, T ; H m−1 ), their sum is in L2 (0, T ; H m ). It is then natural to ask whether, separately, ∂i ∂j u and utt ∈ L2 (0, T ; H m ). However, if this were the case, it would follow that u ∈ L2 (0, T ; H m+2 ); hence, by the trace theorem ((1.7.59) of Theorem 1.7.4), (2.2.19)

u ∈ C([0, T ]; H m+3/2) ∩ C 1 ([0, T ]; H m+1/2 ) .

2.3. Proof of Theorem 2.2.1

81

In turn, this would imply that the conditions u0 ∈ H m+3/2 and u1 ∈ H m+1/2 , as opposed to u0 ∈ H m+1 and u1 ∈ H m only, would be necessary for the solvability of (2.1.1)+(2.1.2). 3) In the special case that f and the coefficients aij , bi , and c are independent of t, the solution to the Cauchy problem for (2.1.1) is in Ym (T ) for all T > 0; in fact, equation (2.1.1) generates a semiflow on the phase space Xm := H m+1 × H m (see, e.g., Milani and Koksch [119, ch. 2, sct. 2.2]). This means that the family S = (S(t))t≥0 of solution operators, with S(t) : Xm → Xm defined by (2.2.20)

S(t)(u0 , u1 ) := (u(t), ut (t)) ,

u being the solution to the Cauchy problem (2.1.1)+(2.1.2), satisfies the following four properties: 1) S(0) = Im , the identity in Xm . 2) For all t, θ ≥ 0, S(t) S(θ) = S(θ) S(t) = S(t + θ) (a semigroup property). 3) For all t ≥ 0, the map Xm  (u0 , u1 ) → S(t)(u0 , u1 ) ∈ Xm is continuous (a pointwise property of continuous dependence on the data). 4) For all (u0 , u1 ) ∈ Xm , the map R≥0  t → S(t)(u0 , u1 ) ∈ Xm is continuous (a pointwise property of continuity with respect to the family parameter). (A trivial example of semiflow in RN is the family (et A )t≥0 of the exponentials of a N × N matrix A.) 

2.3. Proof of Theorem 2.2.1 We prove Theorem 2.2.1 in various steps. First, we establish the a priori estimates (2.2.13) and (2.2.14) for solutions of (2.1.1) in Ym (T ), and use these estimates to prove the well-posedness of the Cauchy problem (2.1.1)+(2.1.2) in Ym (T ). Next, we construct sequences of approximate solutions to (2.1.1), and use estimates analogous to the a priori ones, to show that these approximations converge to a limit, which we verify to be the desired solution of (2.1.1)+(2.1.2) in Ym (T ). Finally, we prove the additional regularity result (2.2.18). As agreed in part 4 of section 1.1, we denote by C, or K, a universal positive constant, which may vary from estimate to estimate, and even within the same estimate. Without loss of generality, we can take C, K ≥ 1. 2.3.1. A Priori Estimates and Well-Posedness. 1. We prove that any solution in Ym (T ) of the Cauchy problem (2.1.1)+ (2.1.2) satisfies the estimates (2.2.13) and (2.2.14). To this end, we first

82

2. Linear Equations

assume that u satisfies the additional regularity condition u ∈ Ym+1 (T ); we then remove this assumption, resorting to a regularization process involving the Friedrichs’ mollifiers of part 5 of section 1.5.1. The starting point of our procedure is the identity (2.3.1)

utt − aij ∂i ∂j u, ut m = f + bj ∂j u + c u, ut m ,

obtained by multiplying (2.1.1) in H m by ut , which is legitimate if u ∈ Ym (T ). However, in order to proceed we need to split the left side of (2.3.1) into the sum (2.3.2)

utt , ut m − aij ∂i ∂j u, ut m ,

and this is no longer legitimate if u ∈ Ym (T ) only, since, as we have remarked earlier, neither of the terms utt and aij ∂i ∂j u is guaranteed to be in H m , even a.e. in t. This explains why we first establish the estimates under the additional regularity assumption that u ∈ Ym+1 (T ), in which case the two terms in (2.3.2) are indeed defined, at least for a.a. t ∈ ]0, T [. 2. Let u ∈ Ym+1 (T ) be a solution of the Cauchy problem (2.1.1)+(2.1.2). By the additional regularity of u, each term of the left side of (2.1.1) is in H m for a.a. t ∈ ]0, T [ (as opposed to only their difference); indeed, this is clear for utt , while for aij ∂i ∂j u it is a consequence of Theorem 1.5.5, ˜ s for a.a. t, and ∂i ∂j u(t, ·) ∈ H m for all t. Thus, we can because aij (t, ·) ∈ H differentiate (2.1.1) with respect to x up to m times; fixing α ∈ NN , with |α| ≤ m, we obtain (2.3.3) (∂xα u)tt − aij ∂i ∂j (∂xα u) = ∂xα (f + bi ∂i u + c u) + Gα (aij , ∂i ∂j u) =: Rα , where Gα is the commutator defined in (1.5.204); that is, G0 = 0 and, if α > 0,    α β α−β (2.3.4) Gα (aij , ∂i ∂j u) = ∂i ∂j u . β ∂x aij ∂x 0 0, [a, b] ⊆ R, u ∈ C([a, b]; R≥0 ), and v, w ∈ L1 (a, b; R≥0 ). Assume that, for all t ∈ [a, b],  t  t ( 2 (2.3.30) u(t) ≤ c + 2 v(θ) u(θ) dθ + w(θ) u(θ) dθ . a

a

Then, u satisfies, in [a, b], the inequality

2  t

 t (2.3.31) u(t) ≤ c + v(θ) dθ exp w(θ) dθ . a

a

Proof. Call R(t) the right side of (2.3.30). Then, R(t) ≥ u(t), R(t) ≥ c2 > ( 0 for all t ∈ [a, b], and the function t → R(t) satisfies the differential inequality √ d √ (2.3.32) 2 R − w R ≤ 2v . dt Integrating (2.3.32), we find that

 t

 t ( 1 (2.3.33) R(t) ≤ c + v(θ) dθ exp w(θ) dθ . 2 a a Squaring this, and keeping in mind that u(t) ≤ R(t), (2.3.31) follows.



Remark. As in the standard Gronwall’s inequality, Proposition 2.3.1 can be generalized to the case in which the constant c2 in (2.3.30) is replaced by a variable c2 (t), the map t → c2 (t) being positive, differentiable and increasing in [a, b].  3. We now proceed to obtain (2.2.13), from which, as we have seen, (2.2.14) follows, assuming that u ∈ Ym (T ) only.  To this end, we regularδ ize u by means of the Friedrichs’ mollifiers ρ δ>0 in the space variables, introduced in part 2 of section 1.4; that is, we set    1 δ δ (2.3.34) u (t, x) := [ρ ∗ u(t, ·)](x) = N ρ x−y u(t, y) dy . δ δ From (2.1.1), we derive that for each δ > 0, uδ solves the equation (2.3.35)

uδtt − aij ∂i ∂j uδ = f δ + bi ∂i uδ + c uδ + F δ + Lδ ,

2.3. Proof of Theorem 2.2.1

87

where, omitting the variables t and x, which are as in (2.3.34), f δ := ρδ ∗ f , and (2.3.36)

F δ := ρδ ∗ (aij ∂i ∂j u) − aij ∂i ∂j uδ ,

(2.3.37)

Lδ := ρδ ∗ (bi ∂i u + cu) − (bi ∂i uδ + c uδ ) .

Since uδ ∈ Ym+1 (T ), it satisfies estimate (2.3.25), with I02 replaced by

(2.3.38)

(I0δ )2 := uδ1 2m + uδ0 2m+1  T f δ 2m + F δ 2m + Lδ 2m dθ ; +3 0

that is, (2.3.39)

uδ (t)2m + D uδ (t)2m ≤ 3(I0δ ψ0 )2 ,

t ∈ [0, T ] .

Since u0 ∈ H m+1 , u1 ∈ H m , and f (t) ∈ H m for a.a. t ∈ [0, T ], by Theorems 1.5.1 and 1.5.6, and Corollary 1.5.4, it follows that (2.3.40) uδ1 m ≤ u1 m ,

uδ0 m+1 ≤ u0 m+1

f δ (t)m ≤ f (t)m ,

as well as (2.3.41)

F δ (t)m ≤ C ∇aij (t)s−1 ∇u(t)m ,

(2.3.42)

Lδ (t)m ≤ C (bj (t)s + c(t)s ) u(t)m .

Moreover, as δ → 0, uδ0 → u0 in H m+1 , uδ1 → u1 in H m , and, for a.a. t ∈ [0, T ], (2.3.43)

f δ (t)m → f (t)m ,

F δ (t)m → 0 ,

Lδ (t)m → 0 .

This allows us to pass to the limit as δ → 0 in (2.3.38), by Lebesgue’s dominated convergence Theorem. Since also (2.3.44)

uδ (t)2m + D uδ (t)2m → u(t)2m + D u(t)2m ,

we deduce from (2.3.39) that, for all t ∈ [0, T ], (2.3.45)

u(t)2m + D u(t)2m ≤ 3 I02 ψ02 ;

that is, u satisfies (2.2.13), as claimed.



4. As the proof of (2.2.14) shows, the regularity of utt is automatically determined by the regularity of f , u and the coefficients, via (2.1.1). We shall exploit this fact in the proof of the second regularity estimate in (2.2.18). In particular, we note that if, in addition to (2.2.2), aij , bj , c ∈ C([0, T ]; H s−1 ), and f ∈ L2 (0, T ; H m ) ∩ C([0, T ]; H m−1 ), then utt ∈ C([0, T ]; H m−1 ). For

88

2. Linear Equations

future reference, we summarize this conclusion, introducing, for m ∈ N, the spaces (2.3.46) (2.3.47)

Vm (T ) := L2 (0, T ; H m+1 ) ∩ C([0, T ]; H m ) , Zm (T ) :=

2 .

C j ([0, T ]; H m+1−j ) ,

j=0

which are Banach spaces with respect to their natural norms, defined by  T 2 uVm (T ) := (2.3.48) u2m+1 dt + max u(t)2m , (2.3.49)

u2Zm (T ) :=

0 2 

0≤t≤T

∂tj u2C([0,T ];H m+1−j ) .

j=0

Note that Zm (T ) → Ym (T ) → Xm (T ) → Vm (T ). Theorem 2.2.1 can then be supplemented by Theorem 2.3.1. Let m ∈ N, with 1 ≤ m ≤ s. If, in addition to the as˜ s−1 ), and f ∈ Vm−1 (T ), sumptions of Theorem 2.2.1, aij , bj , c ∈ C([0, T ]; H then the solution u ∈ Ym (T ) of the Cauchy problem (2.1.1)+(2.1.2) is in Zm (T ), and (2.3.50)

uZm (T ) ≤ I1 ψ1 ,

where I12 := (2I0 )2 + f 2C([0,T ];H m−1 ) . Remark. In subsequent parts of these lectures, we will often establish a priori estimates similar to (2.2.13) and (2.2.14). We shall most often do so only formally, in the sense that we agree to have verified that it is possible to resort to a regularization process like the one followed in parts 2 and 3 above.  5. As a consequence of the a priori estimates (2.2.13) and (2.2.14), we can show that problem (2.1.1)+(2.1.2) is well-posed in Ym (T ). Indeed, (2.2.13) and (2.2.14) imply that, for any solution u ∈ Ym (T ) of (2.1.1)+(2.1.2),

 T (2.3.51) u2Ym (T ) ≤ C (ψ0 + ψ1 )2 u0 2m+1 + u1 2m + f 2m dt . 0

Let u and u ˜ be solutions of (2.1.1), corresponding to data {u0 , u1 , f } and ˜ {˜ u0 , u ˜1 , f }. Then, their difference z := u − u ˜ satisfies the Cauchy problem  ztt − aij ∂i ∂j z = (f − f˜) + bj ∂j z + c z , (2.3.52) z(0) = u0 − u ˜0 , zt (0) = u1 − u ˜1 ;

2.3. Proof of Theorem 2.2.1

89

hence, applying (2.3.51) to z, we deduce that (2.3.53) u − u ˜2Ym (T ) ≤ C

 u0 −

u ˜0 2m+1

+ u1 −

u ˜1 2m

T

+

f − f˜2m dt

,

0

where C depends only on the coefficients and T . This means that the norm of u in Ym (T ) depends continuously on that of the data. In particular, (2.3.53) implies that strong solutions of the Cauchy problem (2.1.1)+(2.1.2) are unique.  2.3.2. Existence of Strong Solutions. We prove the existence of a solution to problem (2.1.1)+(2.1.2), which we construct by means of a double approximation argument, in four steps. In the first, we restrict (2.1.1)+(2.1.2) to a sequence of expanding balls in RN , at the boundary of which we impose homogeneous Dirichlet boundary conditions; in the second, we solve the corresponding restricted initial-boundary value problems by means of a Galerkin sequence of approximations. Next, we extend each of these solutions to the whole space RN , thereby producing a sequence of approximated solutions on RN ; finally, we show that these solutions converge to a limit, which we identify as the desired solution of (2.1.1)+(2.1.2). 1. For  ∈ N, we set Ω := {x ∈ RN | |x| <  + 1} (the open ball of RN of center 0 and radius  + 1), and denote by  · m, and  · , · m, the norm and scalar product on H m (Ω ). We also denote by [ · ]m, the norm N1 on m (Ω ) defined by (1.5.137). For r ∈ ] − 1, 1[, we let HΔ 

− r2 (2.3.54) ϕ(r) := exp , 1 − r2 and define ζ ∈ C0∞ (RN ) by ⎧ 1 ⎪ ⎪ ⎨ (2.3.55) ζ (x) := ϕ(|x| − ) ⎪ ⎪ ⎩ 0

if

|x| ≤  ,

if

 < |x| <  + 1 ,

if

|x| ≥  + 1 .

Then, each ζ is a cut-off function in RN , since 0 ≤ ζ (x) ≤ 1 for all x ∈ RN , ζ (x) ≡ 1 on Ω−1 (we agree that Ω− 1 = {0}), and ζ ≡ 0 off Ω . In addition, for each α ∈ NN there is Cα > 0 such that, for all  ∈ N, (2.3.56)

sup |∂xα ζ (x)| ≤ Cα ,

x∈RN

90

2. Linear Equations

and, since ϕ(j) (r) → 0 as |r| → 1− for all j ∈ N, ∂j

(2.3.57)

∂νj

ζ = 0

on

∂Ω ,

where ν denotes the outward unit normal to ∂Ω . Finally, we abbreviate − Δ := − ΔΩ (the Laplace operator with Dirichlet boundary conditions on ∂Ω , introduced in section 1.5.2), denote by E and R the extension and restriction operators defined in part 4 of section 1.5.1, with Ω = Ω , and, given a function h defined on RN (respectively, on Q), set (2.3.58)

h := R (ζ h)

(resp., h (t, ·) := R (ζ h(t, ·) ) .

Note that (2.3.57) implies that h ∈ H0m (Ω ) if h ∈ H m (RN ), and, by (2.3.56), (2.3.59) with Cm

h H m (Ω ) ≤ Cm hH m (RN ) ,     := max|α|≤m {maxβ≤α αβ } Cα .

2. For each fixed  ∈ N, we consider the initial-boundary value problem ⎧     in ]0, T [ ×Ω , ⎪ ⎪ utt − aij ∂i ∂j u = f + bi ∂i u + c u ⎪ ⎨ u(0, ·) = u0 , ut (0, ·) = u1 on {0} × Ω , (2.3.60) ⎪ ⎪  ⎪ ⎩ u(t, ·) =0 on ]0, T [ ×∂Ω . ∂Ω 

Our goal is to solve (2.3.60) by means of a Galerkin approximation method, with the choice, as a total basis of L2 (Ω ), of the sequence W = (wj )j≥1 of the eigenfunctions of − Δ . As we noted in part 3 of section 1.5.2, W is orthogonal in H0m (Ω ) (endowed with the scalar product induced by [ · ]m, ), and H0m+k (Ω ) -regular, for all m ∈ N and k ∈ N≥1 . For each n ∈ N≥1 , we set (2.3.61)

Wn := span{wj | j ≤ n} ,

and project problem (2.3.60) into each finite-dimensional space Wn . Heuristically, this means that, instead of (2.3.60), we consider the infinite set of scalar equations (2.3.62)

utt − aij ∂i ∂j u, wr 0, = f  + bi ∂i u + c u, wr 0, ,

r ≥1,

each of which is to hold in R, for a.a. t ∈ ]0, T [. To implement Galerkin’s method, we proceed as follows. Since the restricted initial values u0 and u1 are in H0m+1 (Ω ) and H0m (Ω ), respectively, n, by (1.5.147) of section 1.5.2 the sequences (un, 0 )n≥0 and (u1 )n≥0 , defined respectively by n n   n,  (2.3.63) un, := u , w  w , u := u1 , wk 0, wk , k 0, k 0 0 1 k=1

k=1

2.3. Proof of Theorem 2.2.1

91

n, are such that un, 0 , u1 ∈ Wn for all n ≥ 1, and, as n → +∞,  un, 0 → u0

(2.3.64)

 un, 1 → u1

in H m+1 (Ω ) ,

in H m (Ω ) .

In fact, recalling that Pn is an orthogonal projection in H0m+1 (Ω ), by the first of (1.5.139) and by (1.5.25) and (2.3.56), it follows that  [un, 0 ]m+1, ≤ u0 m+1, = ζ u0 m+1 ≤ Cm u0 m+1 ;

(2.3.65) analogously, (2.3.66)

[un, 1 ]m, ≤ Cm u1 m .

Since also [f  (t)]m, ≤ Cm f (t)m , we conclude that there exists M > 0, independent of n and of , such that for all n ≥ 1 and  ≥ 0,  T n, 2 n, 2 (2.3.67) u0 m+1, + u1 m, + f  (t)2m, dt ≤ M 2 . 0

In analogy to the technique of separation of variables, we wish to find functions un, : [0, T ] → Wn , that is, of the form (2.3.68)

un, (t, x) =

n 

γkn (t)wk (x) ,

k=1

which are approximate solutions of (2.3.62), in the sense that they should solve the n equations (2.3.69)

 n,   n, un, + c un, , wr 0, , tt − aij ∂i ∂j u , wr 0, = f + bi ∂i u

for 1 ≤ r ≤ n. In addition, and consistently with (2.3.63), un, should satisfy the initial conditions (2.3.70)

un, (0, ·) = un, 0 ,

n, un, t (0, ·) = u1 .

We rewrite (2.3.69) in the form (2.3.71)

 n,  n, ˜ un, + c un, , wr 0, , tt − ∂j (aij ∂i u ), wr 0, = f + bi ∂i u

where ˜bi := bi − ∂j aij . Since the system W is orthonormal in L2 (Ω ) and contained in H01 (Ω ), (2.3.71) is equivalent to the finite system of n scalar second order ordinary differential equations ⎧ n  ⎪ ⎪  ⎪ γ rn (t) + γkn (t)aij ∂i wk , ∂j wr 0, ⎪ ⎪ ⎪ ⎨ k=1 n  (2.3.72)  ⎪ = f , wr 0, + γkn (t)˜bi ∂i wk + c wk , wr 0, , ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎩ 1 ≤ r ≤ n,

92

2. Linear Equations

in the unknowns (γrn (t))1≤r≤n =: γn (t) ∈ Rn . Comparing (2.3.68) for t = 0 with (2.3.70) and (2.3.63), we attach to system (2.3.72) the initial conditions ⎧  ⎪ ∈ Rn , ⎨ γn (0) = γ0n := u0 , wr 0, 1≤r≤n  (2.3.73) ⎪ ⎩ γn  (0) = γ1n := u1 , wr 0, ∈ Rn . 1≤r≤n

We can solve the Cauchy problem (2.3.72)+(2.3.73) by means of Carath´eodory’s theorem (see, e.g., Coddington and Levinson [37, ch. 1, § 2]), which yields a solution γn ∈ C 1 ([0, T ]; Rn ) of (2.3.72) (for a.a. t ∈ [0, T ]) and (2.3.73), with γ n ∈ AC([0, T ]; Rn ). This solution is global on [0, T ], because (2.3.72) is linear (see the remark at the end of the proof of Lemma 2.3.1 below). We then define the function un, by (2.3.68); clearly, (2.3.74)

un, ∈ C 1 ([0, T ]; Wn ) ,

un, t ∈ AC([0, T ]; Wn ) .

By construction, un, solves (2.3.71), a.e. in [0, T ]; in fact, it also solves (2.3.69). Moreover, since Ω is a ball, Wn ⊂ C ∞ (Ω ), so that (2.3.74) implies that (2.3.75)

un, (t) ∈ H 2 (Ω ) ,

2 un, tt (t) ∈ L (Ω )

for all (respectively, almost all) t ∈ [0, T ], and (2.3.69) makes sense. Moreover, by (2.3.73), (2.3.68), and (2.3.63), un, also satisfies the initial conditions (2.3.70). 3. We now introduce, for k ∈ N, the spaces  2 L (Ω ) , k (2.3.76) W0 (Ω ) := H k (Ω ) ∩ H01 (Ω ) ,

if

k =0,

if

k ≥1,

and (2.3.77)

Xk, (T ) := C([0, T ]; W0k+1 (Ω )) ∩ C 1 ([0, T ]; W0k (Ω )) .

We claim:

  Lemma 2.3.1. The sequence un, n≥1 is in a bounded set of Xk, (T ), 0 ≤ k ≤ m. The bounds on this sequence depend only on the data and the coefficients of (2.1.1)+(2.1.2), via the constants M of (2.3.67) and M1 of (2.2.4), respectively. In particular, these bounds are independent of . Proof. We proceed by induction on k. For k = 0, we multiply equation (2.3.71) by γrn  (t), and then sum the resulting identities for 1 ≤ r ≤ n, to obtain (2.3.78) n, n,  n,  n, ˜ un, + c un, , un, tt , ut 0, + aij ∂i u , ∂j ut 0, = f + bi ∂i u t 0, .

2.3. Proof of Theorem 2.2.1

93

From this we obtain, as in (2.3.15), d  n, 2 ut 0, + Q0 (a , ∇un, ) = ∂t (aij ) ∂i un, , ∂j un, 0, dt (2.3.79) + 2f  + ˜bi ∂i un, + c un, , un, t 0, =: R01 + R02 . Since 0 ≤ ζ ≤ 1, recalling (2.2.3) we can estimate (2.3.80)

|∂t aij (t)|L∞ (Ω ) ≤ |∂t aij (t)|∞ ≤ ∂t aij (t)s−1 ≤ μ1 (t) ;

hence, (2.3.81)

R01 ≤ μ1 (t) ∇un, 20, .

Acting likewise for ˜bi and c , we obtain from (2.3.79) that, as in (2.3.16), d  n, 2 ut 0, + Q0 (a , ∇un, ) dt (2.3.82)  2 n, 2 ≤ f  20, + C (1 + μ1 (t)) un, t 0, + u 1, . We add to (2.3.82) the inequality d 2 n, 2 un, 20, ≤ un, t 0, + u 0, dt (see (2.3.19)), and then integrate; recalling (2.3.22), by Gronwall’s inequality we deduce that, for all t ∈ [0, T ], (2.3.83)

(2.3.84)

2 n, 2 un, t (t)0, + u (t)1,  n, 2 n, 2 2 ≤ a2 u1 0, + u0 1, +

 · exp C



T

(1 + μ1 (t)) dt

T 0

f  20, dt

.

0

Recalling (2.3.67) and (2.2.4), we finally conclude that, for all t ∈ [0, T ] and all n ≥ 1, (2.3.85)

2 n, 2 2 2 C(T +M1 ) un, =: Λ20 . t (t)0, + u (t)1, ≤ a2 M e

This proves the claim of Lemma 2.3.1 for k = 0. We assume then that the claim is true for k − 1, 1 ≤ k ≤ m, and proceed to prove it for k. To this end, we note that, by the first of (2.3.75), un, , which satisfies (2.3.71), also solves (2.3.69). In this equation, we replace wr = λ1k (− Δ )k wr , to obtain r that, for 1 ≤ r ≤ n, (2.3.86)

 n, k un, tt − aij ∂i ∂j u , (− Δ ) wr 0,

= f  + bi ∂i un, + c un, , (− Δ )k wr 0, .

We can integrate by parts in each of the terms of (2.3.86), as per (1.5.110), if k is even, or (1.5.111), if k is odd, because condition (1.5.112) is satisfied.

94

2. Linear Equations

Indeed, for the first term of (2.3.86) this is true because, by (2.3.68), un, tt is a linear combination of w1 , . . . , wn , each of which does satisfy (1.5.112). For the other terms of (2.3.86), it is sufficient to recall that, by (2.3.57), both aij ∂i ∂j un, and f  + bi ∂i un, + c un, ∈ H0k (Ω ). Hence, assuming, e.g., that k is even (the other case is analogous), we obtain from (2.3.86) that k/2 (− Δ )k/2 un, wr 0, tt , (− Δ )

− (− Δ )k/2 (aij ∂i ∂j un, ), (− Δ )k/2 wr 0,

(2.3.87)

= (− Δ )k/2 (f  + bi ∂i un, + c un, ), (− Δ )k/2 wr 0, . By Leibniz’ formula, and recalling further that (− Δ )k/2 wr vanishes on ∂Ω , so that we can integrate by parts, we can rewrite the second line of (2.3.87) as (2.3.88) − aij ∂i ∂j (− Δ )k/2 un, +



Γα (aij , un, ), (− Δ )k/2 wr 0,

|α|=k

=

aij

∂i (− Δ )

k/2 n,

u

, ∂j (− Δ )k/2 wr 0,

+ (∂j aij )∂i (− Δ )k/2 un, −



Γα (aij , un, ), (− Δ )k/2 wr 0, ,

|α|=k

where Γα (aij , un, ) has the form 

Γα (aij , un, ) :=

(2.3.89)

Cαγ ∂xγ aij ∂xα−γ ∂i ∂j un, ,

0 0, we define (2.5.3)

Pm (T ) := {u ∈ C([0, T ]; H m+1 ) | Du ∈ L2 (0, T ; H m )} ,

which is a Banach space with respect to the norm defined by  T 2 2 (2.5.4) uPm (T ) := max u(t)m+1 + Du2m dt . 0≤t≤T

0

In addition to (2.2.1), we assume that (2.5.5)

aij ∈ L∞ (Q) , D aij ∈ L1 (0, T ; H s−1 ) ,

bi , c ∈ L1 (0, T ; H s ) ,

108

2. Linear Equations

and claim: Theorem 2.5.1. Let s and m ∈ N, with s > N2 + 1 and 1 ≤ m ≤ s. Given f ∈ L2 (0, T ; H m ) and u0 ∈ H m+1 , the following holds. (1) Existence: There exists a unique solution u ∈ Pm (T ) of problem (2.1.5)+(2.5.1), which satisfies (2.1.5) in H m for a.a. t ∈ [0, T ], as well as a.e. in Q. (2) A Priori Estimates: u satisfies the estimate  T (2.5.6) u2Pm (T ) = max u(t)2m+1 + Du2m dt ≤ J02 ψ˜02 , 0≤t≤T

0

where J0 depends on the data, and ψ˜0 depends on the coefficients and T . (3) Well-Posedness: As a consequence of these estimates, u depends continuously on the data u0 and f , in the sense that the map (u0 , f ) → u is continuous from H m+1 × L2 (0, T ; H m ) into Pm (T ). That is, problem (2.1.5)+(2.5.1) is well-posed. (4) Global Regularity: If, for some k, 0 ≤ k ≤  m 2 , and all r ∈ {0, . . . , k}, ∂tr f ∈ L2 (0, T ; H m−2r )

(2.5.7) and

∂tr aij , ∂tr bi , ∂tr c ∈ L2 (0, T ; H s−2r ) ,

(2.5.8) then (2.5.9)

∂tr+1 u ∈ L2 (0, T ; H m−2r ) .

Sketch of Proof. 1) The proof follows the same steps of the proof of Theorem 2.2.1; namely, we first prove the a priori estimate (2.5.6), from which the well-posedness of (2.1.5)+(2.5.1) follows. The existence part is again proven by a similar double approximation argument (that is, restriction to a sequence of initial-boundary value problems on expanding balls in RN , and Galerkin approximations for each of the latter problems). For an alternative approach, we refer to Krylov [84, ch. 2]. 2) To establish (2.5.6), at least formally, we differentiate (2.1.5) α times with respect to the space variables, with |α| ≤ m, and multiply the corresponding equations in L2 by 2 ∂xα u and 2 ∂xα ut . Summing with respect to α, we obtain an estimate of  T (2.5.10) max u(t)2m+1 + Du2m dt . 0≤t≤T

0

As we have remarked in section 2.3.1, (2.5.10) can be established rigorously, by a regularization process involving Friedrichs’ mollifiers.

2.5. The Parabolic Cauchy Problem

109

3) The global regularity claim (2.5.9) is again proven by induction on k. For k = 0, (2.5.9) just reads ut ∈ L2 (0, T ; H m ), which does hold if u ∈ Pm (T ). Assuming in the sequel that bi , c ≡ 0 for simplicity, to prove the claim for k + 1, k ≥ 0 and 2(k + 1) ≤ m, it is sufficient to show that (2.5.11)

∂tk+2 u ∈ L2 (0, T ; H m−2(k+1) ) .

This follows from the identity (2.5.12)

∂tk+2 u

=

∂tk+1 f

+

k+1  

k+1   ∂t aij

∂tk+1− ∂i ∂j u ,

=0

at the right side of which all the terms are in L2 (0, T ; H m−2(k+1) ). This is clear for ∂tk+1 f , by (2.5.7) with r = k + 1. In the terms with  ≤ k of the sum, we note that ∂t aij ∈ C([0, T ]; H s−2−1 ) by the trace theorem, and ∂tk+1− ∂i ∂j u ∈ L2 (0, T ; H m−2k−2+2) by the induction assumption; thus, by the Sobolev product properties, ∂t aij ∂tk+1− ∂i ∂j u ∈ L2 (0, T ; H m−2k−2 ). When  = k + 1, we reach the same conclusion, since ∂tk+1 aij ∈ L2 (0, T ; H s−2k−2 ) and ∂i ∂j u ∈ C([0, T ]; H m−1 ). In either case, (2.5.11) follows.  From equation (2.1.5) and the existence part of Theorem 2.5.1, it follows that (2.5.13)

− aij ∂i ∂j u = f + bi ∂i u + c u ∈ L2 (0, T ; H m ) .

If the underlying space domain were bounded, and u were subject to suitable boundary conditions, (2.5.13) would imply, by standard results on elliptic regularity theory, similar to those in Proposition 1.5.5, that u ∈ L2 (0, T ; H m+2 ). When (2.1.5) is considered in all of RN , as in the present situation, we can still recover this result, at least under an additional regularity assumption on the coefficients. Theorem 2.5.2. In addition to (2.5.5), assume that ∇aij ∈ L1 (0, T ; H s ). Let u ∈ Pm (T ) be the solution of (2.1.5)+(2.5.1) determined by Theorem 2.5.1. Then, u ∈ L2 (0, T ; H m+2 ), and its norm in this space depends continuously on the norms of f in L2 (0, T ; H m ) and of u0 in H m+1 . Idea of the Proof. The additional regularity of the coefficients allows us to differentiate (2.1.5) one more time with respect to the space variables; that is, with |α| ≤ m + 1. Multiplying the resulting identities in L2 by ∂xα u, we arrive at an estimate of d (2.5.14) u(t)2m+1 + α0 ∇u(t)2m+1 . dt The only detail to note is that, in establishing (2.5.14), one has to deal with the terms ∂xα (f + bi ∂i u + c u), in which the total order of differentiation can be up to m+1, in contrast to the fact that f (t)+bi (t) ∂i u(t)+c(t) u(t) ∈ H m

110

2. Linear Equations

only, for a.a. t. However, such terms appear in a product, in which we can integrate by parts, thereby reducing the order of differentiation to exactly m. More precisely, when |α| = m + 1, i.e. ∂xα = ∂k ∂xγ with k ∈ {1, . . . , N } and |γ| = m, we estimate (2.5.15)

∂k ∂xγ f, ∂k ∂xγ u = − ∂xγ f, ∂k2 ∂xγ u ≤ ∂xγ f  ∇∂xα u ≤ f m ∇um+1 ≤ Cη f 2m + η ∇u2m+1 ,

and analogously for the other terms involving bi ∂i u and c u. We can then choose η > 0 sufficiently small so as to absorb the last term of (2.5.15) into the corresponding term that appears at the left side of (2.5.14);, we can then integrate, and proceed as before, to obtain an estimate of u in C([0, T ]; H m+1 ) ∩ L2 (0, T ; H m+2 ).  2.5.2. Regularity for t > 0. Let u0 ∈ L2 . It is well known that the Cauchy problem for the linear, homogeneous heat equation, that is (see (0.0.6))  ut − Δ u = 0 , (2.5.16) u(0) = u0 , has a solution u ∈ L2 (0, +∞; H 1 ) ∩ Cb ([0, +∞[; L2 ), which is such that u ∈ C ∞ (R>0 × RN ). This improvement of the regularity of the solution from t = 0 to t > 0 is somewhat typical of parabolic equations, and is sometimes referred to as a consequence of the so-called smoothing effect of parabolic operators. In the same spirit, when f ≡ 0 (or is very smooth), we can establish an improved regularity result for equation (2.1.5). More precisely, we claim: Theorem 2.5.3. Assume that the coefficients of (2.1.5) satisfy the second of (2.2.16), and (2.2.17); that is, (2.5.17)

∂tr aij ,

∂tr bi ,

∂tr c ∈ L2 (0, T ; H s−r )

For 0 ≤ m ≤ s, let u0 ∈ τ ∈ ]0, T [,

H m+1

(2.5.18)

u∈

m .

and f ∈

for

0≤r ≤s.

W m,2 (0, T ; H m , L2 ).

Then, for all

C  ([τ, T ]; H m+1− )

=0

and (2.5.19)

(

(· − τ ) ∂tm+1 u ∈ L2 (τ, T ; L2 ) .

That is, if u0 and the coefficients satisfy the same regularity assumptions as for the hyperbolic equation (2.1.1), then the solution of the parabolic equation (2.1.5) enjoys, for t > 0, the same regularity as the solution to the hyperbolic equation.

2.5. The Parabolic Cauchy Problem

111

Proof. Again, we assume for simplicity that bi , c ≡ 0. By (1.7.58) of Theorem 1.7.4, ∂tr f ∈ L2 (0, T ; H m−r ) ,

(2.5.20)

for

0≤r ≤m.

Let τ ∈ ]0, T [. We wish to prove that, for 0 ≤  ≤ m, (2.5.21)

(

(2.5.22)

∂t u ∈ C([τ, T ]; H m+1− ) , (· − τ ) ∂t+1 u ∈ L2 (τ, T ; H m− ) .

We proceed by induction on . For  = 0, (2.5.21) and (2.5.22) follow from the fact that u ∈ Pm (T ); in fact, when  = 0 we can take τ = 0. Assume then that (2.5.21) and (2.5.22) hold for 0 ≤  ≤ m−1. To prove they hold also for  + 1, we follow a formal procedure, which can, again, be justified by means of a regularization process. We set τ1 := 13 τ , τ2 := 23 τ . Differentiating equation (2.1.5) k + 1 times with respect to t, 0 ≤ k ≤ m − 1, we obtain (2.5.23) ∂tk+2 u − aij ∂i ∂j (∂tk+1 u) = ∂tk+1 f +

k+1  

k+1 r



∂tr aij ∂tk+1−r ∂i ∂j u .

r=1

We multiply this identity in H m−1−k by 2(t − τ2 )∂tk+2 u, to obtain (2.5.24) 2(t − τ2 ) ∂tk+2 u2m−1−k +

d  (t − τ2 ) Qm−1−k (a, ∇∂tk+1 u) dt

= Qm−1−k (a, ∇∂tk+1 u) + 2(t − τ2 ) (Λf + Λ1 + Λ2 + Λ3 + Λ4 ) , where Qm−1−k (a, ·) is defined in (2.3.13), and (2.5.25)

Λf

(2.5.26)

Λ1

:= ∂tk+1 f, ∂tk+2 um−1−k ,  := − ∂j aij ∂i ∂xα ∂tk+1 u, ∂xα ∂tk+2 u , |α|≤m−1−k

(2.5.27)

Λ2 :=



1 2

∂t aij ∂i ∂xα ∂tk+1 u, ∂j ∂xα ∂tk+1 u ,

|α|≤m−1−k

(2.5.28)

Λ3 :=

k+1 

r r=1

(2.5.29)

Λ4 :=

k+1



∂tr aij ∂tk+1−r ∂i ∂j u, ∂tk+2 um−1−k ,    α k α k+2 u , β Aαβ (u), ∂x ∂t

|α|≤m−1−k 0 0, (2.5.35)



Λ4 ≤ C

∂xβ aij 2s−|β| ∂xα−β ∂i ∂j ∂tk+1 u2m−k−2−|α|+|β|

0 0 such that for all t ∈ [τ2 , T ], (2.5.43)



t

(θ − τ2 ) ∂t+2 u2m−1− dθ + (t − τ2 ) ∇∂t+1 u(t)2m−1−

  2τ /3 =: ϕ(t)

 ≤

K02

t

+ C 2τ /3

(∂t aij s−1 + ∇aij 2s−1 ) ϕ(θ) dθ ,

114

2. Linear Equations

from which, by Gronwall’s inequality, (2.5.44)



t 2τ /3

(θ − τ2 ) ∂t+2 u2m−1− dθ + (t − τ2 ) ∇∂t+1 u(t)2m−1−

≤

K02

 exp C



T

(1 +

Daij 2s−1 ) dθ

0

=: K12 .

From this, we first deduce that (2.5.45)





T

(θ τ

− τ ) ∂t+2 u2m−1− dθ 

T

≤

2τ /3

T

≤ τ

(θ − τ2 ) ∂t+2 u2m−1− dθ

(θ − τ2 ) ∂t+2 u2m−1− dθ ≤ K12 ,

which shows that (2.5.22) holds for  + 1. Moreover, since t − τ2 ≥ τ1 if t ≥ τ , (2.5.44) also implies that for all t ∈ [τ, T ], ∇∂t+1 u(t)2m−−1 ≤

(2.5.46)

3 2 K , τ 1

from which ∇∂t+1 u ∈ L∞ (τ, T ; H m−−1 ) .

(2.5.47)

Going back to (2.5.24), (2.5.45) and (2.5.47), together with (2.5.31), . . . , (2.5.35), yield that d  (2.5.48) (· − τ2 ) Qm−1− (a, ∇∂t+1 u) ∈ L1 (τ, T ) ; dt consequently, the function t → Qm−1− (a(t), ∇∂t+1 u(t)) is continuous on [τ, T ]. From this, we can deduce that ∇∂t+1 u ∈ C([τ, T ]; H m−−1 ), with an argument similar to the one used, on the function F (t, t0 ) of (2.3.142), to show (2.3.147). More precisely, we fix t0 ∈ [τ, T ], and compute that α0 ∇∂t+1 u(t) − ∇∂t+1 u(t0 )2m−−1 ≤ Qm−1− (a(t), ∇∂t+1 u(t) − ∇∂t+1 u(t0 )) (2.5.49)

= Qm−1− (a(t), ∇∂t+1 u(t)) + Qm−1− (a(t), ∇∂t+1 u(t0 )) −2

 |α|≤m−1−

aij (t) ∂i ∂xα ∂t+1 u(t), ∂i ∂xα ∂t+1 u(t0 )

 

=: V1 (t) + V2 (t) + V3 (t) .

=: ψα (t)

2.5. The Parabolic Cauchy Problem

115

Since, as previously shown, the functions V1 and V2 are continuous, (2.5.50)

V1 (t) + V2 (t) → V1 (t0 ) + V2 (t0 ) = 2 V1 (t0 ) .

We now show that, in V3 (t), ψα (t) → ψα (t0 ), for each multi-index α. Arguing as in (2.3.144), it is sufficient to show that the map t → ∇∂t+1 u(t) is weakly continuous from [τ, T ] into H m−−1 . This follows from Proposition 1.7.1, recalling (2.5.47), and noting that ∇∂t+1 u ∈ C([τ, T ]; H m−−2 ). To see the latter, we refer to equation (2.5.23), with k =  − 1, which implies that (2.5.51)

∂t+1 u

=

∂t f

+

    r

∂t−r aij ∂tr ∂i ∂j u .

r=0

By the trace theorem, it follows that ∂t f ∈ C([0, T ]; H m−−1) and ∂t−r aij ∈ C([0, T ]; H s−+r−1). The induction assumption (2.5.21), with  replaced by r, r ≤  ≤ m − 1, implies that ∂tr ∂i ∂j u ∈ C([τ, T ]; H m−r−1 ). Since (2.5.52)

s −  + r − 1 ≥ m −  − 1,

m−r−1≥m−−1

and (2.5.53) (s−+r −1)+(m−r −1) = (s−1)+(m−−1) >

N 2

+(m−−1) ,

by the Sobolev product properties it follows that the right side of (2.5.51) is in C([τ, T ]; H m−−1 ). Thus, so is ∂t+1 u, and ∇∂t+1 u ∈ C([τ, T ]; H m−−2 ), as claimed. Consequently, ψα (t) → ψα (t0 ). Then, (2.5.54)

V3 (t) → V3 (t0 ) = − 2 V2 (t0 ) = − 2 V1 (t0 ) ,

and we deduce from (2.5.49) that ∇∂t+1 u ∈ C([τ, T ]; H m−−1 ). Since, as we have shown from (2.5.51), ∂t+1 u ∈ C([τ, T ]; H m−−1 ), we conclude that ∂t+1 u ∈ C([0, T ]; H m− ); that is, (2.5.21) also holds for  + 1, as claimed. This completes the proof of Theorem 2.5.3.  Remarks. 1) As estimate (2.5.46) shows, it is not reasonable to expect that the regularity result (2.5.18) should hold on the whole interval [0, T ]. Indeed, if this were the case, that is, if problem (2.1.5)+(2.5.1) did have a solution m 0 u∈ C  ([0, T ]; H m+1− ), then it would follow that, for 0 ≤  ≤ m, =0

(2.5.55)

v := ∂t u(0) ∈ H m+1− .

On the other hand, the values of these derivatives can be recursively computed from the initial value u0 , by means of equation (2.1.5). More precisely, starting from u0 , we can generate the functions (2.5.56)

u+1 := ∂t f (0) +

   h h=0

∂th aij (0) ∂i ∂j u−l ;

116

2. Linear Equations

for example, (2.5.57)

u1 = f (0) + aij (0) ∂i ∂j u0 ,

(2.5.58)

u2 = ft (0) + aij (0) ∂i ∂j u1 + ∂t aij (0) ∂i ∂j u0 .

Note that the functions u can be defined independently of the existence of a solution of (2.1.5). Thus, if problem (2.1.5)+(2.5.1) had a solution u ∈ m 0 C  ([0, T ]; H m+1−), we would conclude that the identities u = v would =0

have to hold for all  ∈ {1, . . . , k}. Therefore, by (2.5.55), the conditions u ∈ H m+1− ,

(2.5.59)

0≤≤k,

would be necessary in order for the regularity result (2.5.18) to hold on the whole interval [0, T ]. For example, (2.5.59) would require that u1 ∈ H m ; but from (2.5.57) we can only deduce, from the assumptions on u0 , f and the aij (i.e., u0 ∈ H m+1 , f ∈ C([0, T ]; H m−1 ) and aij ∈ C([0, T ]; H s−1 )), that u1 ∈ H m−1 . 2) In fact, conditions (2.5.59) are also sufficient for the regularity result (2.5.18) to hold on [0, T ]. Indeed, in this case we do not need to consider the factor t − τ2 in (2.5.24), and, recalling that, by (2.5.55), ∂tk+1 u(0) = vk+1 = uk+1 , estimate (2.5.36) can be replaced by  t ∂tk+2 u2m−1−k dθ + α0 ∇∂tk+1 u(t)2m−1−k 0

 ≤

a1 ∇uk+1 2m−1−k

(2.5.60) +C

k+1  

t 0

t

+ 0

∂tk+1 f 2m−1−k dθ

∂tr aij 2s−r ∂tk+1−r ∂i ∂j u2m−k+r−1 dθ

r=1 t +C ∇aij 2s−1 ∇∂tk+1 u2m−1−k dθ , 0

in the right side of which the first three terms are finite: the first, if (2.5.59) holds, the second, by (2.5.20), and the third is estimated as the term R4 in (2.5.40). 3) As we remarked at the end of section 2.3 for the hyperbolic problem (2.1.1)+(2.1.2), the regularity results for (2.1.5)+(2.5.1), both global (i.e., in [0, T ]) and for t > 0 (i.e., in [τ, T ]), can be improved in an analogous way. In particular, if the data and the coefficients are C ∞ , the solution of problem (2.1.5)+(2.5.1) is C ∞ as well.  2.5.3. Sobolev and H¨ older Solutions. We conclude by showing that Theorem 2.5.1 is consistent with the classical solvability results of the parabolic Cauchy problem (2.1.5)+(2.5.1) in

2.5. The Parabolic Cauchy Problem

117

the H¨older spaces C 1+α/2,2+α(Q), Q = ]0, T [ ×RN . Indeed, assuming again for simplicity that bj , c ≡ 0, take m = s =  N2  + 2 in Theorem 2.5.1, and assume that (2.5.61)

u0 ∈ H s+1 ,

f ∈ L2 (0, T ; H s ) ,

ft ∈ L2 (0, T ; H s−2 ) ,

as well as (2.5.62)

aij ∈ L2 (0, T ; H s+1 ) ,

∂t aij ∈ L2 (0, T ; H s−2 ) .

Then, by the Sobolev imbedding (1.5.61) and Theorem 1.7.5 (with, respectively, m = s + 1, r = 2, and m = s − 1, r = 0 in their statements), it follows that (2.5.63)

u0 ∈ C 2,α (RN ) ,

aij , f ∈ C α/2,α (Q)

(0 < α ≤ 12 ) .

As a consequence of theorem 8.10.1 of Krylov [83], conditions (2.5.63) are sufficient to yield a unique solution u ˜ ∈ C 1+α/2,2+α (Q) to problem (2.1.5)+(2.5.1). On the other hand, by (2.5.62) we can apply Theorem 2.5.2 and part 4 of Theorem 2.5.1, with k = 1, to deduce that problem (2.1.5)+(2.5.1) also has a unique solution u ∈ L2 (0, T ; H s+2 ), with utt ∈ L2 (0, T ; H s−2 ). Again by Theorem 1.7.5 (with m = s + 1 and r = 1), it follows that u ∈ C 1+α/2,2+α (Q). By the uniqueness of both classical and Sobolev solutions, it follows that u = u ˜, which shows the asserted consistency. Thus, we obtain the commutative diagram Sobolev data (2.5.64)

−→

↓ Sobolev solution

H¨older data ↓

−→

H¨ older solution

where the horizontal arrows refer to the Sobolev imbeddings. In the same way, we can show that, if f and the coefficients are more regular, in the sense of part 4 of Theorem 2.5.1 with k > 1, the H¨older and Sobolev additional regularity of u are also consistent (the former, as per theorem 8.12.1 of Krylov [83]).

Chapter 3

Quasi-linear Equations

3.1. Introduction 1. In this chapter we consider the Cauchy problem for the quasi-linear hyperbolic evolution equation (3.1.1)

utt − aij (Du) ∂i ∂j u = f (t, x) ,

where, again, summation for i, j from 1 to N is understood, and Du = {ut , ∇u}. As in Chapter 2, we take (t, x) in the cylinder Q = ]0, T [×RN , where T > 0 is fixed but arbitrary, and we seek solutions of (3.1.1) which satisfy the initial conditions (2.1.2), i.e., (3.1.2)

u(0, x) = u0 (x) ,

ut (0, x) = u1 (x) ,

where u0 and u1 are given functions on RN . Our goal is to show that the Cauchy problem (3.1.1)+(3.1.2) is locally well-posed (in time), in a suitable class of Sobolev spaces of the type introduced in section 2.2. More precisely, we wish to show that, under suitable assumptions on the data u0 , u1 , f , and the coefficients aij , there exists a τ ∈ ]0, T ], and a unique u ∈ Zs (τ ) (s ∈ N, s > N2 + 1), which is a strong solution of (3.1.1)+(3.1.2) on [0, τ ]. In fact, these solutions are also classical. 2. We shall solve equation (3.1.1) by means of the linearization and fixed point method introduced by Kato [70, 71, 72]. In general terms, this means that we fix a function w and consider, instead of the nonlinear equation (3.1.1), its linearized version (3.1.3)

utt − aij (Dw) ∂i ∂j u = f (t, x) , 119

120

3. Quasi-linear Equations

which we proceed to solve by means of the linear theory developed in Chapter 2. This requires sufficient assumptions on the function w to guarantee that the coefficients aw ij (t, x) := aij (Dw(t, x)) of (3.1.3) satisfy the assumptions of Theorem 2.2.1, so that (3.1.3) can indeed be uniquely solved. The corresponding solution u depends on w; this defines a map Γ : w → u = Γ(w), and ends the linearization part of the method. In this introduction, we summarize the required assumptions on w, by writing that w belongs to some function space C. The fixed point part of the method consists in trying to determine a specific space C, so that Γ maps C into itself, and is a strict contraction with respect to a suitable metric. If we are able to do so, Γ will have a unique fixed point u ∈ C, which is evidently a solution of the original equation (3.1.1). As it turns out, C will be determined as one element of a family (C(τ ))0 0, when the size of the data is sufficiently small.

3.1. Introduction

121

4. Equation (3.1.1) is the simplest example of the type of quasi-linear hyperbolic evolution equations we consider. More generally, the coefficients aij may depend also on t, x, u, as in (0.0.2), and the source term f may depend also on u and Du. As long as we confine ourselves to local solutions, such additional dependence does not increase the difficulty of the problem; but the dependence on u or Du may prevent the extendibility of a local solution to a global one. We could also consider equations in so-called divergence form; that is, similarly to (2.1.3), (3.1.5)

utt − ∂j (aij (Du) ∂i u) = f ,

as well as equations in conservation form (3.1.6)

utt − div (a(∇u)) = f ,

where a : RN → RN is monotone. As in Chapter 2, the results we establish for (3.1.1) are not specifically dependent on the fact that the equation is hyperbolic; in fact, the linearization and fixed point method can be readily adapted to obtain local, strong solutions of the quasi-linear parabolic equation (3.1.7)

ut − aij (∇u) ∂i ∂j u = f ,

as well as of the analogous equations in divergence and conservation form. On the other hand, as we shall see in Chapter 4, the question of extendibility of local solutions of quasi-linear parabolic equations to almost global ones has somewhat more favorable answers, due to the availability of the maximum principle. 5. In a surprising difference with the linear case, we are not able to consider weak solutions of quasi-linear hyperbolic equations, neither in the divergence form (3.1.5), nor in the conservation form (3.1.6). Indeed, results in this direction are almost nonexistent, except for a well-developed theory for quasi-linear systems in conservation form in one space dimension; that is, systems of the form (3.1.8)

ut + (F (u))x = 0 ,

where u = u(t, x) ∈ RM , (t, x) ∈ R>0 × R1 , and F ∈ C 1 (RM ; RM ). Among the most recent references on this topic, we refer, e.g., to Dafermos [39], Lu [105], Bressan [18], and Alinhac [5]. The question of the existence of even a local weak solution to equation (3.1.6) (that is, in the space Y0 (τ ), for some τ ∈ ]0, T ]) is, as far as we know, totally open (unless, of course, a is linear). In contrast, existence, uniqueness, and well-posedness results for weak solutions of parabolic quasi-linear equations in conservation form (3.1.9)

ut − div (a(∇u)) = f ,

122

3. Quasi-linear Equations

with a strongly monotone, are available; see, e.g., Lions [99, ch. 2, sct. 1] or Br´ezis [19]. 6. Throughout this chapter, we keep exactly the same notations and conventions that we have introduced in Chapters 1 and 2. In particular, s will always denote an integer, with s > N2 + 1. This implies that H s−1 is an algebra, as well as the Sobolev imbedding (2.1.6), which, as we have already had occasion to realize in Chapter 2, will play an essential role in the sequel. For notational simplicity, we assume that the imbedding H s−1 → L∞ has unitary norm; that is, that for all u ∈ H s−1 , |u|∞ ≤ us−1 .

(3.1.10)

Likewise, we agree to not write explicitly the constants that appear at the right side of estimates such as the Gagliardo-Nirenberg inequalities (1.5.42) or of Sobolev product estimates like (1.5.74), unless of course the specific value of a constant plays an essential role in the estimate.

3.2. The Hyperbolic Cauchy Problem In this section we prove a local existence and uniqueness result for strong solutions of the quasi-linear hyperbolic Cauchy problem (3.1.1)+(3.1.2). We assume that the coefficients aij in (3.1.1), which are functions of the 1 + N variables ∂t u, ∂1 u, . . . , ∂N u, are smooth, symmetric, and uniformly strongly elliptic. More precisely, we assume that aij ∈ C s (R1+N ), with aij (p) = aji (p) for all p ∈ R1+N , and that there exists α0 > 0 such that, for all p ∈ R1+N and ξ ∈ RN , aij (p) ξ i ξ j ≥ α0 |ξ|2

(3.2.1)

(compare to (2.2.1)). We also assume that the data u0 , u1 , and f satisfy (3.2.2)

u0 ∈ H s+1 ,

u1 ∈ H s ,

f ∈ Vs−1 (T ) ,

for some T > 0. We call Gs (T ) := H s+1 × H s × Vs−1 (T ) the space where the data u0 , u1 and f are taken. For (u0 , u1 , f ) ∈ Gs (T ), we define 1/2  (3.2.3) I1 = I1 (u0 , u1 , f ) := u0 2s+1 + u1 2s + f 2Vs−1 (T ) . Given also M , R > 0, we set (3.2.4)

DM (T ) := {(u0 , u1 , f ) ∈ Gs (T ) | I1 (u0 , u1 , f ) ≤ M } ,

(the closed ball of Gs (T ) of radius M ), and (3.2.5) Bs (T, R) := {u ∈ Zs (T ) | uZs (T ) ≤ R ,

u(0) = u0 ,

ut (0) = u1 } .

3.2. The Hyperbolic Cauchy Problem

123

To see that there do exist values of T and R such that Bs (T, R) is not empty, we first prove Proposition 3.2.1. Let s ≥ 0, u0 ∈ H s+1 , and u1 ∈ H s . There exists u ∈ C(R; H s+1 ) ∩ C 1 (R; H s ), such that u(0) = u0 , ut (0) = u1 , and for all t ≥ 0, (3.2.6)

  ut (t)2s + u(t)2s+1 ≤ 84 u1 2s + u0 2s+1 .

Proof. We follow Shibata and Kikuchi [147, App. 5]. With i = set

√

− 1, we

(3.2.7)

 √ √ 2 2 ψ(t, ξ) := 2 e(i−1) 1+|ξ| t − e2(i−1) 1+|ξ| t u ˆ0 (ξ) −

u √ 1  (i−1)√1+|ξ|2 t ˆ1 (ξ) 2 − e2(i−1) 1+|ξ| t ( . e i−1 1 + |ξ|2

Then, (3.2.8) ψt (t, ξ) := 2 (i − 1) −



(

e(i−1)

 √ √ 2 2 ˆ0 (ξ) 1 + |ξ|2 e(i−1) 1+|ξ| t − e2(i−1) 1+|ξ| t u

√

1+|ξ|2 t

− 2 e2(i−1)

√

1+|ξ|2 t

u ˆ1 (ξ) ,

so that (3.2.9)

ψ(0, ξ) = u ˆ0 (ξ) ,

ψt (0, ξ) = u ˆ1 (ξ) .

Thus, defining u = u(t, x) by u ˆ(t, ξ) = ψ(t, ξ), (3.2.9) implies that u(0) = u0 and ut (0) = u1 , as desired. To show (3.2.6), recalling (1.5.15) we compute from (3.2.7) that  u(t)2s+1 (3.2.10)

(1 + |ξ|2 )s+1 |ψ(t, ξ)|2 dξ  ≤ 20 (1 + |ξ|2 )s+1 |u0 (ξ)|2 dξ  + 4 (1 + |ξ|2 )s |u1 (ξ)|2 dξ

=

≤ 20 u0 s+1 + 4 u1 2s , 2

124

3. Quasi-linear Equations

2 2 2 2 2 having recalled √ that (a + b + c + d) ≤ 4 (a + b + c + d ). Likewise, since also |i − 1| = 2, we obtain from (3.2.8) that  ut (t)2s = (1 + |ξ|2 )s |ψt (t, ξ)|2 dξ  ≤ 64 (1 + |ξ|2 )s+1 |u0 (ξ)|2 dξ (3.2.11)  + 20 (1 + |ξ|2 )s |u1 (ξ)|2 dξ

≤ 64 u0 s+1 + 20 u1 2s . 2 

Adding (3.2.10) and (3.2.11) yields (3.2.6).

As a consequence of Proposition 3.2.1, it follows that for all R > 0 such that   (3.2.12) R2 ≥ 84 u1 2s + u0 2s+1  (with s > N2 , and for all T > 0, the ball Bs (T, R) is not empty. Remark. Proposition 3.2.1 can be generalized to more than two functions; namely, given s + 2 functions uj ∈ H s+1−j , 0 ≤ j ≤ s + 1, it is possible to s+1 0 j find u ∈ C (R; H s+1−j ), such that ∂tj u(0) = uj for all j = 0, . . . s + 1, j=0

and for all t ≥ 0, (3.2.13)

∂tj u(t)2s+1−j

≤C

s+1 

uk 2s+1−k ,

k=0

with C independent of u0 , . . . , us+1 . For a proof, see Shibata and Kikuchi [147].  In the remainder of these lectures, we assume for simplicity that the ellipticity constant in (3.2.1) satisfies (3.2.14)

α0 ≥ 1 .

This choice introduces no real loss of generality, as it will almost always be clear or, otherwise, it will be explicitly shown. 3.2.1. Strong Solutions. The main result of this chapter is concerned with the local solvability of the quasi-linear hyperbolic equation (3.1.1). Theorem 3.2.1. Under the above stated assumptions on the data and the coefficients of (3.1.1) (in particular, aij ∈ C s (R1+N ), s > N2 + 1), the following holds.

3.2. The Hyperbolic Cauchy Problem

125

(1) Local Existence: There exist τ ∈ ]0, T ] and R > 0, such that the Cauchy problem (3.1.1)+(3.1.2) admits a unique strong solution u ∈ Bs (τ, R). In particular, equation (3.1.1) holds in H s−1 for all t ∈ [0, τ ]. In fact, u is a classical solution of (3.1.1) in Qτ :=, ]0, τ [ ×RN ; that is, u ∈ Cb2 (Qτ ), and equation (3.1.1) holds pointwise in Qτ . Both τ and R depend on I1 (u0 , u1 , f ); more precisely, if (u0 , u1 , f ) ∈ DM for some M > 0, then τ and R depend (only) on M , and are, respectively, decreasing and increasing in M. (2) Local Well-Posedness: Given M > 0, let τM ∈ ]0, T ] and RM > 0 be the values of τ and R determined in part 1 above. For 1 ≤ m ≤ s − 1, the map Φ : DM → Bs (τM , RM ) defined in (2.2.15), that is, (3.2.15)

(u0 , u1 , f ) → u =: Φ(u0 , u1 , f ) , is locally Lipschitz continuous from H m+1 × H m × Vm−1 (T ) into Zm (τM ); that is, the estimate

(3.2.16) Φ(u0 , u1 , f ) − Φ(˜ u0 , u ˜1 , f˜)Zm (τM )  1/2 ≤ ΛM u0 − u ˜0 2m+1 + u1 − u ˜1 2m + f − f˜2Vm−1 (T ) holds, with ΛM depending on M , via τM and RM . Thus, there is at most one solution of (3.1.1)+(3.1.2) in Zm (τM ) (and, therefore, in Zs (τM )). In addition, if aij ∈ C s+1 (R1+N ), Φ is continuous from DM into Bs (τM , RM ). (3) Regularity: If, for some k ∈ N≥0 , aij ∈ C s+k (R1+N ), and (3.2.17)

u0 ∈ H s+1+k ,

u1 ∈ H s+k ,

f ∈ Vs−1+k (T ) ,

then the solution u ∈ Bs (τ, R) determined in part 1 above is such that u ∈ Bs+k (τ, R  ), for some R  ≥ R depending on R. That is, increasing the regularity of the data does not decrease the length of the time interval on which u is defined. Remark. The condition f ∈ Vs−1 (T ) can be replaced by the weaker requirement that f ∈ L2 (0, T ; H s ). As in the linear problem, this would still yield a local solution u, but with u ∈ Ys (τ ) only. The stronger requirement allows us to treat all second order derivatives equally, thereby simplifying our proofs considerably.  3.2.2. Preliminary Lemmas. In this section we present some results, based on Proposition 1.5.8, concerning the estimate of the factor aij (Dw) in equation (3.1.3). In turn, ˜ s−1 × H s−1 → H s−1 , these results will imply since, by Theorem 1.5.5, H

126

3. Quasi-linear Equations

corresponding estimates on utt , by means of product estimates like (1.5.94) on the term aij (Dw) ∂i ∂j u in H s−1 (compare to the conclusions of part 4 of section 2.3.1, leading to Theorem 2.3.1). 1. For r ∈ R1+N , we set a ˜ij (r) := aij (r)−aij (0), and denote, as usual, by aij the gradient of the function R1+N  p → aij (p). Note that a ˜ij (0) = 0,   and a ˜ij = aij . Recalling the definition of hm,ϕ in (1.5.155), with C0 as in (1.5.159) we set, for 0 ≤ r ≤ s and λ > 0, ⎧ if r =0, ⎨ C0 max h0,aij (λ) i,j (3.2.18) ψr (λ) := ⎩ C0 max hr,aij (λ) (1 + λr−1 ) if r = 0 , 

i,j

and ψ(λ) := max{ψr (λ) | 0 ≤ r ≤ s} .

(3.2.19)

In accord with (1.1.5), if u = u(t, x) is a smooth function we set D 2 u := (utt , ∇ut , ∂x2 u). Finally, for r > 0 we define Λ(r) := max{1, r}. Lemma 3.2.1. Let τ ∈ ]0, T ]. Then: ˜ s ), and for all t ∈ [0, τ ], 1) If w ∈ Xs (τ ), then aij (Dw) ∈ C([0, τ ]; H aij (Dw(t))s˜ ≤ ψ(|Dw(t)|∞ ) Λ(Dw(t)s ) .

(3.2.20)

2) If also w ∈ Zs (τ ), then D(aij (Dw)) ∈ C([0, τ ]; H s−1 ), and for all t ∈ [0, τ ] (3.2.21) D(aij (Dw(t)))s−1 ≤ ψ(|Dw(t)|∞ ) Λ(Dw(t)s−1 ) D 2 w(t)s−1 . 3) If w, u ∈ Zs (τ ), setting ψ˜0 (t) := ψ(|Dw(t)|∞ + |Du(t)|∞ ) , ψ˜1 (t) := Dw(t)s−1 + Du(t)s−1 , ψ˜2 (t) := Dwt (t)s−1 + Dut (t)s−1 ,

(3.2.22) (3.2.23) (3.2.24) the estimates (3.2.25)

aij (Dw(t)) − aij (Du(t))m ≤ ψ˜0 (t) ψ˜1 (t) Dw(t) − Du(t)m

and

(3.2.26)

∂t (aij (Dw(t)) − aij (Du(t)))m−1 ≤ ψ˜0 (t) ψ˜1 (t) Dwt (t) − Dut (t)m−1 + ψ˜0 (t) ψ˜2 (t) Dw(t) − Du(t)m−1 ,

where 0 ≤ m ≤ s − 1, hold, pointwise in [0, τ ]. These estimates also hold for m = s if aij ∈ C s+1 (R1+N ), with appropriate modification of the definition of ψ in (3.2.19); that is, taking 0 ≤ r ≤ s + 1.

3.2. The Hyperbolic Cauchy Problem

127

4) If w, u ∈ Xs (τ ), then aij (Dw) ∂i ∂j u ∈ C([0, τ ]; H s−1 ), and for all t ∈ [0, τ ], (3.2.27) aij (Dw(t)) ∂i ∂j u(t)s−1 ≤ ψ(|Dw(t)|∞ ) Λ(Dw(t)s−1 ) ∇u(t)s . 5) When w = u, (3.2.27) can be modified into (3.2.28)

aij (Du(t)) ∂i ∂j u(t)s−1 ≤ 2 ψ(|Du(t)|∞ ) Du(t)s .

We explicitly remark that, in (3.2.28), the factor Du(t)s appears with exponent 1; this is crucial in the sequel. Proof. 1) By Proposition 1.7.7, Dw ∈ Cb (Qτ ); thus, by Proposition 1.5.9, ˜ s ). We first estimate aij (Dw) in aij (Dw) = a ˜ij (Dw) + aij (0) ∈ C([0, τ ]; H Cb (RN ); with abuse of notation, we use the same notation | · |∞ for the norm in Cb (RN ). By (1.5.157) of Proposition 1.5.8, (3.2.29) |aij (Dw)|∞ ≤ h0,aij (|Dw|∞ ) ≤ ψ0 (|Dw|∞ ) ≤ ψ(|Dw|∞ ) Λ(Dws ) . Next, by (1.5.159), ∇aij (Dw)s−1 =



|∂xα ∇aij (Dw)|2

|α|≤s−1

(3.2.30)

≤



ψ|α|+1 (|Dw|∞ ) |∂xα ∇Dw|2

|α|≤s−1

≤ ψs (|Dw|∞ ) ∇Dws−1 ≤ ψ(|Dw|∞ ) Λ(Dws ) . Together with (3.2.29), (3.2.30) proves (3.2.20) and part of (3.2.21). 2) To complete the proof of (3.2.21), we write (3.2.31)

∂t (aij (Dw)) = aij  (Dw) · Dwt ;

˜ s ); and since also Dwt ∈ again by Proposition 1.5.9, aij  (Dw) ∈ C([0, τ ]; H s−1 s−1 s−1 s−1 ˜ C([0, τ ]; H ), and H ·H → H by Theorem 1.5.5, it follows that ∂t (aij (Dw)) ∈ C([0, τ ]; H s−1 ) as well. In addition, from (3.2.31), and (3.2.20) with aij and s replaced by aij  and s − 1, (3.2.32)

∂t (aij (Dw))s−1 ≤ aij  (Dw) Dwt s−1 s−1 ≤ ψ(|Dw|∞ ) Λ(Dws−1 ) Dwt s−1 .

Together with (3.2.30), (3.2.32) implies (3.2.21).

128

3. Quasi-linear Equations

3) Estimate (3.2.25) follows from (1.5.180) and (1.5.187); an analogous argument, applied to the identity (3.2.33)

∂t (aij (Dw) − aij (Du)) = aij  (Dw) · (Dwt − Dut ) + (aij  (Dw) − aij  (Du)) · Dut ,

allows us to prove (3.2.26). Note that to carry over these arguments to the case m = s, we need to be able to differentiate s times the difference (3.2.34)



1

aij (Dw) − aij (Du) =

aij  (λ Dw + (1 − λ)Du) · (Dw − Du) dλ ,

0

which explains why the validity of (3.2.25) and (3.2.26) for m = s requires the additional requirement that aij ∈ C s+1 (R1+N ). 4) To show (3.2.27), we proceed as in the estimates of Propositions 1.5.8 and 1.5.9. Let α ∈ NN , with |α| ≤ s − 1. By Leibniz’ formula,   α β α−β (3.2.35) ∂xα (aij (Dw) ∂i ∂j u) = ∂i ∂j u . β ∂x aij (Dw) ∂x β≤α

When β = 0, recalling (3.2.29), (3.2.36)

|aij (Dw) ∂xα ∂i ∂j u|2 ≤ |aij (Dw)|∞ ∇us ≤ ψ(|Dw|∞ ) ∇us .

When β > 0, we deduce from Proposition 1.5.8 that ∂xβ aij (Dw) ∈ H s−1−|β| ; since also ∂xα−β ∂i ∂j u ∈ H s−1−|α|+|β| , and, as we immediately verify, H s−1−|β| · H s−1−|α|+|β| → L2 ,

(3.2.37) we can proceed with (3.2.38)

∂xβ aij (Dw) ∂xα−β ∂i ∂j u ≤ C ∂xβ aij (Dw)s−1−|β| ∂xα−β ∂i ∂j us−1−|α|+|β| ≤ C ∇aij (Dw)s−2 ∇us ≤ ψ(|Dw|∞ ) Dws−1 ∇us . Together with (3.2.36), this implies (3.2.27). 5) Finally, let w = u. We refer again to (3.2.35): when β > 0, we define |β| q ∈ ]1, +∞[ by 1q := |α|+1 , and use the Gagliardo-Nirenberg inequalities to obtain that (3.2.39)

|∂xβ aij (Du)|2 q ≤ ψ|β| (|Du|∞ ) |∂xβ Du|2 q |α|+1

≤ ψs−1 (|Du|∞ ) |∂x

1/q

1−1/q

Du|2 |Du|∞

3.2. The Hyperbolic Cauchy Problem

129

and (3.2.40)

|∂xα−β ∂i ∂j u|2 q  ≤ C |∂x|α|+1 ∇u|2

1−1/q

|∇u|1/q ∞ .

Thus, (3.2.41)

|∂xβ aij (Du) ∂xα−β ∂i ∂j u|2 ≤ C |Du|∞ ψs−1 (|Du|∞ ) Dus .

Noting that λ ψs−1 (λ) ≤ 2 ψs (λ), we see that (3.2.28) follows from (3.2.41) and (3.2.36) (with w = u). This completes the proof of Lemma 3.2.1.  Remark. If (3.1.1)+(3.1.2) has a local solution u ∈ Xs (τ ), then part 4 of Lemma 3.2.1 implies that (3.2.42)

u2 := f (0) + aij (u1 , ∇u0 ) ∂i ∂j u0 ∈ H s−1 .

However, it is important to realize, as in (2.5.57) and (2.5.58), that this result is independent of the actual existence of a solution. More precisely, (3.2.42) is a direct consequence of the assumptions on the data and the coefficients, and its proof follows from Theorem 1.5.5, with  = s, r = s − 1, ˜ s. and aij (u1 , ∇u0 ) ∈ H  With some abuse of notation, we define (3.2.43)

Du0 := (u1 , ∇u0 ) ∈ (H s )2 ,

Du1 := (u2 , ∇u1 ) ∈ (H s−1 )2 .

2. The first part of Lemma 3.2.1 provides an estimate of aij (Dw(t)) in s ˜ ˜ s−1 . We H . We shall also need the following estimate of aij (Dw(t)) in H define (3.2.44) a1 (Du0 ) := max |aij (Du0 )|∞ , i,j

a3 (Du0 ) := max aij (Du0 ) ; s−1 i,j

note that, by (1.5.6), a1 (Du0 ) ≤ a3 (Du0 ). Lemma 3.2.2. Let w ∈ Bs (τ, R), for some τ ∈ ]0, T ] and R > 0. Then, for all t ∈ [0, τ ], (3.2.45)

aij (Dw(t)) ≤ a3 (Du0 ) + t ψ(R) Λ(R) R . s−1

Consequently, if also u ∈ Xs (τ ), (3.2.46)

aij (Dw(t)) ∂i ∂j u(t)s−1 ≤ (a3 (Du0 ) + t ψ(R) Λ(R) R) ∇us .

130

3. Quasi-linear Equations

˜ s−1 ; Proof. By the second part of Lemma 3.2.1, ∂t (aij (Dw(t))) ∈ H s−1 → H thus, recalling (3.2.21) and that w(0) = u0 and wt (0) = u1 , (3.2.47)



t

aij (Dw(t)) ≤ aij (Dw(0)) + ∂t (aij (Dw)) dθ s−1 s−1 s−1 0  t ≤ a3 (Du0 ) + ∂t (aij (Dw))s−1 dθ 0  t ≤ a3 (Du0 ) + ψ(R) Λ(R) R dθ , 0

from which (3.2.45) follows. Finally, (3.2.46) is an immediate consequence ˜ s−1 × H s−1 → H s−1 . of (3.2.45), via the imbedding H  3.2.3. Linear Estimates. We conclude with a result which, in a certain sense, takes the role of the a priori estimates of section 2.3.1. Lemma 3.2.3. Assume that problem (3.1.1)+(3.1.2) has a solution u ∈ Bs (τ, R), for some τ ∈ ]0, T ] and R > 0. There are constants β0 and γ0 , depending respectively on a1 (Du0 ) of (3.2.44) and R, such that, for 0 ≤ m ≤ s and all t ∈ [0, τ ], (3.2.48)



u(t)2 + Du(t)2m ≤ β0 eβ0 γ0 t



t

u0 2m+1 + u1 2m +

f 2m dθ

.

0

Proof. Since u ∈ Bs (τ, R) ⊂ Zs (τ ), we can consider (3.1.1) as a linear equation of type (2.1.1), with known coefficients (3.2.49)

auij (t, x) := aij (Du(t, x)) .

The conditions on the coefficients aij ensure that the auij satisfy the assumptions of Theorem 2.2.1, with m = s and T replaced by τ . In particular, the auij satisfy (2.2.1) and (2.2.2). To see this explicitly, the first inequality of (2.2.1) follows from (3.2.1), while the second follows from the first claim of ˜ s ) ⊂ L∞ (Q), Lemma 3.2.1 and (3.2.29), which imply that auij ∈ C([0, τ ]; H and (3.2.50)

|auij (t, x)| ≤ ψ0 (|Du(t)|∞ ) ≤ ψ0 (Du(t)s−1 ) ≤ ψ0 (R) .

Note also that the definition of a1 (Du0 ) in (3.2.44) is consistent with the definition of a1 in (2.3.18), because, by (3.2.49), (3.2.51)

a1 (Du0 ) = max |auij (0, ·)|∞ ; i,j

3.3. Proof of Theorem 3.2.1

131

consequently, recalling (2.3.13),



Qs (au (0), ∇u0 ) = (3.2.52)

aij (Du0 ) ∂i ∂xα u0 , ∂j ∂xα u0 

|α|≤s

≤ a1 (Du0 ) ∇u0 2s . As for (2.2.2), the second claim of Lemma 3.2.1 shows that D(auij ) ∈ C([0, τ ]; H s−1 ) ⊂ L1 (0, τ ; H s−1 ) ,

(3.2.53)

and, by (3.2.21), for all t ∈ [0, τ ], D auij (t)s−1 ≤ ψ(R) Λ(R) R .

(3.2.54)

As a consequence, u satisfies an estimate analogous to (2.3.23), that is, for 0 ≤ m ≤ s, (3.2.55)



u(t) + 2

Du(t)2m 

+ C a22

t

≤

a22

 u0 2m+1

+

u1 2m

t

+

f 2m dθ

0

  (1 + Dauij (θ)s−1 ) u2m + Du2m dθ ,

0

where a2 = a2 (Du0 ) is defined similarly as in (2.3.22), but with a1 replaced by a1 (Du0 ); that is, recalling (3.2.14), a22 := max{1, a1 (Du0 )} .

(3.2.56)

In the sequel, we shall abbreviate a1 := a1 (Du0 ) and a2 := a2 (Du0 ), and set β0 := C a22 ; note that β0 depends only on the initial values u0 and u1 (more precisely, by the first of (3.2.44), on |aij (Du0 )|∞ ). Recalling (3.2.54), and setting γ0 := 1 + ψ(R) Λ(R) R, we deduce from (3.2.55) that (3.2.57) u(t) + 2

Du(t)2m

 t   2 2 ≤ β0 u0 m+1 + u1 m + f 2m dθ 0



t

+ γ0

(u(t)2m + Du(t)2m ) dθ ,

0

from which (3.2.48) follows, by Gronwall’s inequality.



3.3. Proof of Theorem 3.2.1 In this section we describe the linearization and fixed point technique we follow to prove Theorem 3.2.1. We proceed in five steps. First, we linearize equation (3.1.1); that is, we consider equation (3.1.3), and define the map Γ : w → u =: Γ(w), where u is the unique solution of (3.1.3). Then, we give sufficient conditions of τ and R, which ensure that Γ is a strict

132

3. Quasi-linear Equations

contraction on Bs (τ, R), and that the resulting fixed point is the unique local solution of (3.1.1)+(3.1.2) on [0, τ ]. We next establish the well-posedness estimates (3.2.16), and then the continuity of Φ on DM . Finally, we prove the regularity result stated in the last part of the Theorem.

3.3.1. Step 1: Linearization. 1. We linearize (3.1.1) in the following way. For τ ∈ ]0, T ] and R > 0 to be determined, we fix w ∈ Bs (τ, R) and consider the linear Cauchy problem

(3.3.1)

⎧ ⎨ utt − aw ij (t, x)∂i ∂j u = f (t, x) , ⎩ u(0) = u , 0

ut (0) = u1 ,

with aw ij (t, x) := aij (Dw(t, x)) as in (3.2.49). Reasoning as in Lemma 3.2.3, the fact that w ∈ Bs (τ, R) ⊂ Zs (τ ) implies that the coefficients aw ij satisfy the assumptions of Theorem 2.2.1, with m = s, and T replaced by τ . In particular, they satisfy conditions (2.2.1) and (2.2.2). Thus, we can apply Theorem 2.3.1, and deduce that the Cauchy problem (3.3.1) has a unique solution u ∈ Zs (τ ). The fact that u is uniquely determined by w allows us to define a map Γ : Bs (τ, R) → Zs (τ ), by w → u =: Γ(w). We now proceed to put sufficient restrictions to τ and R, so that Γ maps the corresponding ball Bs (τ, R) into itself, and is a strict contraction, at least with respect to a weaker norm. 2. Our first restriction on τ and R is described in Proposition 3.3.1. Let the map Γ be defined as above. There exist R∗ > 0 and τ0 ∈ ]0, T ], such that for all τ ∈ ]0, τ0 ], Γ maps Bs (τ, R∗ ) into itself. Proof. Fix w ∈ Bs (τ, R), with τ ∈ ]0, T ] and R > 0 to be determined. Proceeding exactly as in the proof of Lemma 3.2.3, and recalling that w(0) = u0 and wt (0) = u1 , so that Dw0 = Du0 , we deduce an estimate analogous to (3.2.48), with m = s. More precisely, recalling the definitions of β0 and γ0 in Lemma 3.2.3, and setting β1 (R) := β0 γ0 = C a22 (1 + ψ(R) Λ(R) R), we obtain that for all τ ∈ ]0, T ], and all t ∈ [0, τ ], (3.3.2)

u(t)2 + Du(t)2s ≤ β0 I02 eβ1 (R) t ,

where I0 is as in (2.3.21), that is,  (3.3.3)

I02 := u0 2s+1 + u1 2s +

T 0

f 2s dθ .

3.3. Proof of Theorem 3.2.1

133

By (3.2.46) of Lemma 3.2.2, (3.3.4) utt (t)s−1 ≤ f (t)s−1 + aw ij (t) ∂i ∂j u(t)s−1 ≤ f (t)s−1 + (a3 (Du0 ) + t ψ(R) Λ(R) R) ∇u(t)s . Set (3.3.5)

ψ1 (Du0 , R, t) := (a3 (Du0 ) + t ψ(R) Λ(R) R)2 .

Adding (3.3.4) to (3.3.2) yields (3.3.6)

u(t)2 + Du(t)2s + utt (t)2s−1 ≤ (1 + 2 ψ1 (Du0 , R, t)) β0 I02 eβ1 (R) t + 2 f (t)2s−1 .

By (3.3.3) and (3.2.3), (3.3.7)

I02 + f (t)2s−1 ≤ u0 2s+1 + u1 2s + f 2Vs−1 (T ) = I12 ;

thus, we deduce from (3.3.6) that, for all t ∈ [0, τ ], u(t)2 + Du(t)2s + utt (t)2s−1 (3.3.8)

≤ (3 + 2 ψ1 (Du0 , R, t)) β0 I12 eβ1 (R) t =: ψ2 (Du0 , R, t) I12 .

We now note that ψ2 is continuous in t, and that, recalling (3.3.5), (3.3.9) ψ2 (Du0 , R, 0) = (3 + 2 ψ1 (Du0 , R, 0)) β0 = (3 + 2(a3 (Du0 ))2 ) β0 . This quantity depends on the norm Du0 s of the data u0 and u1 , but not on R. Thus, if we set (for example) ( (3.3.10) R∗ := 2 I1 β0 (3 + 2(a3 (Du0 ))2 ) , which is a quantity that depends only on u0 s+1 , u1 s and f Vs−1 (T ) , there exists τ0 ∈ ]0, T ] such that for all τ ∈ [0, τ0 ], (3.3.11)

ψ2 (Du0 , R∗ , τ ) I12 ≤ R∗2 .

Since ψ2 is increasing with respect to t, we deduce from (3.3.8) that, for all τ ∈ [0, τ0 ], and all t ∈ [0, τ ], (3.3.12)

u(t)2 + Du(t)2s + utt (t)2s−1 ≤ ψ2 (Du0 , R∗ , τ0 ) I12 ≤ R∗2 .

In turn, this implies that, for all τ ∈ [0, τ0 ], (3.3.13)

uZs (τ ) = Γ(w)Zs (τ ) ≤ R∗ .

This means that, if w ∈ Bs (τ, R∗ ), τ ≤ τ0 , then also u = Γ(w) ∈ Bs (τ, R∗ ). That is, Γ maps Bs (τ, R∗ ) into itself, as claimed. 

134

3. Quasi-linear Equations

3.3.2. Step 2: Contractivity. 1. We proceed to show that, if τ ≤ τ0 is sufficiently small, Γ is a contraction on Bs (τ, R∗ ), with respect to lower order norms. Proposition 3.3.2. Let 0 ≤ m ≤ s − 1 and σ ∈ ]0, 1[. There exists τ1 ∈ ]0, τ0 ] such that for all w, w ˜ ∈ Bs (τ1 , R∗ ), (3.3.14)

Γ(w) − Γ(w) ˜ Xm (τ1 ) ≤ σ w − w ˜ Xm (τ1 ) .

Proof. For τ ∈ ]0, τ0 ] to be determined, let w, w ˜ ∈ Bs (τ, R∗ ), and set u = Γ(w), u ˜ = Γ(w), ˜ z = u−u ˜. Then, by Proposition 3.3.1, both u, u ˜ ∈ Bs (τ, R∗ ), and z solves the Cauchy problem ⎧ ⎨ ztt − aw ij (t, x)∂i ∂j z = g , (3.3.15) ⎩ z(0) = 0 , zt (0) = 0 , where (3.3.16)

g := (aij (Dw) − aij (D w)) ˜ ∂i ∂j u ˜.

Since w, w ˜ and u ˜ ∈ Bs (τ, R∗ ) ⊂ Zs (τ ), it follows that g ∈ L2 (0, τ ; H m ). As in section 3.2.3, we can consider (3.3.15) as a linear problem, and deduce that z satisfies an estimate analogous to (3.2.48); more precisely, for t ∈ [0, τ ],  t

2 2 2 (3.3.17) z(t) + Dz(t)m ≤ β0 gm dθ eβ1 (R∗ ) t , 0

with β1 as in (3.3.2). By the imbedding Lemma 3.2.1,

H s−1

· H m → H m , and (3.2.25) of

gm ≤ aij (Dw) − aij (D w) ˜ m ∂i ∂j u ˜s−1 (3.3.18)

≤ ψ(2 R∗ )(2 R∗ ) Dw − D w ˜ m ∇˜ us ≤ ψ(2 R∗ )(2 R∗2 ) w − w ˜ Xm (τ ) .

Inserting this into (3.3.17), we obtain that, for all t ∈ [0, τ ], (3.3.19) z(t)2 + Dz(t)2m ≤ 4 β0 (ψ(2 R∗ ))2 R∗4 τ eβ1 (R∗ ) τ w − w ˜ 2Xm (τ ) . Thus, given σ ∈ ]0, 1[, if we choose τ1 ∈ ]0, τ0 ] such that (3.3.20)

4 β0 (ψ(2 R∗ ))2 R∗4 τ1 eβ1 (R∗ ) τ1 ≤ σ 2 ,

we deduce from (3.3.19) that, if τ ∈ ]0, τ1 ] and 0 ≤ t ≤ τ , (3.3.21)

z(t)2 + Dz(t)2m ≤ σ 2 w − w ˜ 2Xm (τ1 ) ,

from which (3.3.14) follows.



2. We are now ready to conclude the proof of the existence part of Theorem 3.2.1. Starting from an arbitrary function u0 ∈ Bs (τ1 , R∗ ), we

3.3. Proof of Theorem 3.2.1

135

consider the Picard iterations un+1 = Γ(un ); that is, explicitly, each un+1 is defined as the solution of the Cauchy problem ⎧ ⎨ un+1 − aij (D un (t, x))∂i ∂j un+1 = f (t, x) , tt (3.3.22) ⎩ un+1 (0) = u , un+1 (0) = u . 0 1 t Propositions 3.3.1 and 3.3.2 imply that the sequence (un )n≥0 is bounded in Zs (τ1 ), and a Cauchy sequence in Zm (τ1 ), 0 ≤ m ≤ s − 1. Thus, taking m = s − 1, there is a function u ∈ Zs (τ1 ), such that, as n → ∞, (3.3.23)

un → u

(3.3.24)

unt → ut

(3.3.25)

untt → utt

in L∞ (0, τ1 ; H s+1 ) weak∗ and C([0, τ1 ]; H s ) , in L∞ (0, τ1 ; H s )

weak∗ and C([0, τ1 ]; H s−1 ) ,

in L∞ (0, τ1 ; H s−1 ) weak∗ and C([0, τ1 ]; H s−2 )

(note that s − 2 ≥  N2  ≥ 0; we remark, in passing, that we cannot use Proposition 1.7.9 directly, since the imbedding H s+1 (RN ) → H s (RN ) is not compact). Moreover, for all t ∈ [0, τ1 ], (3.3.26)

Du(t)s−1 = lim Dun (t)s−1 ≤ R∗ .

Again by (3.2.25) of Lemma 3.2.1, (3.3.27)

aij (Dun ) − aij (Du)s−1 ≤ 2 R∗ ψ(2 R∗ ) Dun − Dus−1 .

Then, by (3.3.23) and (3.3.24), aij (Dun ) → aij (Du) in C([0, τ1 ]; H s−1 ). In turn, this implies that (3.3.28)

aij (Dun ) ∂i ∂j un+1 → aij (Du) ∂i ∂j u

in C([0, τ1 ]; H s−2 ) ,

as follows from (3.3.29) aij (Dun ) ∂i ∂j un+1 − aij (Du)∂i ∂j us−2 ≤ aij (Dun ) − aij (Du)s−1 ∂i ∂j un+1 s−2 + aij (Du) ∂i ∂j un+1 − ∂i ∂j us−2 s−1 ≤ 2 R∗2 ψ(2 R∗ ) Dun − Dus−1 + aij (Du) un+1 − us . s−1 The functions un satisfy the equations of (3.3.22) in C([0, τ1 ]; H s−1 ); by (3.3.28) and the second of (3.3.25), we can let n → ∞, and deduce that u (i.e., the fixed point of Γ) solves equation (3.1.1) in C([0, τ1 ]; H s−2 ). On the other hand, the first of (3.3.25) and (3.3.23) also imply, via the Sobolev product estimates and Proposition 1.5.8, that u solves (3.1.1) in H s−1 , for a.a. t ∈ [0, τ1 ]. 3. So far, the solution u we have constructed is such that (3.3.30)

u ∈ L∞ (0, τ1 ; H s+1 ) ∩ C([0, τ1 ]; H s ) ,

(3.3.31)

ut ∈ L∞ (0, τ1 ; H s ) ∩ C([0, τ1 ]; H s−1 ) .

136

3. Quasi-linear Equations

We can now consider (3.1.1) as linear, and argue as in Lemma 2.3.2 to deduce that u ∈ C([0, τ1 ]; H s+1 ), ut ∈ C([0, τ1 ]; H s ), and u(0) = u0 , ut (0) = u1 . Finally, since f ∈ Vs−1 (T ) → C([0, τ1 ]; H s−1 ), by claim 4 of Lemma 3.2.1 it follows that (3.3.32)

utt = f + aij (Du) ∂i ∂j u ∈ C([0, τ1 ]; H s−1 ) .

The function u is also a classical solution on Qτ : in fact, u ∈ Bs (τ, R) ⊂ Zs (τ ), so that, recalling (2.3.47) and Proposition 1.7.7, u, ut and   ∇u ∈ s 1 s−1 1 2 C([0, τ ]; H ) ∩ C ([0, τ ]; H ) → Cb Qτ . Hence, u ∈ Cb Qτ . This concludes the proof of the first part of Theorem 3.2.1, with τ = τ1 and R = R∗ . Note that if, as in (3.2.4), (u0 , u1 , f ) ∈ DM for some M > 0, then (3.3.10) and (3.3.11) imply that R and τ can be determined as, respectively, an increasing and a decreasing function of M ; that is, ultimately, of the size of the data u0 , u1 , and f .  3.3.3. Step 3: Lipschitz Estimates. In this section we prove the Lipschitz estimates (3.2.16). Given M > 0, let (u0 , u1 , f ) and (˜ u0 , u ˜1 , f˜) ∈ DM , and set u := Φ(u0 , u1 , f ), u ˜ := Φ(˜ u0 , u ˜1 , f˜), z := u − u ˜. Then, u, u ˜ and z ∈ Bs (τM , RM ) ⊂ Zs (τM ), and z solves a Cauchy problem analogous to (3.3.15); that is, recalling (3.2.49), ⎧ ⎪ ztt − auij ∂i ∂j z = f − f˜ + (aij (Du) − aij (D˜ u)) ∂i ∂j u ˜, ⎪ ⎪

  ⎪ ⎨ =: g (3.3.33) z(0) = u0 − u ˜0 =: z0 , ⎪ ⎪ ⎪ ⎪ ⎩ zt (0) = u1 − u ˜1 =: z1 . Again, g ∈ Vm−1 (τM ), 1 ≤ m ≤ s − 1. Set  (3.3.34) Dd := z0 2m+1 + z1 2m +

T

1/2 f − f˜2m dt

.

0

As in (3.2.48) of Lemma 3.2.3, z satisfies the estimate

 t 2 2 2 2 (3.3.35) z(t) + Dz(t)m ≤ ψ3 (M ) Dd + gm dθ , 0

for t ∈ [0, τM ], where ψ3 (M ) := β0 eβ1 (RM ) τM .

(3.3.36)

We estimate g as in (3.3.18); that is, (3.3.37)

2 g(t)m ≤ 2 RM ψ(2 RM ) Dz(t)m =: (ψ4 (M ))1/2 Dz(t)m .

Replacing this into (3.3.35) yields (3.3.38)

z(t) + 2

Dz(t)2m

≤ ψ3 (M )



 Dd2

t

+ ψ4 (M ) 0

Dz2m dθ

,

3.3. Proof of Theorem 3.2.1

137

from which, by Gronwall’s inequality, z(t)2 + Dz(t)2m ≤ Dd2 ψ3 (M ) eψ3 (M ) ψ4 (M ) τM ,

(3.3.39)

for all t ∈ [0, τM ]. From (3.3.33), we proceed with (3.3.40) ztt m−1 ≤ f − f˜m−1 + gm−1 + aij (Du)

∇zm  s−1

;

from (3.2.45), (3.3.41)

aij (Du) ≤ a3 (Du0 ) + τM ψ(RM ) Λ(RM ) RM =: ψ5 (M ) ; s−1

finally, as in (3.3.37), g(t)m−1 ≤ (ψ4 (M ))1/2 Dz(t)m−1 .

(3.3.42)

Replacing these estimates in (3.3.40), we obtain that (3.3.43) ztt (t)2m−1 ≤ 2 f (t) − f˜(t)2m−1 + ψ6 (M ) Dz(t)2m , for a suitable constant ψ6 depending on M via ψ4 (M ) and ψ5 (M ). Adding this to (3.3.39), and recalling that we always choose ψk ≥ 1, we deduce that (3.3.44) z(t)2 + Dz(t)2m + ztt (t)2m−1   ≤ (1 + ψ6 (M )) z(t)2 + Dz(t)2m + 2 f (t) − f˜(t)2m−1 ≤ (1 + ψ6 (M )) Dd2 ψ3 (M ) eψ3 (M ) ψ4 (M ) τM + 2 f (t) − f˜(t)2m−1 ≤



Dd2 + f − f˜2C([0,τm ];H m−1 ) Λ2M ,

with (3.3.45)

Λ2M := 2 ψ3 (M )(1 + ψ6 (M )) eψ3 (M ) ψ4 (M ) T .

Since, by (3.3.34), (3.3.46)

Dd2 + f − f˜2C([0,τm ];H m−1 ) ≤ u0 − u ˜0 2m+1 + u1 − u ˜1 2m + f − f˜2Vm−1 (T ) ,

we can conclude from (3.3.44) the proof of the Lipschitz estimates (3.2.16).  Remark. Using interpolation, we can deduce from (3.2.16) further wellposedness estimates in Zs−η (τM ), for η ∈ ]0, 1[. More precisely, the map Φ defined in (3.2.15) is H¨older continuous, with exponent η, from DM into Zs−η (τM ). This results from inserting (3.3.44) into the interpolation estimate (3.3.47)

Dzs−η ≤ C Dzs1−η Dzηs−1 ≤ C (2 RM )1−η Dzηs−1

and in the corresponding estimate of ztt s−1−η , obtained as in (3.3.40). 

138

3. Quasi-linear Equations

3.3.4. Step 4: Strong Well-Posedness. In this section we prove the continuity of the map Φ with respect to the norm of Zs (τM ). Note that our proof of (3.2.16) does not carry over to the highest value m = s, because, if u ˜ ∈ Zs (τM ), we can only deduce that ∂i ∂j u ˜ ∈ C([0, τM ]; H s−1 ), so that the function g defined in (3.3.33) need not be in C([0, τM ]; H s ). We overcome this difficulty, regularizing u ˜ by means of the Friedrichs’ mollifiers, as described in part 3 of section 1.7. 1. We follow a method first proposed by Beir˜ao da Veiga (see, e.g., [12]) and then extended in [120]. Our goal is to show that, for any (u0 , u1 , f ) ∈ DM and ε > 0, there is δ > 0 such that for any (˜ u0 , u ˜1 , f˜) ∈ DM , if (3.3.48)

u0 − u ˜0 2s+1 + u1 − u ˜1 2s + f − f˜2Vs−1 (T ) ≤ δ 2 ,

then (3.3.49)

Φ(u0 , u1 , f ) − Φ(˜ u0 , u ˜1 , f˜)Zs (τM ) ≤ ε .

As in section 3.3.3, let u := Φ(u0 , u1 , f ) and u ˜ := Φ(˜ u0 , u ˜1 , f˜). Then, u and u ˜ ∈ Bs (τM , RM ), and solve the equations utt − aij (Du) ∂i ∂j u = f , u ˜tt − aij (D˜ u) ∂i ∂j u ˜ = f˜ .

(3.3.50) (3.3.51)

For |α| ≤ s − 1, we can differentiate each of these equations α times with respect to the space variables. We obtain (3.3.52)

(∂xα u)tt − aij (Du) ∂i ∂j (∂xα u) = ∂xα f + ϕα ,

(3.3.53)

(∂xα u ˜)tt − aij (D˜ u) ∂i ∂j (∂xα u ˜) = ∂xα f˜ + ψα ,

where (3.3.54)

ϕα :=

   α β

∂xβ aij (Du) ∂xα−β ∂i ∂j u ,

0 N2 +1), the following holds. (1) Local Existence: There exist τ ∈ ]0, T ] and R > 0, such that the Cauchy problem (3.1.7)+(3.4.1) admits a unique strong solution ˜s (τ, R). In particular, equation (3.1.7) holds in H s−1 , for u ∈ B a.a. t ∈ ]0, τ [. Both τ and R depend on J1 (u0 , f ); more precisely, if ˜ M for some M > 0, then τ and R depend (only) on M , (u0 , f ) ∈ D and are, respectively, decreasing and increasing in M . In addition, if aij ∈ C s+1 , then u ∈ L2 (0, τ ; H s+2 ), and equation (3.1.7) holds in H s , for a.a. t ∈ ]0, τ [. (2) Local Well-Posedness: Given M > 0, let τM ∈ ]0, T ] and RM > 0 be the values of τ and R determined in part 1 above. For 0 ≤ ˜ :D ˜M → B ˜s (τM , RM ), defined by (u0 , f ) → m ≤ s − 1, the map Φ ˜ Φ(u0 , f ) := u is locally Lipschitz continuous with respect to the norms induced by H m+1 × L2 (0, T ; H m ) and Pm (τM ); in particular, there is at most one solution of (3.1.7)+(3.4.1) in Ps (τM ). ˜ is continuous from D ˜ M into In addition, if aij ∈ C s+1 (RN ), Φ ˜s (τM , RM ). B (3) Regularity: If, for some k ∈ N, aij ∈ C s+k (RN ), u0 ∈ H s+1+k ˜s (τ, R) determined and f ∈ L2 (0, T ; H s+k ), then the solution u ∈ B  ˜ in part 1 above is such that u ∈ Bs+k (τ, R ), for some R  ≥ R depending on R. That is, increasing the regularity of the data does not decrease the length of the time interval on which u is defined. Sketch of Proof. We proceed as in the proof of Theorem 3.2.1. ˜s (τ, R), and 1. For τ ∈ ]0, T ] and R > 0 to be determined, we fix w ∈ B consider the linear Cauchy problem  u t − aw ij (t, x) ∂i ∂j u = f , (3.4.7) u(0) = u0 , where aw ij (t, x) := aij (∇w(t, x)). These coefficients satisfy properties similar to those of Lemma 3.2.1; in particular, by Proposition 1.7.7, aw ij ∈   s−1 ) → L1 (0, τ ; H s−1 ), Cb Qτ → L∞ (Qτ ); in addition, ∇ aw ∈ C([0, τ ]; H ij 2 (0, τ ; H s−1 ) → L1 (0, τ ; H s−1 ). Thus, by Theorem 2.5.1, and ∂t aw ∈ L ij problem (3.4.7) has a unique solution u ∈ Ps (τ ). This defines a map ˜:B ˜s (τ, R) → Ps (τ ), of which we seek a fixed point. Γ 2. To this end, we prove Proposition 3.4.1. There exist R∗ > 0, τ0 ∈ ]0, T ], and τ∗ ∈ ]0, τ0 ], such ˜ maps B ˜s (τ, R∗ ) into itself, and is a strict contraction that for all τ ∈ ]0, τ0 ], Γ

3.4. The Parabolic Cauchy Problem

147

˜s (τ∗ , R∗ ), with respect to the weaker norm of P0 (τ∗ ); that is, for all w, on B ˜s (τ∗ , R∗ ), w ˜∈B ˜ ˜ w) (3.4.8) Γ(w) − Γ( ˜ P (τ ) ≤ 1 w − w ˜ P (τ ) . 0

∗

2

0

∗

˜ ˜ w) Proof. We proceed formally, assuming that f , u := Γ(w) and u ˜ := Γ( ˜ are more regular than they actually are. Again, this can be justified by means of a regularization process, as long as all the estimates we established are in terms of the correct norms of f , u and u ˜. 1) We differentiate the equation of (3.4.7) α times with respect to the space variables, |α| ≤ s, and multiply in L2 by 2 ∂xα ut . Recalling the definition of Qk in (2.3.13), we obtain d Qs (aw , ∇u) dt  = 2 f, us + aij  (∇w) · ∇wt ∂i ∂xα u, ∂j ∂xα u

2 ut 2s +

−2

(3.4.9)



|α|≤s

aij  (∇w) · ∇∂j w ∂i ∂xα u, ∂xα ut 

|α|≤s

+2



α Gα (aw ij , ∂i ∂j u), ∂x ut 

|α|≤s

=: 2 f, us +



˜ 0,α + R ˜ 1,α + R ˜ G,α R

,

|α|≤s

where

G0 (aw ij , ∂i ∂j u)

(3.4.10)

:= 0 and, if α > 0,    α β w α−β ˜ α (aw , u) , Gα (aw ∂i ∂j u =: G ij , ∂i ∂j u) = β ∂x aij ∂x 0 0, (3.4.34)

2f, us+1 ≤ 2 f s (u0 + Δus ) ≤ Cη f 2s + u2s+1 + η ∇u2s+1 .

For the last term, we estimate instead 2aij  (∇w) · ∇∂j w ∂i ∂xα u, ∂xα u (3.4.35)

≤ 2 |aij  (∇w)|∞ |∇∂j w|∞ ∇us+1 us+1 ≤ κ(R) ∇ws ∇us+1 us+1 ≤ ω(R) u2s+1 + η ∇u2s+1 ,

with k ∈ K and ω ∈ K0 . Taking η small allows us to absorb the last terms of (3.4.34) and (3.4.35) into the quadratic term at the left side of (3.4.32).

3.4. The Parabolic Cauchy Problem

151

We can act in a similar way for the other terms of the right side of (3.4.32); using the previously obtained estimates of u in Ps (τ∗ ), we finally arrive at an estimate of the type d (3.4.36) u2s+1 + ∇u2s+1 ≤ σ(R, t∗ ) , dt σ ∈ K. In turn, integration of (3.4.36) allows us to obtain the desired estimate of u in L2 (0, τ∗ ; H s+2 ). ˜ as well as the additional regularity 5. The continuity properties of Φ, of u on the same interval [0, τ ], claimed in the second and third parts of Theorem 3.4.1, can be proven in a similar way as we did in the proof of the analogous claims in Theorem 3.2.1. This ends our sketch of the proof of Theorem 3.4.1.  Remark. If f is such that, in addition to the second of (3.4.2), ft ∈ L2 (0, T ; H s−2 ), then the local solution u ∈ Ps (τ ) provided by Theorem 3.4.1 is also in C 1+α/2,2+α (Qτ ), Qτ = ]0, τ [×RN , for some α ∈ ]0, 1[, as follows from the result of section 2.5.3, considering equation (3.1.7) as linear. In fact, while the first of conditions (2.5.62) need not hold (because aij (0) = 0), the fact that u ∈ Ps (τ ) still implies that the coefficients auij = aij (∇u) ∈ C α/2,α (Qτ ). Indeed, arguing as in Theorem 1.7.5, and keeping in mind that aij ∈ Cb1 (RN ), by Theorem 1.7.5 it follows that if u ∈ C([0, τ ]; H s+1 ) and ut ∈ L2 (0, τ ; H s ), then auij ∈ C γ/2,γ (Qτ ), for some γ ∈ ]0, 1[. Analogous considerations hold for the additional regularity of u in ]0, τ ]; for an example of the type of results that can be obtained in this direction, see, e.g., Theorem 4.6.3 of Chapter 4. 

Chapter 4

Global Existence

4.1. Introduction 1. In this chapter we assume that the source term f of the hyperbolic equation (3.1.1) or of the parabolic equation (3.1.7) is defined on all of [0, +∞[, and we consider some situations in which the corresponding Cauchy problems have a global strong solution. More precisely, in the hyperbolic case, we introduce, in analogy to (2.3.47), the spaces (4.1.1)

Zm (∞) :=

2 .

C j ([0, +∞[; H m+1−j ) ,

j=0

(4.1.2)

Zm,b (∞) :=

2 .

Cbj ([0, +∞[; H m+1−j ) ,

j=0

and say that u is a global (respectively, global bounded) strong solution of problem (3.1.1)+(3.1.2) if u ∈ Zs (∞) (respectively, if u ∈ Zs,b (∞)). Note that (4.1.3)

u ∈ Zs (∞)

⇐⇒

u ∈ Zs (T ) ∀ T > 0 .

Analogous definitions hold in the parabolic case; see section 4.6. We shall see that, in general, global strong solutions will exist only if some additional restrictions are assumed of the equation, or the data, or even the space dimension N . In general, the data have to be small, as measured by their norm in a suitable data space Gs (∞), analogous to the space Gs (T ) defined in the beginning of section 3.2. More precisely, in analogy to (2.3.46), for m ≥ 0 we define (4.1.4)

Vm (∞) := L2loc (0, +∞; H m+1 ) ∩ C([0, +∞[; H m ) , 153

154

4. Global Existence

and set Gs (∞) := H s+1 × H s × Vs−1 (∞). In this context, a related question is whether, given any T > 0, we can determine δ > 0 so that, if (4.1.5)

(u0 , u1 , f )Gs (T ) ≤ δ ,

the corresponding problem (3.1.1)+(3.1.2) has a solution u ∈ Zs (T ). We call this an almost global existence result. Conversely, we can start with small data, that is, assume that (4.1.5) holds, with T = ∞, for some given δ > 0, and ask for how long the corresponding solution will exist; that is, whether we can give an estimate for the largest value of T , such that (3.1.1)+(3.1.2) has a strong solution u ∈ Zs (T ). We call this a life-span estimate result. 2. The situation is somewhat similar to the question of global existence versus blow up in finite time for solutions of nonlinear ODEs. To illustrate this, it is sufficient to consider the simple example of the Cauchy problem   y = y2 , (4.1.6) y(0) = y0 > 0 . As we know, (4.1.6) cannot have a global solution in [0, +∞[, since y develops a singularity at Tc = y10 > 0. However, we can establish (sharp) almost global existence and life-span estimates results for (4.1.6). Indeed, given 9 any T > 0, if we let (e.g.) δ = 10T we see that for any y0 ∈ [0, δ] the corresponding solution of (4.1.6) will be defined on [0, T ]. Conversely, if y0 ∈ ]0, δ] for some δ > 0, then the corresponding solution of (4.1.6) will exist on [0, T ], as long as T < 1δ . Finally, in this example it is also worth mentioning that, given any τ > 0, there are solutions of (4.1.6) which blow up before τ ; indeed, this is the case if y0 > τ1 . Similar remarks are valid for the hyperbolic problem (3.1.1)+(3.1.2). In section 4.3 we will give an example, that shows that global existence is in general not to be expected for such problem, while in section 4.4 we give an almost global existence result, in the same spirit of (4.1.6); that is, given T > 0 we determine δ > 0 such that if the smallness condition (4.1.5) holds, then the Cauchy problem (3.1.1)+(3.1.2) has a solution u ∈ Zs (T ). We also give a life-span estimate for such solutions (which is far from optimal if N > 1). 3. To show that global existence may hold if the equation satisfies certain specific types of structural conditions, we modify problem (4.1.6) by adding dissipation; that is, we consider the perturbed problem   y + σ y = y2 , (4.1.7) y(0) = y0 ≥ 0 , with σ > 0. In this case, the interval [0, σ] is invariant (see section 4.3.3 below); therefore, if 0 ≤ y0 ≤ σ (that is, if y0 is sufficiently small), (4.1.7) does have a global, bounded solution on [0, +∞[, with 0 ≤ y(t) ≤ σ for all t ≥ 0. On the other hand, if y0 > σ, the solution of (4.1.7) will still blow up at a finite positive time Tc , with Tc → +∞ as y0 → σ + (we find Tc =

4.2. Life Span of Solutions

155

 ln 1 + y0σ−σ . In a sense, this reflects the fact that, if y is small, the linear part of the equation dominates the nonlinear part, while the opposite happens if y is large (in which case, figuratively speaking, the solution grows up too fast for the linear part to catch up). Finally, if 0 < y0 < σ, the stability result y(t) → 0+ as t → +∞ holds, as we can show by means of the following a priori estimate argument. Let [0, Tc [ be the maximal time of existence of the solution of (4.1.7). Multiplying the equation by 2 y(t), we first see that, for all t ∈ [0, Tc [, 1 σ

d 2 y = 2 y y  = 2 y(y 2 − σ y) = 2 y 2 (y − σ) . dt Thus, y is decreasing as long as 0 ≤ y(t) ≤ σ, and this confirms the invariance of the interval [0, σ] if 0 < y0 < σ (if y0 = 0, or y0 = σ, then y(t) ≡ 0, or y(t) ≡ σ). We can then proceed from (4.1.8) with (4.1.8)

d 2 y + 2 σy 2 = 2 y 3 ≤ 2 y0 y 2 , dt from which we deduce the exponential inequality (4.1.9)

d 2 y + 2(σ − y0 ) y 2 ≤ 0 . dt Since σ − y0 > 0, (4.1.10) implies that Tc = +∞, and y decays exponentially to 0. We conclude by remarking that the inequality 0 ≤ y0 ≤ σ can be interpreted in the sense that the larger the dissipation (as measured by σ), the larger the data (in this case, only the initial value y0 ) can be taken, to have global solutions. In sections 4.5.2 and 4.5.3 we will use a very similar argument to prove some global existence results for the dissipative Cauchy problem (3.1.4)+(3.1.2), as well as for the parabolic Cauchy problem (3.1.7)+(3.4.1) in section 4.6. For the latter, a slightly better situation occurs, in that the availability of the maximum principle allows us to prove a global existence result with no restrictions on the size of the data (see section 4.6.5).

(4.1.10)

For general results on global existence of smooth solutions to either the hyperbolic or the parabolic problem (in general for small data), or on the estimate of their life-span, see, e.g., Racke [136], or Zheng and Chen [169] and the references therein; more specific results for general nonlinear hyperbolic equations can be found in H¨ormander [55, 56] and Li Ta-Tsien [90].

4.2. Life Span of Solutions 1. As in the case of ODEs, local solutions of (3.1.1)+(3.1.2) can be extended to later times by repeated applications of the local existence Theorem 3.2.1.

156

4. Global Existence

Definition 4.2.1. Let 0 < τ < θ ≤ +∞, m ∈ N, and u ∈ Zm (τ ). A function v : [0, θ] → H m+1 is an extension of u to [0, θ] if v ∈ Zm (θ), and the restriction of v to [0, τ ] coincides a.e. with u. Let T > 0 be given. Our goal is to start with a local solution u ∈ Zs (τ ), s > N2 + 1, of (3.1.1)+(3.1.2), defined on some interval [0, τ ] ⊂ [0, T ], and extend it to a strong solution u ˜, defined on a largest possible [0, τm ] ⊆ [0, T ]. According to Definition 4.2.1, this requires that u ˜ ∈ Zs (τm ). Since the strong solutions of (3.1.1)+(3.1.2) obtained by means of Theorem 3.2.1 are unique wherever they are defined, extensions are uniquely defined as well; thus, for simplicity, we do not distinguish between u and u ˜. Since u ∈ Zs (τ ), the functions v0 := u(τ, ·) and v1 := ut (τ, ·) are well defined in, respectively, H s+1 and H s . By Theorem 3.2.1, the Cauchy problem ⎧ ⎨ vtt − aij (Dv)∂i ∂j v = f (t + τ, x) , (4.2.1) ⎩ v(0) = v vt (0) = v1 , 0 has a unique solution v ∈ Zs (θ), for some θ ∈ ]0, T − τ ]. The function u ˜ defined by  u(t) if 0≤t≤τ, (4.2.2) u ˜(t) := v(t − τ ) if τ ≤t≤τ +θ, is then an extension of u to [0, τ + θ], and a solution of (3.1.1)+(3.1.2) in Zs (τ + θ). A repeated application of this process leads to an increasing sequence (τn )n≥0 ⊂ [0, T ], with τ0 := τ , and a corresponding sequence of extensions (un )n≥0 , with u0 := u, and un ∈ Zs (τn ) for each n. As we have noted, each un+1 is an extension of un to [0, τn+1 ]; hence, this process yields an extension of the local solution u, originally defined on [0, τ ], to the expanding intervals [0, τn ], n ≥ 1. If lim τn = T , this process produces a solution of (3.1.1)+(3.1.2) in Zs (τ ), for all τ ∈ ]0, T [ . If this occurs for arbitrary T , by (4.1.3) we deduce that u ∈ Zs (∞); that is, u is the desired global solution. However, it may well be that τ∗ := lim τn < T ; in which case, we shall see that u cannot be extended to a global solution. 2. Let u ∈ Zs (τ ) be a local solution of (3.1.1)+(3.1.2), for some τ > 0. We set (4.2.3) T (u) := {T ∈ R>0 | (3.1.1)+(3.1.2) admits a solution u ∈ Zs (T )} . This set is not empty, since τ ∈ T (u), and connected. We define then (4.2.4)

Tc (u) := sup T (u) ,

and call Tc (u) the maximal time of existence (or life-span, or blow-up time, or critical time) of u. When there is no danger of ambiguity, we abbreviate

4.2. Life Span of Solutions

157

Tc (u) =: Tc . By (4.1.3), u can be extended to a global solution u ∈ Zs (∞) if and only if Tc (u) = +∞; if instead Tc (u) is finite, we say that u blows up in finite time. Proposition 4.2.1. The set T (u) has no maximum; in addition, if Tc (u) is finite, then (4.2.5)

lim

τ →Tc (u)−

uZs (τ ) = +∞ .

Proof. 1) If there were T0 ∈ T (u) such that Tc (u) = T0 , then u could not be extended to any interval [0, T0 + ε], ε > 0, with u ∈ Zs (T0 + ε). However, the solution of the Cauchy problem (4.2.1) with initial data v(0) = u(T0 ), vt (0) = ut (T0 ) provides an extension of u to the right of T0 , contradicting the fact that T0 = max T (u). 2) We abbreviate Tc := Tc (u), which is assumed to be finite. If (4.2.5) were not true, there would be M0 > 0, and an increasing sequence (tn )n∈N ⊂ [0, Tc [, such that tn → Tc− and uZs (tn ) ≤ M0

(4.2.6)

for all n. Then, the sequences (u(tn ))n∈N and (ut (tn ))n∈N are bounded in H s+1 and H s , so they admit subsequences, still denoted (u(tn ))n∈N and (ut (tn ))n∈N , which converge weakly in H s+1 and H s , respectively, to some elements u ˜0 ∈ H s+1 and u ˜1 ∈ H s . We define then the functions u ˜ : [0, Tc ] → s+1 H and v : [0, Tc ] → H s by  u(t) if 0 ≤ t < Tc , u ˜(t) := (4.2.7) u ˜0 if t = Tc ,  (4.2.8)

v(t)

=

ut (t)

if

0 ≤ t < Tc ,

u ˜1

if

t = Tc ,

and show that (4.2.9)

u ˜ ∈ L∞ (0, Tc ; H s+1 ) ∩ C([0, Tc ]; H s ) ,

(4.2.10)

v ∈ L∞ (0, Tc ; H s ) ∩ C([0, Tc ]; H s−1 ) .

This is a consequence of the fact that the function t → uZs (t) is nondecreasing. In fact, given t ∈ [0, Tc [, let n ∈ N be such that t ≤ tn . Then, by (4.2.6), (4.2.11)

u(t)s+1 ≤ uZs (tn ) ≤ M0 ,

158

4. Global Existence

which shows that u ∈ L∞ (0, Tc ; H s+1 ). To show that u ∈ C([0, Tc ]; H s ), given t, t  ∈ ]0, Tc [, with, e.g., t < t  , choose n ∈ N such that t  ≤ tn . Then, u(t  )



t

− u(t)s ≤

 ut (θ)s dθ ≤

t

(4.2.12)

 ≤

t t

t

t

uZs (θ) dθ

uZs (tn ) dθ ≤ M0 (t  − t) .

This shows that u is Lipschitz continuous from [0, Tc [ into H s ; thus, u admits an extension u ¯ to [0, Tc ], with u ¯ ∈ C([0, Tc ]; H s ). Since (4.2.12) also shows that u(tn ) → u ˜0 strongly in H s , it follows that u ¯(Tc ) = u ˜0 . Hence, u ¯ = u ˜, and this completes the proof of (4.2.9). The proof of (4.2.10) is analogous. By Proposition 1.7.1, u ˜ ∈ Cw ([0, Tc ]; H s+1 ), and s v ∈ Cw ([0, Tc ]; H ). Clearly, v = u ˜t as distributions in D  (]0, Tc [; H s ); hence, ∞ s by (4.2.10), u ˜t ∈ L (0, Tc ; H )∩C([0, Tc ]; H s−1 ) → Cw ([0, Tc ]; H s ). We now proceed as in the proof of Lemma 2.3.2. Since u solves (3.1.1) on [0, Tc [, it satisfies the identity  d  aij (Du)∂i ∂xα u, ∂j ∂xα u ut 2s + dt |α|≤s

  =: Esu (t)

(4.2.13)

= 2f, ut s + +2





∂t aij (Du) ∂i ∂xα u, ∂j ∂xα u

|α|≤s

− ∂j aij (Du) ∂i ∂xα u + Gα (u), ∂xα ut  ,

|α|≤s

which is formally obtained by differentiating equation (3.1.1) α times with respect to the space variables, |α| ≤ s (when |α| = s, this can be justified by regularization), multiplying the resulting identities in L2 by 2 ∂xα ut , and then summing with respect to α (compare to (3.3.87)). In (4.2.13), Gα is defined in (3.3.92). In Lemma 4.4.1 below, we shall prove that the right side of (4.2.13) can be bounded in terms of the constant M0 of (4.2.6); from this, it follows that, for 0 < t < t  < Tc , (4.2.14)

|Esu (t  ) − Esu (t)| ≤ κ(M0 ) (t  − t) ,

for suitable κ ∈ K. Arguing exactly as in the proof of Lemma 2.3.2, we conclude that u ˜ ∈ C([0, Tc ]; H s+1 ) ∩ C 1 ([0, Tc ]; H s ). But then, the solution of the Cauchy problem (4.2.1) with initial data v(0) = u ˜(Tc ), vt (0) = u ˜t (Tc ) would provide an extension of u ˜ to the right of Tc , contradicting the fact that Tc is a supremum. This ends the proof of Proposition 4.2.1. 

4.3. Non Dissipative Finite Time Blow-Up

159

4.3. Non Dissipative Finite Time Blow-Up In this section we consider the homogeneous hyperbolic equation in one space dimension utt − a(ux )uxx = 0 ,

(4.3.1)

which is of type (3.1.1), and we show that the maximal time of existence of its strong solutions must be finite, no matter how small, or how smooth, the initial values are. This result shows that global existence for quasi-linear hyperbolic equations cannot in general be expected. On the other hand, we also show that almost global existence does hold; that is, strong solutions of (4.3.1) do exist on all of a given interval [0, T ], if the initial values are sufficiently small, with the essential restriction that the smallness of the data is determined by T . For other explicit blow-up results on semi-linear hyperbolic equations, or quasi-linear systems of conservation laws of the type (3.1.8), we refer to Alinhac [4]. 4.3.1. Lax’s Example. In accord with the framework introduced in section 3.2, in (4.3.1) we assume that a ∈ C 2 (R) and that it satisfies the uniform ellipticity condition (3.2.1), which now reads (4.3.2)

∃ α0 > 0

∀p ∈ R,

a(p) ≥ α0 .

We also assume that a  (0) = 0; in fact, for convenience, we assume that a  (0) > 0. We attach to (4.3.1) the initial conditions (4.3.3)

u(0, ·) = u0 ,

ut (0, ·) = u1 ,

with u0 ∈ H 3 , u1 ∈ H 2 . By Theorem 3.2.1, the Cauchy problem (4.3.1)+ (4.3.3) has a unique local solution u ∈ Z2 (τ ), for some τ > 0 determined by the size of u0 3 and u1 2 . We ask then, whether this solution can be extended to a global one, i.e., to a solution u ∈ Z2 (∞). We proceed to show that the answer may well be negative, no matter how small (or how smooth) the initial conditions (4.3.3) are. To this end, we fix δ0 ∈ ]0, 1] so small that 1  a  (r) ≥ if |r| ≤ δ0 . a (0) 10 √ Then, setting k := a, we note that the function  r (4.3.5) R  r → H(r) := k(z) dz

(4.3.4)

0

is invertible, with continuous inverse that  − 1  1  H  ≤ δ0 (4.3.6) 2r

H − 1. if

Hence, there is δ1 ∈ ]0, δ0 ] such |r| ≤ δ1 .

160

4. Global Existence

Finally, we set (4.3.7)

k0 := max k(z) , |z|≤1

C0 := 2(1 + k0 ) ,

and introduce the function (4.3.8)

R  r → ψ1 (r) := u1  (r) + k(u0  (r)) u0  (r) .

This function is well defined, because, by the Sobolev imbedding theorem, u0 ∈ Cb2 (R) and u1 ∈ Cb1 (R). In fact, ψ1 ∈ Cb (R). We claim: Theorem 4.3.1. Let δ ∈ ]0, 1[ be such that C0 δ ≤ δ1 , and assume that (4.3.9)

u0 3 + u1 2 ≤ δ .

Assume also that ψ1 (ξ0 ) > 0 for some ξ0 ∈ R. Then, there is no u ∈ Z2 (∞), solution of (4.3.1)+(4.3.3). Proof. We adapt an argument by Lax [87], as revisited by Klainerman and Majda [79] (see also § 3.3 of Majda [108, ch. 3]); for a geometrical interpretation of the condition ψ1 (ξ0 ) > 0, see section 4.3.2 below. In essence, the proof proceeds by contradiction. Assuming that the Cauchy problem (4.3.1)+(4.3.3) did have a global solution u ∈ Z2 (∞), we proceed to construct a differentiable function t → z(t), defined in terms of the derivatives Du and D 2 u, and satisfying, for t ≥ 0, the nonlinear Bernoulli ODE (4.3.10)

z  = γ(t) z 2 ,

with γ continuous and positive, and z(0) > 0. Since z must blow up at a finite positive time, it follows that at least one of the derivatives of u must also develop a singularity at a finite time. Hence, we reach a contradiction. 1. To proceed, it is convenient to transform (4.3.1) into the first order system  v t − wx = 0, (4.3.11) wt − a(v) vx = 0 , formally obtained from (4.3.1) by setting ux =: v, ut =: w. This system can be written as







v v 0 0 −1 (4.3.12) + = , − a(v) 0 w t w x 0

  =: A(v)

and the strong ellipticity condition (4.3.2) assures that system (4.3.12) is hyperbolic and genuinely nonlinear, in the sense that (see, e.g., Smoller [149, ch. 20], or Lax [88, ch. 10]), for each v ∈ R, the matrix A(v) has the two √ real, distinct eigenvalues λ± (v) = ± k(v), with k = a. Correspondingly,

4.3. Non Dissipative Finite Time Blow-Up

161

system (4.3.12) admits two families C± of characteristic curves, each defined by the solution of the nonlinear Cauchy problems     x = − k(v(t, x)) , x = + k(v(t, x)) , (4.3.13) x(0) = ξ, x(0) = η, the family C− being parametrized by ξ, and C+ by η ∈ R. We denote the solutions of (4.3.13) by x = κ− (t, ξ) and x = κ+ (t, η), respectively; note that each such solution is globally defined, because the right sides of the ODEs in (4.3.13) are globally Lipschitz continuous in x, uniformly on compact time intervals [0, T ], for all T > 0. This follows from the fact that (4.3.14)

∂x (k(v(t, x))) = k  (v(t, x)) vx (t, x) = k  (ux (t, x)) uxx (t, x) ,

which is bounded in x, because, if t ∈ [0, T ], recalling that u ∈ Z2 (T ), (4.3.15)

|uxx (t, x)| ≤ |uxx (t)|∞ ≤ uxx (t)1 ≤ uZ2 (T ) ,

and analogously for |ux (t, x)|. We then consider the change of dependent variables (v, w) → (r, s), defined by  v  v (4.3.16) r := w + k(z) dz , s := w − k(z) dz , 0

which is invertible, with inverse   (4.3.17) v = H − 1 12 (r − s) ,

0

w=

1 2

(r + s) ,

with H as in (4.3.5). The new coordinates (r, s) are known as the Riemann invariants of system (4.3.11); they are characterized by the property that they are constant along their correspondent characteristic curves. More precisely, if, in (4.3.16), we consider r and s as functions of (t, x), through the composition (t, x) → (v, w) → (r, s), then, using (4.3.11) and recalling that k 2 = a, we deduce that the identities (4.3.18)

rt − k rx ≡ 0 ,

st + k sx ≡ 0

hold for all (t, x) ∈ R≥0 × R. In turn, from (4.3.18) and (4.3.13) it follows that, for all ξ and η ∈ R, the functions (4.3.19)

t → ρ(t, ξ) := r (v(t, κ− (t, ξ)), w(t, κ− (t, ξ))) ,

(4.3.20)

t → σ(t, η) := s (v(t, κ+ (t, η)), w(t, κ+(t, η))) ,

are constant. To see this, we compute that, for example, ρt = rv (vt + vx x  ) + rw (wt + wx x  ) (4.3.21)

= k(v)(vt − vx k(v)) + (wt − wx k(v)) = k(v)(vt − wx ) + (wt − a(v) vx ) ≡ 0

162

4. Global Existence

(in (4.3.21), the functions v, w, etc., are evaluated at (t, x) = (t, κ− (t, ξ))). An analogous computation shows that σt ≡ 0. From this, recalling (4.3.16) we obtain that, for all t ≥ 0 and ξ, η ∈ R,  u0  (ξ) (4.3.22) ρ(t, ξ) = ρ(0, ξ) = r(v(0, ξ), w(0, ξ)) = u1 (ξ) + k(z) dz , 0



u0  (η)

(4.3.23) σ(t, η) = σ(0, η) = s(v(0, η), w(0, η)) = u1 (η) −

k(z) dz . 0

For future reference, we note that, since by (4.3.9), for all r ∈ R, (4.3.24)

|u0  (r)| + |u1 (r)| ≤ |u0  |∞ + |u1 |∞ ≤ u0  1 + u1 1 ≤ δ ,

we deduce from (4.3.22), (4.3.23) and (4.3.7) that, for all t ≥ 0 and ξ, η ∈ R, (4.3.25)

|ρ(t, ξ)| + |σ(t, η)| ≤ 2 δ(1 + max k(z)) = 2(1 + k0 ) δ = C0 δ . |z|≤1

We also note the crucial fact that, for all t ≥ 0, and all ξ ∈ R, there is η ∈ R such that (4.3.26)

κ− (t, ξ) = κ+ (t, η) .

Indeed, arguing as in (4.3.14) and (4.3.15), we see that the backward Cauchy problem for the unknown θ → x(θ; t, ξ),  dx = k(v(θ, x)) , dθ (4.3.27) x(θ = t) = κ− (t, ξ) , has a unique global solution, defined for θ ≤ t; thus, η = x(0; t, ξ). 2. We now set (4.3.28)

μ(t, ξ) := k(v(t, κ− (t, ξ))) ,

ν(t, ξ) := k  (v(t, κ− (t, ξ))) ,

where κ− is defined by the characteristic system (4.3.13), and proceed to show that there exists ν0 > 0 such that ν(t, ξ) ≥ ν0 for all t ≥ 0 and ξ ∈ R. To this end, given such (t, ξ), let η ∈ R be determined as in (4.3.26). Then, recalling the first of (4.3.17), as well as (4.3.19) and (4.3.20), we compute that   v(t, κ− (t, ξ)) = H − 1 12 r(v(t, κ− (t, ξ)), w(t, κ−(t, ξ)))  − s(v(t, κ− (t, ξ)), w(t, κ−(t, ξ)))   = H − 1 12 r(v(t, κ− (t, ξ)), w(t, κ−(t, ξ))) (4.3.29)  − s(v(t, κ+ (t, η)), w(t, κ+(t, η)))   = H − 1 12 (ρ(t, ξ) − σ(t, η))   = H − 1 12 (ρ(0, ξ) − σ(0, η)) .

4.3. Non Dissipative Finite Time Blow-Up

163

Recalling then (4.3.25) and (4.3.6), and that, by assumption, C0 δ ≤ δ1 , (4.3.29) yields that    (4.3.30) |v(t, κ− (t, ξ))| ≤ H − 1 1 δ1  ≤ δ0 ; 2

hence, by (4.3.4), ν(t, ξ) = k  (v(t, κ− (t, ξ)) ≥

(4.3.31)

a  (0) >0, 20 k0

as claimed. 3. We further set (4.3.32) (4.3.33)

( μ(t, ξ) rx (v(t, κ− (t, ξ)), w(t, κ− (t, ξ))) , ν(t, ξ) γ(t, ξ) := , 2 (μ(t, ξ))3/2 z(t, ξ) :=

and proceed to show that, for each ξ ∈ R, z(·, ξ) satisfies the ODE (4.3.10). To this end, we first note that (4.3.34)

μ  = k  (v) (vt − k(v) vx ) = ν (vt − μ vx ) ,

where v, vt and vx are evaluated at (t, κ− (t, ξ)). From (4.3.16), recalling that wx = vt , we derive that (4.3.35)

rx = vt + k(v) vx ,

(4.3.36)

rxt = vtt + k(v) vxt + k  (v) vt vx

(4.3.37)

rxx = vtx + k(v) vxx + k  (v) (vx )2 ,

where all functions are evaluated at (t, x). By (4.3.28), along the characteristic x = κ− (t, ξ) we have that (4.3.38)

k(v(t, κ− (t, ξ))) = μ(t, ξ) ,

k  (v(t, κ− (t, ξ))) = ν(t, ξ) ;

thus, (4.3.39)

z  − γ z2 =

μ νμ √ (rx )2 . √ (vt + μ vx ) + μ(rxt − μ rxx ) − 2 μ 2 μ3/2

By (4.3.36) and (4.3.37), and recalling also (4.3.34), we obtain ν z  − γ z2 = √ ((vt )2 − μ2 (vx )2 − (vt + μ vx )2 ) 2 μ √ + μ(vtt + ν vt vx − μ2 vxx − μν(vx )2 ) (4.3.40) √ = μ (vtt − μ2 vxx − 2μν(vx )2 ) . From (4.3.11), recalling that a = k 2 , we compute that, for generic (t, x), (4.3.41) vtt = wtx = a(v) vxx + a  (v)(vx )2 = a(v) vxx + 2 k(v) k  (v)(vx )2 ; thus, along the characteristic C− , (4.3.42)

vtt = μ2 vxx + 2 μ ν (vx )2 .

164

4. Global Existence

Replacing (4.3.42) into (4.3.40) yields (4.3.10). 4. By (4.3.2), the denominator of γ cannot vanish; thus, recalling (4.3.31), we can define the strictly increasing functions  t (4.3.43) R≥0  t → Γ(t, ξ) := γ(θ, ξ) dθ . 0

These functions are also unbounded as t increases, because, by (4.3.31) and (4.3.7), (4.3.44)

γ(θ, ξ) ≥

a  (0) 5/2

.

40 k0

Equation (4.3.10) has the explicit solution (4.3.45)

z(t, ξ) =

z(0, ξ) , 1 − z(0, ξ) Γ(t, ξ)

from which we see that z(t, ξ) has the same sign of z(0, ξ) for as long as 1 z(0, ξ) < Γ(t,ξ) . If z(0, ξ) ≤ 0, this condition is satisfied, the denominator of (4.3.45) is positive, and z(·, ξ) does not develop singularities in [0, +∞[. If instead z(0, ξ) > 0, z(·, ξ) is also positive, and develops a singularity at the time tξ , uniquely defined by (4.3.46)

Γ(tξ , ξ) =

1 . z(0, ξ)

Recalling (4.3.32), (4.3.28), (4.3.35), and that κ− (0, ξ) = ξ, we compute that ( z(0, ξ) = k(v(0, ξ)) (vt (0, ξ) + k(v(0, ξ)) vx (0, ξ)) ( = k(ux (0, ξ)) (uxt (0, ξ) + k(ux (0, ξ)) uxx (0, ξ)) (4.3.47) ( = k(u0  (ξ)) (u1  (ξ) + k(u0  (ξ)) u0  (ξ)) ( = k(u0  (ξ)) ψ1 (ξ) =: ψ(ξ) . By assumption, z(0, ξ0 ) = ψ(ξ0 ) > 0; hence, z(·, ξ0 ) develops a singularity at the positive time t0 := tξ0 , defined by (4.3.46). From this, it follows that no local solution u of (4.3.1)+(4.3.3) can be continued beyond t0 . In fact, u will have a finite blow-up time Tc ≤ t0 , because if t0 < Tc , it would follow that u ∈ Z2 (t0 ). But then, by the Sobolev imbedding theorem, the functions t → u(t, ξ0 ), t → Du(t, ξ0 ), and t → D 2 u(t, ξ0 ) would remain bounded in [0, t0 ]. For example, as in (4.3.15), (4.3.48)

|uxx (t, ξ0 )| ≤ uxx (t)1 ≤ max u(t)3 ≤ uZ2 (t0 ) . 0≤t≤t0

In turn, this would imply that z(·, ξ0 ) would also be bounded in [0, t0 ]. This ends the proof of Theorem 4.3.1. 

4.3. Non Dissipative Finite Time Blow-Up

165

Remarks. 1) As the last argument of its proof indicates, Theorem 4.3.1 guarantees the finite time blow-up only of at least one of the second order derivatives of u, leaving open the possibility that u and its first order derivatives are globally defined and bounded as t → +∞. In fact, for the specific system (4.3.11), at least ut and ux remain bounded, as a consequence of the fact that (4.3.11) admits a bounded, positively invariant region, as we show in section 4.3.3 below. 2) If, with ψ(ξ) defined in (4.3.47) and tξ in (4.3.46), we set (4.3.49)

t∗ := inf{tξ | ψ(ξ) > 0} ,

then t∗ > 0. To see this, note first that (4.3.46) and (4.3.47) imply that tξ > 0 for all ξ such that ψ(ξ) > 0; hence, t∗ ≥ 0. If t∗ = 0, we could choose an infinitesimal sequence (tξn )n≥0 ⊂ ]0, 1]. Then, recalling (4.3.28) and (4.3.2), and acting as in (4.3.15) to estimate |v|∞ = |ux |∞ , that is, (4.3.50)

|ux (t)|∞ ≤ u(t)Z1 (1) =: p0 ,

we deduce that, for all t ∈ [0, 1] and all ξ ∈ R, (4.3.51)

0 < γ(t, ξ) ≤

1

sup |k  (p)| =: γ∗ .

3/4 2 α0 |p|≤p0

From this, it follows from (4.3.43) that, for all n ≥ 0, (4.3.52)

0 < Γ(tξn , ξn ) ≤ γ∗ tξn ;

hence, Γ(tξn , ξn ) → 0, so that, by (4.3.46), z(0, ξn ) → +∞. But this would contradict the fact that the sequence (z(0, ξn ))n≥0 is bounded above, as we see from (4.3.47), keeping in mind that ψ1 ∈ Cb (R). Thus, t∗ > 0, as claimed. 3) In fact, arguing as in the last step of the proof of Theorem 4.3.1 we can see that Tc satisfies the upper bound Tc ≤ t∗ . It is then natural to ask, whether Tc = t∗ . While we do not know the answer to this question, we note that this equality may be prevented by the fact that, while at least one of the second order derivatives utt , utx or uxx develops a singularity at some point (Tc , ξc ), it is possible that the linear combination yielding the factor rx of z in (4.3.32), that is, recalling (4.3.35), rx = utx + k(ux ) uxx , remains bounded as (t, x) → (Tc , ξc ). 4) For another explicit example of blow-up in finite time for quasi-linear equation of the form (4.3.1), see John [65] (or Racke [136, ch. 1], and Klainerman [75]), where they consider equation (4.3.1) with a(p) = (1 + p)2 , and construct C0∞ initial values (4.3.3), so that the corresponding solution u develops a singularity in uxx (i.e., again in one of the second order derivatives) in finite time. Note that, in this example, the uniform ellipticity condition (4.3.2) is satisfied only with α0 = 0; on the other hand, since the solutions

166

4. Global Existence

considered are supposed to be small, it is sufficient that (4.3.2) be satisfied for p = 0 (see (4.5.176) below, in the third remark after Theorem 4.5.2).  4.3.2. Geometrical Interpretation. Following Keller and Ting [73] (see also Klainerman and Majda [79]), we can give a geometrical interpretation to the condition ψ1 (ξ0 ) > 0 of Theorem 4.3.1. 1. Let γξ be a generic characteristic curve of the family C− , defined by the first of (4.3.13). The argument is based on the observations that the condition ψ1 (ξ0 ) > 0 implies the existence of some t0 , with the property that ∂ κ− (t, ξ0 ) < 0 ∂ξ

(4.3.53)

for all t in a neighborhood U of t0 , and that, in turn, (4.3.53) implies that the characteristic curves γξ1 and γξ2 starting from points ξ1 < ξ2 near ξ0 will intersect at some point (t∗ , x∗ ), with t∗ ∈ U (see Figure 1). Indeed,

x6

ξ2 ξ1

0

t∗

-

t

Figure 1. Intersection of characteristics.

the function t → κ− (t, ξ1 ) − κ− (t, ξ2 ) =: δ(t) is such that δ(0) = ξ1 − ξ2 < 0, and, by (4.3.53), δ(t) > 0 for t sufficiently close to t0 , provided that ξ1 and ξ2 are sufficiently close to ξ0 . Hence, δ(t) = 0 for some t =: t∗ ∈ ]0, t0 ]. But then, at least one of the derivatives utt , utx or uxx (that is, one of the derivatives wt , wx = vt , or vx ) must become singular at (t∗ , x∗ ). To see this, we argue by contradiction. Assuming otherwise, by (4.3.19), the function t → ρ(t, ξ) would be constant in [0, t∗ ]; and then, from the fact that x∗ = κ− (t∗ , ξ1 ) = κ− (t∗ , ξ2 ) it would follow that (4.3.54)

ρ(t∗ , ξ1 ) = r (v(t∗ , x∗ ), w(t∗ , x∗ )) = ρ(t∗ , ξ2 ) .

4.3. Non Dissipative Finite Time Blow-Up

167

In turn, by (4.3.22), (4.3.54) would imply that ρ(0, ξ1 ) = ρ(0, ξ2 ), which would further yield that  u0  (ξ2 ) (4.3.55) u1 (ξ1 ) − u1 (ξ2 ) = k(z) dz . u0  (ξ1 )

By the mean value theorems, we deduce from (4.3.55) that, for some ξ¯1 and ξ¯2 ∈ [ξ1 , ξ2 ], and z¯ between u0  (ξ1 ) and u0  (ξ2 ), (4.3.56) u1  (ξ¯1 )(ξ1 − ξ2 ) = k(¯ z )(u0  (ξ2 ) − u0  (ξ1 )) = k(¯ z ) u0  (ξ¯2 ) (ξ2 − ξ1 ) , which implies that (4.3.57)

u1  (ξ¯1 ) + k(¯ z ) u0  (ξ¯2 ) = 0 .

Since u0  is continuous, z¯ → u0  (ξ0 ) as ξ1 and ξ2 → ξ0 ; thus, we deduce from (4.3.57) that (4.3.58)

u1  (ξ0 ) + k(u0  (ξ0 )) u0  (ξ0 ) = ψ1 (ξ0 ) = 0 ,

contradicting the assumption ψ1 (ξ0 ) > 0. Thus, one of the second order derivatives of u must blow up, as claimed. 2. We now show that (4.3.53) holds, if ψ1 (ξ0 ) > 0. To this end, we set (4.3.59) φ(t, ξ) :=

∂ κ− (t, ξ) , ∂ξ

r˜(ξ) := ρ(0, ξ) ,

s˜(t, ξ) := s(t, κ− (t, ξ)) ,

where s = s(t, x) through the composition (t, x) → (v(t, x), in  w(t,x))  ˜ (4.3.16); we also set, for λ ∈ R and H as in (4.3.5), k(λ) := k H − 1 12 λ . Then, ∂φ ∂ ∂ ∂m ∂ ∂ (4.3.60) (t, ξ) = κ− (t, ξ) = κ− (t, ξ) =: (t, ξ) , ∂t ∂t ∂ξ ∂ξ ∂t ∂ξ and, by the first of (4.3.13), recalling (4.3.17), and that ρ(t, ξ) = ρ(0, ξ) = r˜(ξ) along the characteristic γξ , (4.3.61) ∂m ∂ ˜ (t, ξ) = − k(˜ r(ξ) − s˜(t, ξ)) ∂ξ ∂ξ  = − k˜  (˜ r(ξ) − s˜(t, ξ)) ρξ (0, ξ) − sx (t, κ− (t, ξ)) ∂ξ κ− (t, ξ) . From (4.3.22), recalling (4.3.8), (4.3.62)

ρξ (0, ξ) = u1  (ξ) + k(u0  (ξ)) u0  (ξ) = ψ1 (ξ) ,

and, from the second of (4.3.18), (4.3.63)

sx (t, x) = −

1 st (t, x) . k(v(t, x))

168

4. Global Existence

Replacing (4.3.62) and (4.3.63) into (4.3.61), and recalling (4.3.60) and (4.3.59), we obtain  ∂φ r(ξ) − s˜(t, ξ)) ψ1 (ξ) (t, ξ) = − k˜  (˜ ∂t (4.3.64) 1 + st (t, κ− (t, ξ)) φ(t, ξ) . ˜ r(ξ) − s˜(t, ξ)) k(˜ We now set

 1/2 ˜ r(ξ) − s˜(t, ξ)) k1 (t, ξ) := k(˜ ,

(4.3.65)

and, recalling the first of (4.3.13) and the second of (4.3.18), we compute that (omitting the reference to the variables (t, ξ) and (t, κ− (t, ξ))) r − s˜) k˜  (˜ (− st − sx (κ− )t ) ˜ 2 k(˜ r − s˜) k˜  (˜ r − s˜) = − (s − sx k(v)) ˜ r − s˜) t 2 k(˜

∂ ln(k1 (t, ξ)) = ∂t (4.3.66)

= −

k˜  (˜ r − s˜) (2 st ) ; ˜ 2 k(˜ r − s˜)

that is, explicitly, (4.3.67)

∂ r(ξ) − s˜(t, ξ)) k˜  (˜ st (t, κ− (t, ξ)) . ln(k1 (t, ξ)) = − ˜ ∂t k(˜ r(ξ) − s˜(t, ξ))

From this, it follows that (4.3.64) can be rewritten as

∂ (4.3.68) φt (t, ξ) − r(ξ) − s˜(t, ξ)) , ln(k1 (t, ξ)) φ(t, ξ) = − ψ1 (ξ) k˜  (˜ ∂t which we interpret as a family of ODEs in the unknowns t → φ(t, ξ). Since φ(0, ξ) = ∂ξ κ− (0, ξ) = 1, integration of (4.3.68) yields    t ˜ k1 (t, ξ) k (˜ r(ξ) − s˜(θ, ξ)) (4.3.69) φ(t, ξ) = 1 − ψ1 (ξ) k1 (0, ξ) dθ . k1 (0, ξ) k1 (θ, ξ) 0 By (4.3.31), k˜  (˜ r − s˜) ≥ 201k0 a  (0) > 0; moreover, (4.3.2) and (4.3.25) imply that both k1 and its reciprocal are bounded. Consequently, the function  t ˜ k (˜ r(ξ) − s˜(θ, ξ)) (4.3.70) t → dθ k1 (θ, ξ) 0 is strictly increasing and unbounded. Thus, since ψ1 (ξ0 ) > 0, there is t0 > 0 such that  t ˜ k (˜ r(ξ0 ) − s˜(θ, ξ0 )) (4.3.71) ψ1 (ξ0 ) k1 (0, ξ0 ) dθ > 1 k1 (θ, ξ0 ) 0

4.3. Non Dissipative Finite Time Blow-Up

169

for all t in a neighborhood U of t0 . Hence, we deduce from (4.3.69) that φ(t, ξ0 ) = ∂ξ κ− (t, ξ0 ) < 0 for all t ∈ U , as claimed in (4.3.53).  4.3.3. Invariant Regions. In this section we briefly recall from Smoller [149, ch. 14, sct. B] (see also Bardos [11], and Chueh, Conley and Smoller [32]), the notion of a positively invariant region for a quasi-linear evolution equation, and show that system (4.3.11) does admit such a region. As a consequence, it will follow that the first order derivatives of the solution of (4.3.1) remain bounded for all t ≥ 0. We consider the system (4.3.72)

Ut + A(U ) Ux = f (U ) ,

in the unknown U = U (t, x) ∈ RN , t ≥ 0, x ∈ R, and assume that the matrix A and the function f are so smooth that (4.3.72) admits classical solutions, corresponding to sufficiently smooth initial values U (0). Definition 4.3.1. A subset R ⊂ RN (not necessarily bounded, nor open) is positively invariant for system (4.3.72) if for any smooth function U0 taking values in R, the solution U of the Cauchy problem for (4.3.72), with initial value U (0) = U0 , takes values in R whenever it is defined. If equation (4.3.72) admits a bounded positively invariant region R, the map (t, x) → U (t, x), a priori defined for small |t| only, is bounded; in particular, since the bound is independent of t, U can be extended into a bounded map, defined on all of [0, +∞[ × R, and with values in R. We now show that system (4.3.11) does admit a bounded, positively invariant region in R2 . To this end, with δ as in (4.3.9), we set (4.3.73)

c0 := max{|r(v, w)|, |s(v, w)| | |v| ≤ δ ,

|w| ≤ δ} ,

where r and s are the Riemann invariants defined in (4.3.16), and consider the region R ⊂ R2 bounded by the curves r(v, w) = ± c0 , s(v, w) = ± c0 ; that is (see Figure 2) (4.3.74)

R := {(v, w) ∈ R2 | |r(v, w)| < c0 ,

|s(v, w)| < c0 } .

We claim: Proposition 4.3.1. The region R defined in (4.3.74) is positively invariant for system (4.3.11). In fact, it is positively invariant also for the dissipative system  v t − wx = 0, (4.3.75) σ >0. wt − a(v) vx = − σ w ,

170

4. Global Existence

6w

c0 r(v, w) = c0

s(v, w) = c0

h0

-

v

− h0 r(v, w) = − c0

s(v, w) = − c0 − c0

Figure 2. Invariant region for (4.3.11) (h0 := H − 1 (c0 )).

System (4.3.75) corresponds to the dissipative equation (4.3.76)

utt + σ ut − a(ux ) uxx = 0 ,

which we will consider in section 4.5. Sketch of Proof. We consider system (4.3.75) for σ > 0; the result for system (4.3.11) follows by letting σ → 0 (as in theorem 14.11 of Smoller [149]). 1) To prove that R is positively invariant, we must show that the graph of every solution (t, x) → (v(t, x), w(t, x)), starting in R, that is, with (4.3.77)

(v(0, x), w(0, x)) ∈ R

for all x ∈ R ,

remains in R for all t ∈ R>0 . Assuming otherwise, there would be a solution (v, w) of (4.3.75), and a point P0 = (v(t0 , x0 ), w(t0 , x0 )) on its graph, such that P0 ∈ ∂ R. Assume for example that P0 is in the interior of the fourth quadrant. Then, v(t0 , x0 ) > 0 > w(t0 , x0 ), and, letting p(t, x) := (v(t, x), w(t, x)) and s˜(t, x) := s(p(t, x)), we can assume that (4.3.78)

s˜(t0 , x0 ) = − c0 ,

(4.3.79)

s˜(t, x) ≥ − c0

(4.3.80)

s˜(t, x0 ) > − c0

∀ (t, x) ∈ [0, t0 [×R , ∀ t ∈ [0, t0 [ .

More precisely, (4.3.78) translates the fact that P0 ∈ ∂ R; (4.3.79) means that the graph of the solution (v, w) is in R for 0 ≤ t < t0 , while (4.3.80) means that the orbit t → p(t, x0 ) remains in the interior of R for 0 ≤ t < t0 , and intersects ∂ R for the first time at t = t0 . Note that t0 > 0, for, if t0 = 0,

4.4. Almost Global Existence

171

(4.3.78) would contradict (4.3.77) for x = x0 ; in fact, (4.3.77) implies that s˜(0, x) > − c0 for all x ∈ R. Now, (4.3.78) and (4.3.80) imply that (4.3.81)

s˜t (t0 , x0 ) ≤ 0 ,

because s˜(t, x0 ) − s˜(t0 , x0 ) s˜(t, x0 ) + c0 = 0 × R,

(4.3.82)

s˜t = sv vt + sw wt = − k(v) vt + wt (4.3.83)

= − k(v) vt + a(v) vx − σ w = − k(v) (wx − k(v) vx ) − σ w = − k(v) s˜x − σ w ;

thus, since k(v(t0 , x0 )) > 0, σ > 0, and w(t0 , x0 ) < 0, we deduce from (4.3.83) and (4.3.81) that 1 (4.3.84) s˜x (t0 , x0 ) = − (˜ st (t0 , x0 ) + σ w(t0 , x0 )) > 0 . k(v(t0 , x0 )) The function x → s˜(t0 , x) is then increasing near x0 ; therefore, recalling (4.3.78), there is ε > 0 (depending on t0 and x0 ), such that, if 0 < x0 − x ˜ < ε, (4.3.85)

s˜(t0 , x ˜) < s˜(t0 , x0 ) = − c0 .

This implies that there is η > 0 (depending on t0 and x ˜), such that s˜(t, x ˜) < − c0 for |t − t0 | < η. For 0 < t0 − t < η, this contradicts (4.3.79). 2) The other cases, corresponding to P0 being in the interior of the other quadrants, are treated similarly; if instead P0 is on one of the axes (that is, if P0 is at one of the corners of ∂R), one can use an approximation argument, as in the proof of theorem 14.11 of Smoller [149]. This concludes the proof of Proposition 4.3.1.  As a consequence, returning to the original equations (4.3.1) and (4.3.76), we deduce that the functions (t, x) → v(t, x) = ux (t, x) and (t, x) → w(t, x) = ut (t, x) remain bounded for all t ≥ 0, as claimed. 

4.4. Almost Global Existence In this section we prove an almost global existence result for the Cauchy problem (3.1.1)+(3.1.2), under the same assumptions on the functions aij as in section 3.2 (that is, aij ∈ C s (R1+N ), s ∈ N, s > N2 + 1), and that the data u0 , u1 and f are sufficiently small. Essentially, the size of the data is determined by the length of the interval [0, T ]. We keep the same notations of section 3.2; in particular, for the set Bs (T, R), introduced in (3.2.5), and

172

4. Global Existence

the numbers a1 (Du0 ), a2 (Du0 ), a3 (Du0 ), defined in (3.2.44) and (3.2.56). We sometimes abbreviate these numbers by a1 , a2 , a3 . 1. We first prove a refinement of the linear estimate given in Lemma 3.2.3. Lemma 4.4.1. Let s > N2 + 1, and assume that the Cauchy problem (3.1.1)+(3.1.2) has a solution u ∈ Zs (T ), for some T > 0. There is κ ∈ K, such that for all t ∈ [0, T ],

 T Du(t)2s ≤ a22 (Du0 ) Du0 2s + f 2s dt 0     2 · exp t a2 (Du0 ) sf + κ max Du(θ)s−1 (4.4.1) 0≤θ≤t   · max f (θ)s−1 + max Du(θ)s , 0≤θ≤t

0≤θ≤t

where sf := 0 if f ≡ 0, and sf := 1 otherwise. Proof. We establish direct a priori estimates on u, proceeding as in parts 2 and 3 of section 3.3.5. We start from identity (4.2.13), that is,  d u (4.4.2) (R0,α + R1,α + RG,1,α ) , Es (t) = 2f, ut s + dt |α|≤s

where the terms at the right side are defined in (3.3.89), (3.3.90) and (3.3.91). At first, 2f, ut s ≤ f 2s + ut 2s .

(4.4.3)

Next, as in (3.3.95) and (3.3.96), with k = 0, (4.4.4)

R0,α ≤ |aij  (Du)|∞ |Dut |∞ ∇u2s ,

(4.4.5)

R1,α ≤ 2 |aij  (Du)|∞ |D∇u|∞ ∇us ut s .

Likewise, as from (3.3.100), (4.4.6)

|Gα (u)|2 ≤ h(|Du|∞ ) |∇Du|∞ Dus ,

with h ∈ K; thus, (4.4.7)

RG,1,α ≤ h(|Du|∞ ) |∇Du|∞ Du2s .

Putting this, together with (4.4.3), (4.4.4) and (4.4.5), into (4.4.2), we obtain  d  aij (Du)∂i ∂xα u, ∂j ∂xα u ut 2s + dt (4.4.8) |α|≤s ≤ f 2s + (sf + h(|Du|∞ ) |D 2 u|∞ ) Du2s .

4.4. Almost Global Existence

173

Integrating (4.4.8) on [0, t], 0 < t < T , and recalling the definition of a2 (Du0 ) in (3.2.56),

 T 2 2 2 2 2 Du(t)s ≤ a2 u1 s + ∇u0 s + f s dt 0 (4.4.9)  t + a22

0

(sf + h(|Du|∞ ) |D 2 u|∞ ) Du2s dθ .

From this, by the Sobolev and Gronwall inequalities,

 T 2 2 2 2 2 Du(t)s ≤ a2 u1 s + ∇u0 s + f s dt 0 (4.4.10)  t

·exp a22

B(u(θ)) dθ , 0

where B(u) := sf + h(Dus−1 ) D 2 us−1 .

(4.4.11)

Recalling that D 2 u = {utt , ∇ut , ∂x2 u}, D 2 us−1 ≤ Dus + utt s−1 ;

(4.4.12)

using equation (3.1.1), and recalling also (3.2.28) of Lemma 3.2.1, utt s−1 ≤ f s−1 + 2 h(|Du|∞ ) Dus ;

(4.4.13)

thus, from (4.4.12), D 2 us−1 ≤ f s−1 + (1 + 2 h(Dus−1 ) Dus .

(4.4.14)

Replacing this into (4.4.11), B(u) (4.4.15)

≤

sf + h(Dus−1 ) (f s−1 + 3 h(Dus−1 ) Dus )

≤: sf + κ(Dus−1 ) (f s−1 + Dus ) .

Inserting this into (4.4.10) yields (4.4.1).



2. We can now state our almost global existence result. Given T and δ > 0, we let Dδ (T ) be as in (3.2.4); that is, (4.4.16)

  Dδ (T ) := (u0 , u1 , f ) ∈ Gs (T ) | u0 2s+1 + u1 2s + f 2Vs−1 (T ) ≤ δ 2 .

We claim: Theorem 4.4.1. Assume that (u0 , u1 , f ) ∈ Gs (∞). For all T > 0, there is δ > 0 such that if (u0 , u1 , f ) ∈ Dδ (T ), the local solution of the corresponding Cauchy problem (3.1.1)+(3.1.2) can be extended to a solution u ∈ Zs (T ).

174

4. Global Existence

Proof. Proceeding by contradiction, assume there is T0 > 0 such that for all δ > 0, there are data (u0 , u1 , f ) ∈ Dδ (T0 ), with the property that the local solution of the corresponding Cauchy problem (3.1.1)+(3.1.2) blows up at some time Tδ ∈ ]0, T0 [. Recalling (4.2.5), this means that (4.4.17)

lim uZs (t) = +∞ ;

t→Tδ−

in turn, (4.4.17) implies that also (4.4.18)

lim Du(t)s = +∞ .

t→Tδ−

In fact, if we could bound Du(t)s in [0, Tδ ], we could also bound u(t)s and utt (t)s−1 : the former, via the identity  t (4.4.19) u(t) = u0 + ut (θ) dθ , 0

and the latter by means of Lemma 3.2.1, as in (4.4.13). Hence, a bound on Du(t)s in [0, Tδ ] would yield a bound on uZs (Tδ ) , contradicting (4.4.17). Recalling the first of (3.2.44), we see that there is δ0 > 0 such that, for all δ ≤ δ0 , (4.4.20)

a1 (Du0 ) ≤ 2 max |aij (0)|∞ ; i,j

consequently, by (3.2.56), there is A2 ≥ 1 such that, for all δ ∈ ]0, δ0 ] and (u0 , u1 , f ) ∈ Dδ (T0 ), (4.4.21)

a2 (Du0 ) ≤ A2 .

Fix now δ ∈ ]0, δ0 ], and consider the corresponding solution that blows up at Tδ . Recalling that (u0 , u1 , f ) ∈ Dδ (T ), and A2 ≥ 1, (4.4.18) implies that there is θδ ∈ ]0, Tδ [, such that, for all t ∈ [0, θδ ], (4.4.22)

2

Du(t)s ≤ Du(θδ )s = 2 A2 eA2 T0 /2 δ =: A3 δ .

By (4.4.1) and (4.4.21), we deduce then that, if δ ≤ δ0 ,     Du(t)2s ≤ A22 δ 2 exp A22 sf + κ(A3 δ) (δ + A3 δ) t  ≤ A22 δ 2 exp A22 (1 + κ(A3 δ)(1 + A3 )δ) T0 (4.4.23)   2 = A22 δ 2 eA2 T0 exp A22 κ(A3 δ)(1 + A3 )δ0 T0 . Recalling (4.4.22), (4.4.23) implies that, for all t ∈ [0, θδ ],   1 (4.4.24) Du(t)2s ≤ Du(θδ )2s exp A22 κ(A3 δ)(1 + A3 )δ0 ) T0 . 4 Taking t = θδ , we deduce from (4.4.24) that   (4.4.25) 4 ≤ exp A22 κ(A3 δ)(1 + A3 )T0 δ ,

4.5. Global Existence for Dissipative Equations

175

from which, letting δ → 0, we reach a contradiction. This end the proof of Theorem 4.4.1.  Remark. When f ≡ 0, the conclusion of Theorem 4.4.1 can be restated by saying that if the initial values u0 and u1 satisfy the smallness condition u0 2s+1 + u1 2s ≤ δ 2 ,

(4.4.26)

δ ≤ δ0 ,

then the life-span of the corresponding solution is at least of order 1δ . Indeed, if f ≡ 0, sf = 0 in (4.4.23); hence, we can repeat the previous argument, with θδ in (4.4.22) redefined by the condition Du(θδ )s = 2 A2 δ ,

(4.4.27)

and a corresponding modification of (4.4.23) into   (4.4.28) Du(t)2s ≤ A22 δ 2 exp A22 κ(2A2 δ)(1 + 2A2 )Tδ δ . Proceeding as in the proof of Theorem 4.4.1, we reach, as in (4.4.25), the inequality   (4.4.29) 4 ≤ exp A22 κ(2A2 δ0 )(1 + 2A2 )Tδ δ , from which (4.4.30)

Tδ ≥

ln 4 2 A2 (1 + 2A2 )κ(2A2 δ0 )



1 =O δ



1 . δ

If N ≥ 3 and the initial values are sufficiently smooth, this result can be significantly improved. In fact, writing (3.1.1) in the form (4.4.31)

utt − aij (0) ∂i ∂j u = (aij (Du) − aij (0)) ∂i ∂j u =: F (Du, ∂x2 u) ,

and observing that the map (p, q) → F (p, q) is at least quadratic near the origin (of R(N +1)(N +2)/2 ), it is possible to show  that Tδ = +∞ (i.e., the solution is global) if N ≥ 4, while Tδ ≥ O e1/δ if N = 3 (see Racke [136, ch. 10] and Ponce [132]). 

4.5. Global Existence for Dissipative Equations In this section we prove some results analogous to that of example (4.1.7); namely, a result on the existence of global and global bounded solutions for the hyperbolic dissipative Cauchy problem (3.1.4)+(3.1.2), that is,  utt + σut − aij (Du)∂i ∂j u = f , (4.5.1) u(0) = u0 , ut (0) = u1 , under the sole assumption that the data u0 , u1 and f are sufficiently small, their size being determined by the value of σ and the ellipticity constant α0 of (3.2.1). If f decays to 0, we also find a stability result analogous to the decay of y to 0 for (4.1.7). When f ≡ 0, this result was essentially proven by

176

4. Global Existence

Matsumura [111], and later extended (see, e.g., Racke [135]) to nonlinear dissipative wave equations of the form (4.5.2)

utt + ut − ∂j (aij (t, x) ∂i u) = F (t, x, u, ut , ∇u, ∇ut , ∂x2 u) .

In the non homogenous case, the large time behavior of f plays an essential role in the type of global existence results one can obtain for (4.5.1); in particular, in order to obtain solutions that remain bounded as t → +∞, mere boundedness of f is not sufficient, as we can already see from the linear equation (4.5.3)

utt + 2 ut − Δu = f

(see section 4.5.1 below). Rather, an additional integrability condition at infinity seems to be required, either with respect to t, or with respect to x. The corresponding results are different; for example, in the second case they also depend on the dimension N . At present, we do not know whether these additional integrability conditions are actually necessary, although our results on (4.5.3) seem to indicate that they almost are. Another indication in this direction is that, as we shall see in section 4.6, similar conditions seem to be needed for the parabolic Cauchy problem (3.1.7)+(3.4.1) as well.

4.5.1. The Linear Dissipative Equation. In this section we report from Milani and Volkmer [121] some results on the linear equation (4.5.3) which may shed some light on the role played by the asymptotic properties of the source term f . 1. The Solution Kernel. Given a smooth function g : RN → R, we consider the homogeneous Cauchy problem (4.5.4)

utt + 2 ut − Δu = 0 ,

with initial data (4.5.5)

u(0) = 0 ,

ut (0) = g .

By standard Fourier transform techniques, we immediately verify that its solution is (4.5.6)

u(t, x) = [Wd (t, ·) ∗ g](x) ,

4.5. Global Existence for Dissipative Equations

177

where Wd is the so-called dissipative wave kernel, defined by (4.5.7)

( ⎧ sinh( 1 − |ξ|2 t) ⎪ − t ⎪ ( e ⎪ ⎪ ⎪ 1 − |ξ|2 ⎪ ⎪ ⎨ −t te ˆ d (t, ξ) := h(t, ξ) := W ⎪ ⎪ ( ⎪ ⎪ ⎪ ⎪ − t sin(( |ξ|2 − 1 t) ⎪ ⎩ e |ξ|2 − 1

if

|ξ| < 1 ,

if

|ξ| = 1 ,

if

|ξ| > 1 .

More generally, the solution to the homogenous Cauchy problem for (4.5.4) is (4.5.8)

uhom (t, x) = [Wd (t, ·) ∗ (2u0 + u1 )](x) + ∂t ([Wd (t, ·) ∗ u0 ])(x) ,

and, by Duhamel’s formula (variation of parameters), that of the non homogenous one, i.e., for (4.5.3), by  t hom (4.5.9) u(t) = u (t) + Wd (t − θ) ∗ f (θ) dθ . 0

We can then use the following properties of the solution kernel h to determine the asymptotic behavior of (4.5.8) and (4.5.9). Proposition 4.5.1. For all t ≥ 0 and ξ ∈ RN , √ 1 5 (4.5.10) |h(t, ξ)| ≤ , |ξ| |h(t, ξ)| ≤ , 2 2

|ht (t, ξ)| ≤

3 . 2

In addition, if |ξ| ≤ 1, (4.5.11)

0 ≤ h(t, ξ) ≤ 2 e− t|ξ|

2 /2

,

while if |ξ| ≥ 1, (4.5.12)

|h(t, ξ)| ≤ t e− t ,

|ξ| |h(t, ξ)| ≤

Proof. 1) If |ξ| < 1, noting that the function x → we immediately have that (4.5.13)

0 ≤ h(t, ξ) ≤ e− t sinh t ≤

If |ξ| = 1, h(t, ξ) = t e− t ≤ (4.5.14)

√

2 (1 + t) e− t .

sinh x x

is increasing in R>0 ,

1 . 2

≤ 12 ; if instead |ξ| > 1, ( |ξ|2 − 1|)| 1 1 − t | sin(t ( |h(t, ξ)| ≤ e ≤ t e− t ≤ ≤ . 2 e 2 |ξ| − 1 1 e

178

4. Global Existence

This √ proves the first of (4.5.10). To prove the second, we use the first if |ξ| ≤ 5, and ( | sin(t |ξ|2 − 1|)| |ξ| 1√ −t ( (4.5.15) |ξ| |h(t, ξ)| ≤ e |ξ| 5 ≤( ≤ 2 |ξ|2 − 1 |ξ|2 − 1 √ if |ξ| ≥ 5. The third of (4.5.10) follows immediately from the identity ht (t, ξ) = − h(t, ξ) + e− t Cos(t, ξ) ,

(4.5.16) where

( cosh( 1 − |ξ|2 t) , ( cos( |ξ|2 − 1 t) ,



(4.5.17)

Cos(t, ξ) :=

using the bounds |h(t, ξ)| ≤

1 2

if

|ξ| < 1 ,

if

|ξ| > 1 ,

and |Cos(t, ξ)| ≤ et . √

2) We proceed to prove (4.5.11). If |ξ| ≤ 38 , from (4.5.7) √ 2 e− t(1− 1−|ξ| ) 3 2 1 (4.5.18) h(t, ξ) ≤ ≤ e− t |ξ| /2 . 2 8 2 1− 9 If instead (4.5.19)

√ 8 3

≤ |ξ| ≤ 1, since sinh x ≤ x ex if x ≥ 0, √ 2 2 h(t, ξ) ≤ t e− t(1− 1−|ξ| ) ≤ 2 e− t |ξ| /2 ,

where the second inequality follows by maximizing the function t → t e− at ( 2 on [0, +∞[, with a = 1 − 1 − |ξ|2 − |ξ|2 . 3) Finally, √ the first of (4.5.12) is contained in (4.5.14); as for the second, if 1 ≤ |ξ| ≤ 2, by the inequality | sin x| ≤ |x| we obtain √ (4.5.20) |ξ| |h(t, ξ)| ≤ 2 t e− t , √ while if |ξ| ≥ 2, (4.5.21)

√ |ξ| |ξ| |h(t, ξ)| ≤ e− t ( ≤ 2 e− t . |ξ|2 − 1 

In either case, (4.5.12) follows.

Remarks. 1) More generally, instead of (4.5.4), we could consider, as in (4.5.1), the equation (4.5.22)

utt + σ ut − Δu = 0 ,

with σ > 0. Then, as in (4.5.6), the solution to the Cauchy problem (4.5.22)+(4.5.5) would be given by (4.5.23)

(σ)

u(t, x) = [Wd (t, ·) ∗ g](x) ,

4.5. Global Existence for Dissipative Equations

179

(σ)

for a suitable dissipative wave kernel Wd , defined as in (4.5.7), and satisfying bounds similar to those of Proposition 4.5.1. In this notation, the (2) kernel Wd of (4.5.7) corresponds to Wd of (4.5.23). 2) When N = 1, the inverse Fourier transform of the function ξ → h(t, ξ) can be determined in terms of Bessel functions (Erd´ely et al. [46]). In fact, equation (4.5.4), which when N = 1 reads utt + 2ut − uxx = 0 ,

(4.5.24)

can be solved explicitly, i.e., without resorting to the Fourier transform, by the Riemann’s method of integration (see, e.g., Ludford [106]). To this end, it is convenient to first transform (4.5.24) into wtt − wxx − w = 0 ,

(4.5.25)

where w(t) = et u(t). The Riemann function corresponding to (4.5.25) is ( (4.5.26) R(t, x; η, ξ) := I0 (t − η)2 − (x − ξ)2 , where I0 is the zero-order modified Bessel function of the first kind. By (7) of Ludford [106], the solution of (4.5.25) is (4.5.27) 1 w(t, x) = u0 (x − t) + u0 (x + t) + 2



x+t 

u1 (ξ)I0 (ρ(t, x; ξ))

x−t

 t u0 (ξ) I1 (ρ(t, x; ξ)) dξ , ρ(t, x; ξ) ( where I1 = I0  , and ρ(t, x; ξ) := t2 − (x − ξ)2 . Then, the solution of − t (4.5.24) is u(t, x) := e w(t, x). Of course, this solution coincides with the one given by (4.5.8). In particular, if u0 = 0, we find that  x+t ( 1 −t (Wd ∗ u1 )(x) = 2 e u1 (ξ) I0 ( t2 − (x − ξ)2 ) dξ −

(4.5.28) =

1 −t 2e



x−t t

u1 (x − z) I0 (

−t

(

t2 − z 2 ) dz ;  

=: Φ(t,z)

that is, u is given by the convolution u(t, x) = e− t [Φ(t, ·) ∗ u1 ](x), t ≥ 0. The explicit formula (4.5.28) yields additional information on u, that would not be available from (4.5.6). For example, since I0 (x) > 0 for all x ∈ R, we deduce from (4.5.28) that if u0 = 0 and u1 ≥ 0, then u(t, x) ≥ 0 for all x ∈ R.1  1 We

are grateful to Prof. H. Volkmer for the material reported in this second Remark.

180

4. Global Existence

2. Linear Decay Estimates for the Homogeneous Equation. The explicit formula (4.5.6) allows us to show that solutions of (4.5.4) satisfy the following decay estimates. Proposition 4.5.2. Given k, m ∈ N, and q ∈ [1, 2], set  (4.5.29) νq (k, m) := N4 2q − 1 + k + 12 m . Let g ∈ H k+m−1 ∩ Lq . Then, for any multi-index α, with |α| = m,  (4.5.30) ∂tk ∂xα (Wd (t) ∗ g) ≤ C (1 + t)− νq (k,m) ∂xk+m−1 g + |g|q , if k + m − 1 ≥ 0, while if k = m = 0, (4.5.31)

Wd (t) ∗ g ≤ C (1 + t)− νq (0,0) (g−1 + |g|q ) .

In (4.5.30) and (4.5.31), the constant C depends on N , k, m, and q. The same results hold for the equation (4.5.32)

wtt + σ wt − aij ∂i ∂j w = 0 ,

where σ > 0 and A = [aij ] is a positive definite matrix with constant entries. Proof. 1) We follow Matsumura [110], where estimates (4.5.30) are stated together with similar estimates on |∂tk ∂xα [Wd (t) ∗ g]|∞ , but only the proof of the latter are explicitly given (and only for q = 1, 2). Fix k, m, and q. By Parseval’s formula (1.5.9), setting u(t) := 12 e− t [Wd (t) ∗ g], (4.5.33)

∂tk ∂xα u(t)2 = [F (∂tk ∂xα u)](t)2  ≤ |ξ|2m |∂tk u ˆ(t, ξ)|2 dξ =: I(t) .

We split the integral into the sum (4.5.34)

I(t) =

4  j=1

Ij (t) :=

4   j=1

|ξ|2m |∂tk u ˆ(t, ξ)|2 dξ , Ωj

where Ω1 := { |ξ| > 2 }, Ω2 := { 1 < |ξ| < 2 }, Ω3 := { 12 < |ξ| < 1 }, and Ω4 := { |ξ| < 12 }. In analogy to (4.5.17), we set (  sinh( 1 − |ξ|2 t) , if |ξ| < 1 , (4.5.35) Sin(t, ξ) := ( sin( |ξ|2 − 1 t) , if |ξ| > 1 . 2) Assume first that k + m − 1 ≥ 0. We recall from (4.5.6) and (4.5.7) that (4.5.36)

u ˆ(t, ξ) = (2π)N/2 h(t, ξ) gˆ(ξ) .

4.5. Global Existence for Dissipative Equations

181

By Leibniz’ formula, when |ξ| = 1, ∂tk h(t, ξ) =

(

=

(

1 | |ξ|2 − 1 |

(4.5.37)

Setting R(ξ) :=

k  k

1 | |ξ|2 − 1 |

j

∂tk−j e− t ∂tj Sin(t, ξ) .

j=0

( |ξ|2 − 1, we compute that, in Ω1 ∪ Ω2 , 

(4.5.38)

∂tk (e− t Sin(t, ξ))

|∂tj Sin(t, ξ)|

=

(R(ξ))j | sin(R(ξ) t)|

if

j

is even ,

(R(ξ))j | cos(R(ξ) t)|

if

j

is odd .

To estimate I1 (t), we use the fact that, in Ω1 , 1 ≤ (4.5.39) |I1 (t)| ≤ C e

−2 t

 |ξ|2m Ω1

≤Ce

−2 t



1 2

|ξ| ≤ R(ξ) ≤ |ξ|. Then,

(R(ξ))2k |ˆ g(ξ)|2 dξ (R(ξ))2

|ξ|2(m+k−1) |ˆ g(ξ)|2 dξ ≤ C e−2 t ∂xk+m−1 g2 .

Next, (4.5.40) |I2 (t)| ≤ C1 e

− 2t 2

t

k   j=0 j even

+ C2 e− 2t

k   j=0 j odd

|ξ|

2m

Ω2

   sin(t R(ξ)) 2   |ˆ (R(ξ))  g (ξ)|2 dξ t R(ξ)  2j

|ξ|2m (R(ξ))2j−2 | cos(t R(ξ))|2 |ˆ g (ξ)|2 dξ ; Ω2

and since | sinx x | ≤ 1 for x > 0, and, in Ω2 , both |ξ| ≥ 1 and 0 < R(ξ) ≤ |ξ| ≤ 2, k  

|I2 (t)| ≤ 4 C1 e− 2t t2

j=0 j even

(4.5.41) + C2 e

− 2t

|ξ|2m |ξ|2j−2 |ξ|2 |ˆ g(ξ)|2 dξ Ω2

k  

j=0 j odd

|ξ|2m |ξ|2j−2 |ˆ g (ξ)|2 dξ Ω2

182

4. Global Existence

from which (4.5.42)



|I2 (t)| ≤ (4 C1 + C2 )

k +1 2

≤ C e− 2t (1 + t)2

e

− 2t

 (1 + t)

|ξ|2m+2k−2 |ˆ g (ξ)|2 dξ

2 Ω2

 |ξ|2(m+k−1) |ˆ g (ξ)|2 dξ .

( 3) For the estimate of I3 (t), we set√S(ξ) := 1 − |ξ|2 , and use the facts x that sinh ≤ ex for x > 0 and S(ξ) ≤ 23 in Ω3 . Thus, x (4.5.43) |I3 (t)| ≤ C1 e

k  

− 2t 2

t

j=0 j even

+ C2 e

− 2t

|ξ| Ω3

k   j=0 j odd

≤ C e− t/4 (1 + t)2

2m

   sinh(t S(ξ)) 2   |ˆ (S(ξ))  g (ξ)|2 dξ t S(ξ)  2j

|ξ|2m (S(ξ))2j−2 |cosh(t S(ξ))|2 |ˆ g (ξ)|2 dξ Ω3



|ξ|2m |ˆ g (ξ)|2 dξ Ω3  2k−2 − t/4 2 ≤ 2 Ce (1 + t) |ξ|2m+2k−2 |ˆ g (ξ)|2 dξ Ω3  ≤ C e− t/4 (1 + t)2 |ξ|2(m+k−1) |ˆ g(ξ)|2 dξ , having also used the inequalities e−2+2 |ξ| < 1.

√

1−|ξ|2

≤ e−2+

√ 3

4) For the estimate of I4 (t), we need the inequalities (4.5.44)

− |ξ|2 ≤ − 1 +

(

1 − |ξ|2 ≤ −

1 2 |ξ| , 2

which are immediate, as well as, for α, β ∈ R>0 ,  ∞ 2 (4.5.45) rα e− β r dr ≤ Cα β −(α+1)/2 , 0

√ which follows by the change of variable s = β r, and  2 (4.5.46) sup rα e− β r ≤ Cα β − α/2 . r>0

≤ e−1/4 for

1 2


2, so

2/p 



I41 (t) ≤

|ˆ g (ξ)| dξ

|ξ|

p

Ω4

2 cp (m+2k) − cp t |ξ|2

e

1−2/p dξ

.

Ω4

By the Hausdorff-Young inequality, 

2/p |ˆ g(ξ)| dξ p

(4.5.55)

≤ |ˆ g |2p ≤ C |g|2q ;

Ω4

moreover, by (4.5.45) again,  2 (4.5.56) |ξ|2 cp (m+2k) e− cp t |ξ| dξ ≤ C (1 + t)− (N +2cp (m+2k))/2 , Ω4

so that we deduce from (4.5.54) that (4.5.57)

I41 (t) ≤ C |g|2q (1 + t)− (N +2cp (m+2k))/(2cp ) = C |g|2q (1 + t)− 2 νq (k,m) ,

where the last step follows from the identity

1 cp

=

p−2 p

=

2 q

− 1.

4c) If q = 2, we use (4.5.46) to estimate

(4.5.58)

  2(m+2k) −(1+t)|ξ|2 I41 (t) ≤ C sup |ξ| |ˆ g (ξ)|2 dξ e |ξ|≤ 12

≤ C (1 + t)−(m+2k) g2 = C |g|22 (1 + t)− 2 ν2 (k,m) . In conclusion, we deduce that, if k+m−1 ≥ 0, (4.5.30) follows from (4.5.34), . . . , (4.5.58). 5) To show (4.5.31), we first recall from (1.5.15) of part 2 of section 1.5.1 that, if g ∈ H − 1 , the integral  1 (4.5.59) G := |ˆ g (ξ)|2 dξ = g2− 1 1 + |ξ|2 is finite. Then, we replace estimates (4.5.39) and (4.5.42) with, respectively,  |ˆ g(ξ)|2 − 2t |I1 (t)| ≤ C e dξ 2 Ω1 |ξ| − 1  (4.5.60) |ˆ g (ξ)|2 ≤ 2 C e− 2t dξ ≤ C1 e− 2t G 2 Ω1 1 + |ξ|

4.5. Global Existence for Dissipative Equations

and |I2 (t)| ≤ C e

− 2t 2





t

Ω2

(4.5.61) ≤ 5C e

− 2t 2



t

Ω2

sin(t R(ξ)) t R(ξ)

185

2 |ˆ g (ξ)|2 dξ

|ˆ g (ξ)|2 dξ ≤ C2 e− t G . 1 + |ξ|2

x For I3 (t), using also the fact that the function x → sinh is increasing on x ]0, +∞[, we replace (4.5.43) with     sinh(t S(ξ)) 2 − 2t 2   g (ξ)|2 dξ |I3 (t)| ≤ C e t  t S(ξ)  |ˆ Ω3  √ 2    sinh t 3    2 2   |ˆ √ ≤ C e− 2t t2   g (ξ)| dξ t 3 Ω3   (4.5.62) 2  2 √ 1 + |ξ| ≤ C e−(2− 3)t |ˆ g (ξ)|2 dξ 2 Ω3 1 + |ξ|  |ˆ g (ξ)|2 − t/4 ≤ Ce dξ = C3 e− t/4 G . 1 + |ξ|2

Next, we replace (4.5.50) with



e− 2(1+S(ξ))t |ˆ g (ξ)|2 dξ

I42 (t) ≤ C Ω4

(4.5.63) ≤ Ce

−2t

 Ω4

|ˆ g (ξ)|2 dξ ≤ C4 e− 2t G . 1 + |ξ|2

The estimate of I41 (t) remains the same as above, with k = m = 0, since it involves only the norm of g in Lq , and not that of g in H − 1 . 6) Finally, the result also holds for (4.5.32), since this equation can be transformed into (4.5.4) (with the coefficient of ut replaced by σ), by the change of variables u(t, x) = w(t, A1/2 x). In this case, the value of C in (4.5.30) and (4.5.31) also depends on σ. This completes the proof of Proposition 4.5.2.  Remarks. The decay rates (4.5.30) show that each derivative of u with respect to t yields the same contribution as any combination of two derivatives with respect to x. Indeed, νq (k, m + 2) = νq (k + 1, m). This is a typically parabolic feature, exemplified by the linear heat equation in (2.5.16), read as ut = Δ u. We shall return to this phenomenon in Chapter 5. In part 5 below, we show that the rates (4.5.30) are optimal. Finally, we mention that additional estimates on the linear dissipative equation (4.5.4) can be found in Caviglia and Morro [23]; more precise decay results in Lp (R3 ) can be found, e.g., in Nishihara [129]. Decay estimates for solutions of equation (4.5.4) in an exterior domain of RN can be found in Racke [134], and

186

4. Global Existence

for equations with a time-dependent dissipation, in Reissig [137]; for other types of equations, with stronger dissipativity terms, such as, e.g., − Δut instead of ut , see also Karch [68], or Shibata [146].  Corollary 4.5.1. Let u be a smooth solution of the Cauchy problem for (4.5.32); let k, m ∈ N, α ∈ NN , with |α| = m, and q ∈ [1, 2]. Then, u satisfies the estimates (4.5.64)

∂tk ∂xα u(t) ≤ C (1 + t)− νq (k,m) Ck,m,q (u, 0) ,

(4.5.65)

∂tk u(t)m ≤ C (1 + t)− νq (k,0) C˜k,m,q (u, 0) ,

(4.5.66)

∂tk Du(t)m ≤ C (1 + t)− νq (k,1) C˜k+1,m,q (u, 0) ,

where (4.5.67) Ck,m,q (u, 0) := ∂xk+m u(0) + |u(0)|q + ∂xk+m−1 ut (0) + |ut (0)|q , (4.5.68) C˜k,m,q (u, 0) := u(0)k+m + |u(0)|q + ut (0)k+m−1 + |ut (0)|q . In addition, as t → +∞, u(t) → 0 as well. Proof. Estimates (4.5.65) and (4.5.66) are a consequence of (4.5.30), applied to the decomposition (4.5.8), and of the inequalities νq (k, m) ≥ νq (k, 1) if m ≥ 1, and νq (k + 1, m) > νq (k, m + 1) if m ≥ 0. If 1 ≤ q < 2, the decay of u(t) follows from (4.5.65) with k = 0, noting that νq (0, 0) > 0 if q < 2. If q = 2, we refer again to (4.5.8): at first, by (4.5.30) with k = 1 and m = 0, ∂t ([Wd (t) ∗ u(0)]) ≤ C (1 + t)− 1 u(0) ;

(4.5.69)

next, we note that, by the first of (4.5.10), there is a constant C such that, for w0 := 2u0 + u1 ,  2 Wd (t) ∗ w0  ≤ 4 C |w ˆ0 (ξ)|2 |h(t, ξ)|2 dξ  (4.5.70) ≤ C |w ˆ0 (ξ)|2 dξ . Since h(t, ξ) → 0 as t → +∞ for all ξ = 0, by Lebesgue’s dominated convergence theorem we conclude that u(t) → 0 as t → +∞. In fact, if u(0) and ut (0) ∈ L2 ∩ L1 , (4.5.31) implies that u(t) ≤ C (1 + t)− N/4 ,

(4.5.71)

which proves the last claim of Corollary 4.5.1.



We note in passing that (4.5.30) with q = 2 implies that solutions of the homogeneous equation (4.5.4) are bounded in the phase space H m+1 × H m , as they satisfy the estimate (4.5.72)

sup (u(t)m+1 + ut (t)m ) ≤ C (u(0)m+1 + ut (0)m ) . t≥0

4.5. Global Existence for Dissipative Equations

187

3. Bounded Solutions of the Non Homogeneous Equation.2 We now present some sufficient conditions for the boundedness of solutions of the non homogeneous equation (4.5.3) in H m+1 . We first show that u(t) ∈ H m+1 for all t ≥ 0. Recalling (4.5.9), we use (4.5.72) to bound uhom (t); thus, it is sufficient to consider the case u0 , u1= 0. Then, the solution of (4.5.3) is formally given by u(t, x) = F −1 v(t, ·) (x), where  (4.5.73)

v(t, ξ) = (2 π)

t

N/2

fˆ(t − θ, ξ) h(θ, ξ) dθ ,

0

with h as in (4.5.7). In particular, the right side of (4.5.73) is well defined if f ∈ L∞ (0, +∞; H m ) for some m ∈ N; in fact, in this case, v(t) ∈ L2 and u(t) ∈ H m+1 for all t ≥ 0. To see this, let first a, b ∈ R≥0 , with a < b. Since fˆ ∈ L∞ (0, +∞; L2 ), and the function (t, ξ) → |fˆ(t, ξ)|2 is measurable and positive, we can define χ : RN → [0, +∞] by  (4.5.74)

b

χ(ξ) :=

|fˆ(t, ξ)|2 dt .

a

By Tonelli’s theorem,   b  b 2 ˆ χ(ξ) dξ = |f (t, ξ)| dξ dt = fˆ(t)2 dt a a (4.5.75)  b = f (t)2 dt ≤ (b − a) f 2L∞ (0,+∞;H m ) ; a

hence, by Fubini’s theorem, the function t → fˆ(t, ξ) is locally integrable for almost all ξ ∈ RN . To see that v(t) ∈ L2 , recalling the first of (4.5.10) we compute that  

 |v(t, ξ)| dξ = (2 π) 2

t

N

2 fˆ(t − θ, ξ) h(θ, ξ) dθ

0

 

t

≤ C 0

(4.5.76)

≤ ≤ ≤

2 Most

C 4 C 4 C 4

 t t 



t

|fˆ(t − θ, ξ)|2 dθ

dξ

|h(θ, ξ)| dθ dξ 2

0

|fˆ(t − θ, ξ)|2 dξ dθ

0 t

t

f (τ )2 dτ

0

f 2L∞ (0,+∞;L2 ) t2 .

of the results of this and the next paragraph are due to Prof. H. Volkmer.

188

4. Global Existence

To see that u(t) ∈ H m+1 , recalling (4.5.73) we compute instead that (4.5.77)

 I(t) := (1 + |ξ|2 )m+1 |ˆ u(t, ξ)|2 dξ  = (1 + |ξ|2 )m+1 |v(t, ξ)|2 dξ 



t

2 |fˆ(t − θ, ξ)| |h(θ, ξ)| dθ

= (2π) (1 + |ξ| ) 0  ≤ C (1 + |ξ|2 )m f1 (t, ξ) f2 (t, ξ) dξ , N

2 m+1

where (4.5.78)

 

(4.5.79)

t

f1 (t, ξ) :=

dξ

|fˆ(t − θ, ξ)|2 dθ ,

0 t

f2 (t, ξ) :=

(1 + |ξ|2 )|h(θ, ξ)|2 dθ .

0

By the first and second of (4.5.10), (1 + |ξ|2 )|h(θ, ξ)|2 ≤ 14 + 54 ; thus,  t I(t) ≤ C t (1 + |ξ|2 )m |fˆ(t − θ, ξ)|2 dξ dθ 0  t ≤ Ct (4.5.80) f (t − θ)2m dθ 0

≤ C f 2L∞ (0,+∞;H m ) t2 . This ends the proof that u(t) ∈ H m+1 ; note, however, that (4.5.80) leaves open the possibility that the map t → u(t)m+1 may be unbounded as t → +∞. Our next results give sufficient conditions on f so that this map, or at least the map t → ∇u(t)m , is bounded. These results depend on the asymptotic properties of f , which in turn may depend on the space dimension N . More precisely, we find that ∇u(t)m remains bounded if N ≥ 3, while boundedness of u(t)0 only holds if N ≥ 5. As follows from Propositions 4.5.5 and 4.5.6 below, the latter condition is sharp. Proposition 4.5.3. Let f ∈ L∞ (0, +∞; H m ), for some m ∈ N, and suppose there are positive constants λ and Λ such that for all t ≥ 0,  |fˆ(t, ξ)|2 (4.5.81) dξ ≤ Λ . 4+λ |ξ|≤1 |ξ| Then, the function t → u(t)m+1 is bounded on [0, +∞[.  In particular, 2 condition (4.5.81) holds for all λ ∈ ]0, λp [, λp := N p − 1 − 4, if f ∈

4.5. Global Existence for Dissipative Equations

L∞ (0, +∞; H m ∩ Lp ), with 1 ≤ p < λp > 0).

2N N +4

189

(which requires N ≥ 5 and implies

Proof. By Schwarz’ inequality, we deduce from (4.5.73) that, for all t ≥ 0 (neglecting a non essential multiplicative constant), 

t

|v(t, ξ)| ≤ 2

(4.5.82)



t

|fˆ(t − θ, ξ)|2 |h(θ, ξ)| dθ

0

|h(θ, ξ)| dθ .

0

If |ξ| ≥ 1, by (4.5.12) we obtain from (4.5.82) that (4.5.83)



t

(1 + |ξ| ) |v(t, ξ)| ≤ 2

2



0



t

+ 

t



t

|fˆ(t − θ, ξ)|2 |ξ| |h(θ, ξ)| dθ

|fˆ(t − θ, ξ)|2 θ e− θ dθ



t

θ e− θ dθ

0



t

+ 2

|fˆ(t − θ, ξ)|2 (1 + θ)e− θ dθ



0 t

≤

|ξ| |h(θ, ξ)| dθ

0

0



|h(θ, ξ)| dθ

0

0

≤

t

|fˆ(t − θ, ξ)|2 |h(θ, ξ)| dθ

t

(1 + θ) e− θ dθ

0

|fˆ(t − θ, ξ)|2 θ e− θ dθ + 4



0

t

|fˆ(t − θ, ξ)|2 (1 + θ) e− θ dθ

0



t

≤ 5

|fˆ(t − θ, ξ)|2 (1 + θ) e− θ dθ .

0

By Fubini’s theorem, (4.5.83) implies that  (1 + |ξ|2 )m+1 |v(t, ξ)|2 dξ |ξ|≥1

(4.5.84)



t

≤5

(1 + θ) e− θ

 |ξ|≥1

0

≤ 10

sup f (t)2m t≥0

(1 + |ξ|2 )m |fˆ(t − θ, ξ)|2 dξ dθ

.

If 0 < |ξ| ≤ 1, recalling that, by (4.5.11) of Proposition 4.5.1, 

t

(4.5.85) 0



t

|h(θ, ξ)| dθ ≤ 2 0

e− θ|ξ|

2 /2

dθ ≤

4 , |ξ|2

190

4. Global Existence

we obtain from (4.5.82) that |v(t, ξ)|2

 t 8 2 |fˆ(t − θ, ξ)|2 e− θ |ξ| /2 dθ |ξ|2 0  t 8 2 |fˆ(t − θ, ξ)|2 |ξ|2+λ e− θ |ξ| /2 dθ . 4+λ |ξ| 0

≤

(4.5.86) =

If 0 ≤ t ≤ 1, we proceed with |v(t, ξ)|2 ≤

(4.5.87)

8 |ξ|4+λ



t

|fˆ(t − θ, ξ)|2 dθ ;

0

while if t > 1, we decompose (4.5.88) 8 |v(t, ξ)| ≤ 4+λ |ξ|



1

2

8 + |ξ|4+λ



t

|fˆ(t − θ, ξ)|2 dθ

0

|fˆ(t − θ, ξ)|2 θ−1−λ/2 (θ |ξ|2 )1+λ/2 e− θ |ξ|

2 /2

dθ .

1

Thus, by (4.5.87) and (4.5.88), |v(t, ξ)|2 ≤

(4.5.89)

8 Mλ |ξ|4+λ



t

|fˆ(t − θ, ξ)|2 min(1, θ−1−λ/2 ) dθ ,

0

where Mλ := max{1, maxr≥0 (r1+λ/2 e− r/2 )}. Again by Fubini’s theorem, (4.5.90)  |ξ|≤1

(1 + |ξ|2 )m+1 |v(t, ξ)|2 dξ

≤2

 t m+4

Mλ

|ξ|≤1 t

0



|fˆ(t − θ, ξ)|2 min(1, θ−1−λ/2 ) dξ dθ |ξ|4+λ

min(1, θ−1−λ/2 ) dθ ≤ 2m+4 Mλ Λ (1 + 2 λ− 1 ) .

≤ 2m+4 Mλ Λ 0

Together with (4.5.84), (4.5.90) implies the asserted boundedness of the function t → u(t)m+1 as t → +∞. The second part of the proposition is proved as in the second part of Proposition 1.5.4, noting that, now, the p number r appearing in the proof of that proposition, i.e., r = 2−p , satisfies N the inequality r < 4 , and the smallness condition on λ guarantees that the function | · |−4−λ is in Lrloc (RN ).  In an analogous way, we can also prove

4.5. Global Existence for Dissipative Equations

191

Proposition 4.5.4. Let f ∈ L∞ (0, +∞; H m ), for some m ∈ N, and suppose there are positive constants λ and Λ such that  |fˆ(t, ξ)|2 (4.5.91) dξ ≤ Λ 2+λ |ξ|≤1 |ξ| for all t ≥ 0. Then, the function t → ∇u(t, ·)m is bounded on[0, +∞[. In 2 particular, inequality (4.5.91) holds for all λ ∈ ]0, λp [, λp := N p − 1 − 2, if f ∈ L∞ (0, +∞; H m ∩ Lp ), with 1 ≤ p < implies λp > 0).

2N N +2

(which requires N ≥ 3 and

4. The Autonomous Case. If f is independent of t, propositions 4.5.3 and 4.5.4 can be refined as follows. Proposition 4.5.5. Let f ∈ H m , for some m ∈ N. Then, u(t) ∈ H m+2 ; moreover, the functions t → u(t, ·)m+2 (respectively, t → ∇u(t, ·)m+1) are bounded on [0, +∞[ if and only if (4.5.81) (respectively, (4.5.91)) holds with λ = 0; that is, if, respectively,  |fˆ(ξ)|2 Λ1 := (4.5.92) dξ < +∞ , 4 |ξ|≤1 |ξ|  (4.5.93)

Λ2 :=

|ξ|≤1

|fˆ(ξ)|2 dξ < +∞ . |ξ|2

In particular, condition (4.5.92) holds if f ∈ H m ∩ Lp , with 1 ≤ p < N2N +4 (which requires N ≥ 5). If N ≤ 4, there are functions f ∈ H m ∩ L1 , such that (4.5.92) fails, and u(t)m+2 is unbounded. Likewise, condition (4.5.93) holds if f ∈ H m ∩ Lp , with 1 ≤ p < N2N +2 (which requires N ≥ 3). If N ≤ 2, m 1 there are functions f ∈ H ∩ L , such that (4.5.93) fails, and ∇u(t)m+1 is unbounded. Proof. If f does not depend on t, (4.5.73) reads (neglecting again the non essential multiplicative constant (2 π)− N )  t ˆ (4.5.94) v(t, ξ) = f (ξ) h(θ, ξ) dθ =: fˆ(ξ) H(t, ξ) . 0

Now, (4.5.95)

1 H(t, ξ) = 2 |ξ|



 1 − e− t

Sin(t, ξ) Cos(t, ξ) + ( |1 − |ξ|2 |

 ,

where the functions Cos and Sin are defined in (4.5.17) and (4.5.35). It is straightforward to deduce from (4.5.95) that, for all t ≥ 0 and ξ ∈ RN \ {0}, 3 1 (4.5.96) |H(t, ξ)| ≤ 2 , lim H(t, ξ) = 2 . t→+∞ |ξ| |ξ|

192

4. Global Existence

Together with (4.5.94), the first of (4.5.96) implies that, for all t ≥ 0,  |fˆ(ξ)|2 (4.5.97) u(t)2m+2 ≤ 9 (1 + |ξ|2 )m+2 dξ . |ξ|4 RN Splitting the integral over the regions |ξ| ≤ 1 and |ξ| ≥ 1, we deduce from (4.5.97) that, if (4.5.92) holds, (4.5.98)

u(t)2m+2 ≤ 36 (2m Λ1 + f 2m ) ;

that is, u(t)m+2 is bounded in t. Conversely, assume that (4.5.92) does not hold. Then, for all n ∈ N≥1 , by (4.5.94),  2 (4.5.99) u(n)m+2 = (1 + |ξ|2 )m+1 |fˆ(ξ)|2 |H(n, ξ)|2 dξ ; thus, by the second of (4.5.96) and Fatou’s lemma,  |fˆ(ξ)|2 2 lim inf u(n)m+2 ≥ (1 + |ξ|2 )m+2 dξ n→+∞ |ξ|4 RN (4.5.100)  |fˆ(ξ)|2 ≥ dξ = +∞ , |ξ|4 RN contradicting the assumption that u(t)m+2 is bounded. Finally, the func2 tion x → e− |x| is in H m ∩ L1 for every m ∈ N, but fails to satisfy (4.5.92) if N ≤ 4. The claims on ∇u(t)m+1 are proven similarly.  Remark. Comparing Proposition 4.5.5 with Propositions 4.5.3 and 4.5.4, the natural question arises whether we can take λ = 0 in (4.5.81) or (4.5.91). In this case, our proof would allow for logarithmic growth of u(t)m+1 or ∇u(t)m as t → +∞, as we see from the last line of (4.5.90), because when λ = 0,  t (4.5.101) min{1, θ− 1 } dθ = ln t , t≥1. 1

However, we do not know if this can truly happen.



We conclude with an example where (4.5.92) fails, without the   condition 2N explicit requirement that f ∈ Lp , p ∈ 1, N +4 . Proposition 4.5.6. Let N ≤ 4, and assume that f ∈ L2 is such that |fˆ(ξ)| ≥ α if |ξ| ≤ β, for some α, β > 0. Let u be the corresponding solution of (4.5.3) with u0 = 0, u1 = 0. Then, the function t → u(t)0 is not bounded. Proof. Since, as in (4.5.100), (4.5.102)

lim inf u(n)20 n→+∞

 ≥

1 ˆ 2 |f (ξ)| dξ , |ξ|4

4.5. Global Existence for Dissipative Equations

193

it is sufficient to show that condition (4.5.92) fails. To show this, we compute that  ˆ 2  β  |f (ξ)| |fˆ(ξ)|2 2 (4.5.103) dξ ≥ dξ ≥ α rN −5 dr , 4 |ξ|4 |ξ| 0 |ξ|≤β and this last integral is finite if and only if N ≥ 5.



5. Optimality of Decay Rates. In this section we briefly consider the question, whether the decay rates of estimates (4.5.30) are optimal. As a partial answer, we claim: Theorem 4.5.1. Let k, m ∈ N, g ∈ S, and set g¯ := gˆ(0). Then, for any α ∈ NN , with |α| = m, and all t ≥ 1, (4.5.104)

∂tk ∂xα [Wd (t) ∗ g] = CN g¯ t− ν1 (k,m) + ω(t) ,

where CN depends only on N , k and m, and ω is a continuous, positive function on [1, +∞[, such that ω(t) = o(t− ν1 (k,m) ) as t → +∞. As a consequence, the decay rate ν1 (k, m) in (4.5.30) is optimal, if and only if g¯ = 0. Proof. We follow Volkmer [161]. We only consider the case k = m = 0; the others can be treated in a similar way. Thus, we wish to show that, for all t ≥ 1, (4.5.105)

Wd (t) ∗ g = CN g¯ t− N/4 + ω(t) ,

for suitable constant CN and function ω, with ω(t) = o(t− N/4 ) as t → +∞. 1) Let u(t) := Wd (t) ∗ g. Then, (4.5.106)

u(t) = ˆ u(t) = (2 π) 2

2

 N

|ˆ g (ξ)|2 |h(t, ξ)|2 dξ ,

with h as in (4.5.7). We consider the Taylor expansion of gˆ at ξ = 0; that is, (4.5.107)

gˆ(ξ) = gˆ(0) + ∇ξ gˆ(0) · ξ + 12 D 2 gˆ(0)ξ · ξ + O(|ξ|3 ) .

Recalling the definition (1.5.7) of the Fourier transform, we compute that, in addition to g¯ = gˆ(0), for 1 ≤ j, k ≤ N ,  ∂ˆ g −i xj g(x) dx =: − i gj , (4.5.108) (0) = ∂ξj (2π)N/2  ∂ 2 gˆ −1 xj xk g(x) dx =: − gjk . (4.5.109) (0) = ∂ξj ∂ξk (2π)N/2 Consequently, adopting the usual summation convention over repeated indices, we obtain from (4.5.107) that     (4.5.110) gˆ(ξ) = g¯ − 12 gjk ξj ξk + O(|ξ|4 ) − i gj ξj + O(|ξ|3 ) ,

194

4. Global Existence

from which we can write |ˆ g (ξ)|2 = g¯2 − g¯ gjk ξj ξk + (gj ξj )2 + ρ(ξ) |ξ|4 (4.5.111) = g¯2 + (gj gk − g¯ gjk ) ξj ξk + ρ(ξ) |ξ|4 , where ρ is a continuous function on RN . We decompose (4.5.112)   −N 2 2 2 (2 π) |ˆ u(t)|2 = |ˆ g (ξ)| |h(t, ξ)| dξ + |ˆ g (ξ)|2 |h(t, ξ)|2 dξ |ξ|≥ 12

|ξ|≤ 12

=: Φ1 (t) + Φ2 (t) . Recalling (4.5.11), the first of (4.5.12), and the second of (4.5.44), we obtain that, for |ξ| ≥ 12 and t ≥ 1, √ 2 |h(t, ξ)| ≤ t e(−1+ 1−|ξ| )t + t e− t (4.5.113) 2 ≤ t e− t |ξ| /2 + t e− t ≤ 2 t e− t/8 . Thus, Φ1 (t) ≤ 2 t e− t/8 |ˆ g |22 = 2 t e− t/8 |g|22 .

(4.5.114)

2) To estimate Φ2 (t), we further decompose (4.5.115)



Φ2 (t) =

|ˆ g(ξ)|2

|ξ|≤ 12

1 − t |ξ|2 /2 2e

 2  2 + h(t, ξ) − 12 e− t |ξ| /2 dξ

  =: (t,ξ)



 2 |ˆ g(ξ)|2 (t, ξ) (t, ξ) + e− t |ξ| /2 dξ

= |ξ|≤ 12

+





1 4

|ˆ g (ξ)|2 e− t |ξ| dξ 2

|ξ|≤ 12

=: Φ3 (t) + Φ4 (t) .

Again by (4.5.7) and the second of (4.5.44), if |ξ| ≤ 12  √ √ e− t 2 2 ( e 1−|ξ| t − e− 1−|ξ| t 0 ≤ h(t, ξ) = 2 1 − |ξ|2 (4.5.116) √ ≤

Thus, |(t, ξ)| ≤ (4.5.117)

1 √2 (−1+ 2 3e

≤ e− t |ξ|

2 /2

1−|ξ|2 )t

.

1 − t |ξ|2 /2 , 2e

and, from (4.5.115),  2 3 Φ3 (t) ≤ 2 |ˆ g(ξ)|2 |(t, ξ)| e− t |ξ| /2 dξ . |ξ|≤ 12

We will later show that (4.5.118)

|(t, ξ)| ≤

1 2

|ξ|2 (1 + t |ξ|2 ) e− t |ξ|

2 /2

+

√1 e− t 3

;

4.5. Global Existence for Dissipative Equations

195

assuming this for the moment, we proceed from (4.5.117) with  2 Φ3 (t) ≤ C |ˆ g (ξ)|2 |ξ|2 (1 + t |ξ|2 ) e− t |ξ| dξ |ξ|≤ 12

+Ce

(4.5.119)

−t



|ˆ g(ξ)|2 e− t |ξ|

2 /2

|ξ|≤ 12

dξ

=: Φ31 (t) + Φ32 (t) . Recalling (4.5.45), (4.5.120)



Φ31 (t) ≤ C

|ˆ g|2∞

1/2

r

N +1 − t r 2

e





1/2

dr + t

r

0

N +3 − t r 2

e

dr

0

 ≤ C |g|21 (1 + t)−(N +2)/2 + (1 + t)−(N +4)/2+1 ≤ C |g|21 (1 + t)−(N/2)−1 . Since also Φ32 (t) ≤ C e− t |ˆ g |2∞ (1 + t)− N/2 ≤ C |g|21 e− t ,

(4.5.121)

we conclude from (4.5.119) that Φ3 (t) ≤ C |g|21 (1 + t)−(N/2)−1 ,

(4.5.122)

as long as (4.5.118) holds. To show the latter, we set again S(ξ) := and decompose

(

1 − |ξ|2 ,

(4.5.123) (t, ξ) = =

1 (−1+S(ξ))t 2S(ξ) e 1 (−1+S(ξ))t 2e

−



−

1 S(ξ)

1 −(1+S(ξ))t 2S(ξ) e

−1 +

1 −(1+S(ξ))t 2S(ξ) e

1 2

− 12 e− t |ξ|

2 /2

 2 e(−1+S(ξ))t − e− t |ξ| /2

=: 1 (t, ξ) + 2 (t, ξ) − 3 (t, ξ) .

Using the inequalities (4.5.124)

1 − 1 ≤ |ξ|2 , S(ξ)

0 ≤ 1 − S(ξ) − 12 |ξ|2 ≤

for |ξ| ≤ 12 , again by the second of (4.5.44) we obtain that (4.5.125)

0 ≤ 1 (t, ξ) ≤

1 2

|ξ|2 e− t |ξ|

2 /2

,

1 2

|ξ|4 ,

196

4. Global Existence

and |2 (t, ξ)| = (4.5.126)

= ≤



  − 1 1 2

1 − t |ξ|2 /2  (−1+S(ξ)+ 21 |ξ|2 )t e 2e

 1 − e(−1+S(ξ)+ 2 |ξ| )t  4 2 1 − t |ξ|2 /2 1 − e− |ξ| t/2 ≤ 14 |ξ|4 t e− t |ξ| /2 . 2e 1 − t |ξ|2 /2 2e

Since also |3 (t, ξ)| ≤

(4.5.127)

√1 e− t 3

if |ξ| ≤ 12 , we see that (4.5.118) does hold. 3) To summarize, by (4.5.112), (4.5.115), (4.5.114), and (4.5.122) we have found that (2 π)− N u(t)2 = Φ1 (t) + Φ2 (t) = Φ4 (t) + Φ1 (t) + Φ3 (t)  = Φ4 (t) + O (1 + t)−(N/2)−1 .

(4.5.128)

To evaluate Φ4 (t), we go back to (4.5.111), by which, recalling (4.5.115), we deduce that    2 4 Φ4 (t) = (gj gk − g¯ gjk ) ξj ξk + ρ(ξ) |ξ|4 e− t |ξ| dξ |ξ|≤ 12  (4.5.129) 2 + g¯2 e− t |ξ| dξ =: Φ5 (t) + Φ6 (t) . |ξ|≤ 12

√ By the change of coordinates η = t ξ,  C 2 |Φ5 (t)| ≤ N/2+1 |gj gk − g¯ gjk | |η|2 e− |η| dη √ max 1 t |η|≤ 2 t j,k  1 2 + N/2+2 max |ρ(ξ)| |η|4 e− |η| dη √ 1 t |ξ|≤ 2 |η|≤ 12 t (4.5.130)

 C 2 ≤ N/2+1 max |gj gk − g¯ gjk | + 1 |η|2 e− |η| dη j,k t ≤ C1 t−(N/2)−1 . Finally, Φ6 (t) = (4.5.131) =

g¯2



tN/2 g¯2  tN/2

e

− |η|2



 dη −

π N/2 + ω ˜ (t) ,

|η|≥ 12

√

e t

− |η|2

dη

4.5. Global Existence for Dissipative Equations

197

with ω ˜ (t) = o(t− N/2−1 ) as t → +∞. Inserting (4.5.131) and (4.5.130) into (4.5.129), and then back into (4.5.128), we finally obtain that, for t ≥ 1,  

N/2 π 1 1 −N 2 2 (4.5.132) (2 π) u(t) = g¯ N/2 + O N/2 , 4 t t +1 from which (4.5.105) follows. This concludes the proof of Theorem 4.5.1, at least for k = m = 0.  4.5.2. Bounded Global Existence. In the next section, we shall see how the results on the linear equation (4.5.3) can be applied to obtain a global existence result for the quasi-linear problem (4.5.1), when f is assumed to be bounded. In this section we present a result on the existence of a global, bounded solution to (4.5.1), under the more restrictive assumption that f be also integrable as t → +∞. Theorem 4.5.2. Let N ≥ 3, s > u0 ∈ H s+1 , u1 ∈ H s , and

N 2

+ 1, and aij ∈ C s (R1+N ). Assume that

f ∈ L1 (0, +∞; H s ) ∩ Cb ([0, +∞[; H s−1 ) .

(4.5.133)

There exists δ0 > 0 such that, if (4.5.134)

u0 s+1 + u1 s + f L1 (0,+∞;H s ) + f Cb ([0,+∞[;H s−1 ) ≤ δ0 ,

the Cauchy problem (4.5.1) has a unique solution u ∈ Zs,b (∞). In addition, if f ∈ L∞ (0, +∞; H s ), then lim Du(t)s = 0 ;

(4.5.135)

t→+∞

and if also lim f (t)s−1 = 0, then t→+∞

(4.5.136)

lim utt (t)s−1 = 0

t→+∞

as well. The size of δ0 depends on σ and the ellipticity constant α0 of (3.2.1). Proof. 1) Since f ∈ L1loc (0, +∞; H s ), Theorem 3.2.1 can be readily adapted, to yield a local solution u ∈ Zs (τ ) of (4.5.1), for some τ > 0. Indeed, we only need to recall the remark at the end of part 2 of section 2.3.1, and observe that the linear energy estimate corresponding to (2.3.29), with m = s, would 't now have, at its left side, the additional term 2σ 0 ut 2s dθ (see (4.5.158) below), and that the estimate of utt s−1 in (2.3.26) also contains the term σut s−1 (as in (4.5.147) below). Since the first additional term is positive, and the second is already estimated, in (2.3.25), we can proceed in exactly the same way as in Theorem 3.2.1. We extend the local solution u to its maximal time of existence Tc , and show that Tc cannot be finite. To this

198

4. Global Existence

end, we establish (formal) a priori estimates, which supplement the standard energy estimates we gave in the nondissipative case. We set  +∞ (4.5.137) F1 := f s dt , F2 := sup f (t)s−1 , t≥0

0

and, with auij as in (3.2.49) and Qk as in (2.3.13), abbreviate Qs (∇u) := Qs (au , ∇u); that is,  (4.5.138) Qs (∇u) := aij (Du) ∂i ∂xα u, ∂j ∂xα u . |α|≤s

2) At first, proceeding as in (4.4.2), we establish the identity  d  ut 2s + Qs (∇u) + 2 σ ut 2s dt  (4.5.139) = 2f, ut s + (R0,α + R1,α + RG,1,α ) , |α|≤s

for 0 ≤ t < Tc , where R0,α , R1,α and RG,1,α are defined in (3.3.89), (3.3.90), and (3.3.91). Next, we multiply the equation of (4.5.1) in H s also by σu, to obtain  d  σu, ut s + 12 σ 2 u2s − σ ut 2s + σ Qs (∇u) dt   (4.5.140) = σ f, us + (R2,α + RG,2,α ) , |α|≤s

where R2,α and RG,2,α are defined in (3.3.97) and (3.3.98). (4.5.139) and (4.5.140) yields  d  ut 2s + σu, ut s + 12 σ 2 u2s + Qs (∇u) dt  

Summing

=: Nsu (t)

(4.5.141)

+ σ ut 2s + σ Qs (∇u) = f, 2ut + σus  + (R0,α + R1,α + σ R2,α + RG,1,α + σ RG,2,α ) . |α|≤s

Note that Nsu (t) is the square of an equivalent norm for (u(t), ut (t)) in the phase space H s+1 × H s ; in fact, by the Cauchy-Schwarz weighted inequality σu, ut s ≤

(4.5.142)

1 3

σ 2 u2s + 34 ut 2s ,

we see that (4.5.143)

1 6

σ 2 u2s + 14 ut 2s + Qs (∇u) ≤ Nsu (·)

and (4.5.144)

Nsu (·) ≤

5 6

σ 2 u2s + 74 ut 2s + Qs (∇u) .

4.5. Global Existence for Dissipative Equations

199

3) In the sequel, we denote by κ, κ1 , . . . , various functions in K. We first estimate (4.5.145) f, 2ut + σus ≤ f s 2 ut + σ us  1/2 ≤ f s 4(ut 2s + 12 σ 2 u2s + σu, ut s ) ( ≤ 2 f s Nsu (t) . Next, we proceed as in (4.4.4), to obtain R0,α ≤ C |aij  (Du)|∞ |Dut |∞ ∇u2s

(4.5.146)

≤ κ1 (Dus−1 ) D 2 us−1 Qs (∇u) .

From the equation of (4.5.1) we obtain, using (3.2.28), that, with κ = 2 ψ, utt s−1 ≤ f s−1 + σ ut s−1 + κ(|Du|∞ ) Dus .

(4.5.147)

Thus, recalling the second of (4.5.137), D 2 us−1 (4.5.148)

≤

utt s−1 + Dus

≤

F2 + (κ(Dus−1 ) + 1 + σ) Dus

=: κ2 (F2 , Dus ) . From (4.5.146), then,  (4.5.149)

|α|≤s

R0,α

≤

κ1 (Dus ) κ2 (F2 , Dus ) Qs (∇u)

=: κ3 (F2 , Dus ) Qs (∇u) ;

note that κ3 (0, 0) = 0. As for the terms R2,α , we first consider the term with

N/2 → LN α = 0. Setting p¯ := N2N −2 , recalling (1.5.46), as well as that H (see (1.5.54)), σ R2,0 ≤ σ |aij  (Du)|∞ |∂j Du|2 |∂i u|N |u|p¯ ≤ κ(|Du|∞ ) Du1 ∇u N/2 ∇u0

(4.5.150)

≤ κ(Dus ) Du2s Du . For the other terms, proceeding as in (4.4.5) we obtain   R1,α + σ R2,α |α|≤s

(4.5.151)

≤ ≤

0 0,  (RG,1,α + σ RG,2,α ) 0 N2 + 1, and assume that aij ∈ C s (RN ), u0 ∈ H s+1 , f ∈ L1 (0, +∞; L2 ) ∩ L2 (0, +∞; H s ). There exists δ0 > 0 such that, if  ∞

2  ∞ 2 (4.6.18) u0 s+1 + f  dt + f 2s dt ≤ δ02 , 0

0

the Cauchy problem (4.6.1) has a unique solution u ∈ Ps,b (∞). Moreover, if also f ∈ L∞ (0, +∞; H s ), then lim ∇u(t)s = 0 ;

(4.6.19)

t→+∞

and if in addition lim f (t)s−1 = 0, then also t→+∞

lim ut (t)s−1 = 0 .

(4.6.20)

t→+∞

If aij ∈ C s+1 (RN ), then ∇u ∈ L2 (0, +∞; H s+1 ), and u ∈ L2loc (0, +∞; H s+2 ). The size of δ0 depends on the coefficient of ut , which in (4.6.1) is 1, and on the ellipticity constant α0 of (3.2.1) (see remark (2) after the proof of Theorem 4.5.2). Sketch of Proof. We proceed as in the proof of Theorem 4.5.2. 1) Assuming again that the life-span Tc of the solution of (4.6.1) is finite, set (4.6.21) (4.6.22)

 F3 :=

∞

 f  dθ ,

+∞

F4 := 4 0 ( 0 Δ := 2 max{u0 s+1 , 4 F3 , F4 } .

f 2s dt ,

We claim that, if Δ is sufficiently small, then, for all t ∈ [0, Tc [, (4.6.23)

u(t)s ≤ Δ .

Arguing by contradiction, assume there are T ∈ ]0, Tc [, and γ ∈ ]1, 2], such that, for all t ∈ [0, T ], (4.6.24)

u(t)s ≤ γ Δ = u(T )s .

We differentiate the equation of (4.6.1) α times with respect to x, |α| ≤ s, then multiply in L2 by 2 ∂xα u and sum the resulting identities for |α| ≤ s, to obtain  d (4.6.25) (R2,α + RG,2,α ) , u2s + 2 Qs (∇u) = 2 f, us + dt |α|≤s

with Qs (∇u), R2,α and RG,2,α defined as in (4.5.138), (3.3.97) and (3.3.98), but with the coefficients auij of (3.2.49) replaced by auij (t, x) := aij (∇u(t, x)).

218

4. Global Existence

At first, (4.6.26)

2 f, us = 2f, u + 2∇f, ∇us−1 ≤ 2 f  u + 4 f 2s + 14 ∇u2s .

Next, estimating R2,α and RG,2,α as in (4.5.152) and (4.5.153), (4.6.27)

R2,α + RG,2,α ≤ h(|∇u|∞ ) |∂x2 u|∞ ∇us ∇us−1 ≤ ω(∇us−1 ) ∇u2s ≤ ω(Δ) ∇u2s ,

with ω ∈ K0 . We then choose Δ so small that 4 ω(Δ) ≤ 1 ,

(4.6.28)

and deduce from (4.6.25) that, for all t ∈ [0, T ], d u2s + 32 ∇u2s ≤ 2 f  u + 4 f 2s . dt Note that (4.6.28) shows that the smallness of the data does depend on α0 (recall that we assume (3.2.14)). Recalling (4.6.21) and (4.6.24), we first deduce from (4.6.29) that  t 2 3 2 u(t)s ≤ u(t)s + 2 ∇u2s dθ (4.6.30) 0 ≤ u0 2s + 4 Δ F3 + F4 ≤ Δ2 , (4.6.29)

which contradicts (4.6.24) for t = T . Thus, u satisfies (4.6.23). 2) We now claim that, for all t ∈ [0, Tc [,  t (4.6.31) Du2s dθ + ∇u(t)2s ≤ C0 Δ2 , 0

with C0 independent of t; of course, this implies that Tc = +∞. Indeed, we first note that (4.6.30) also yields that  t (4.6.32) ∇u2s dθ ≤ Δ2 , 0

for all t ∈ [0, Tc [. Next, we multiply the differentiated equations in L2 by 2 ∂xα ut , and obtain, instead of (4.6.25),  d (4.6.33) 2 ut 2s + (R0,α + R1,α + RG,1,α ) , Qs (∇u) = 2 f, ut s + dt |α|≤s

with R0,α , R1,α and RG,1,α defined as in (3.3.89), (3.3.90) and (3.3.91), but, again, with the coefficients auij of (3.2.49) replaced by auij (t, x) = aij (∇u(t, x)). At first, (4.6.34)

2 f, ut s ≤ 3 f 2s + 13 ut 2s ;

4.6. The Parabolic Problem

219

then, recalling (4.6.23), R0,α ≤ h(|∇u|∞ ) |∇ut |∞ ∇u2s ≤ h(∇us−1 ) ut s ∇u2s

(4.6.35)

≤ κ(Δ) ∇u4s + 13 ut 2s as well as R1,α + RG,1,α ≤ h(|∇u|∞ )|∇∂x u|∞ ∇us ut s ≤ h(∇us−1 ) ∇u2s ut s

(4.6.36)

≤ κ(Δ) ∇u4s + 13 ut 2s . Inserting (4.6.34), (4.6.35) and (4.6.36) into (4.6.33) yields d Qs (∇u) ≤ 3 f 2s + κ(Δ) ∇u4s . dt Integrating this, recalling the second of (4.6.21) and (4.6.22), we obtain that  t ut 2s dθ + Qs (∇u(t)) ut 2s +

(4.6.37)

0

(4.6.38)

 T  t ≤ a22 ∇u0 2s + 3 f 2s dθ + κ(Δ) ∇u4s dθ 0 0  t ≤ C Δ2 + κ(Δ) ∇u4s dθ . 0

By (4.6.32), we deduce from (4.6.38), via Gronwall’s inequality, that

 t  t 2 2 2 2 ut s dθ + ∇u(t)s ≤ C Δ exp κ(Δ) ∇us dθ (4.6.39) 0 0 ≤ C Δ2 eκ(Δ) Δ

2

≤ 2 C Δ2 ,

having assumed that Δ is so small that, in addition to (4.6.28), κ(Δ) Δ2 ≤ ln 2. Adding (4.6.39) to (4.6.32) and (4.6.23), we deduce that, for all t ∈ [0, Tc [,  t (4.6.40) Du2s dθ + u(t)2s+1 ≤ C Δ2 , 0

from which we conclude that u ∈ Ps,b (∞), as claimed. 3) The proof of the stability result (4.6.19) is also similar. Indeed, we first note that the map t → Qs (∇u(t)) decays to 0, as t → +∞, as a consequence of Proposition 1.7.4. Indeed, this map is absolutely continuous, has a bounded integral (by (4.6.32)), and a derivative bounded above (the latter follows from (4.6.37)). Then, the map t → ∇u(t)2s also decays to 0, and (4.6.19) follows. In turn, as in the hyperbolic case, (4.6.19) implies (4.6.20), via equation (4.6.1).

220

4. Global Existence

4) If aij ∈ C s+1 (RN ), we can differentiate the equation of (4.6.1) once more, and repeat all the above estimates with s replaced by s + 1. This leads to an estimate analogous to (4.6.29), namely d u2s+1 + 32 ∇u2s+1 ≤ 2f, us+1 . dt Using integration by parts,

(4.6.41)

(4.6.42)

f, us+1 = f, u0 + ∇f, ∇us = f, u0 − f, Δus ≤ |f, u| + f 2s + 14 ∇u2s+1 ;

replacing this into (4.6.41), we obtain d u2s+1 + ∇u2s+1 ≤ 2 f  u + 2 f 2s , dt which is analogous to (4.6.29). Proceeding then as in the first part of this proof, we can conclude that ∇u ∈ L2 (0, +∞; H s+1 ); together with (4.6.30), we conclude that u ∈ L2loc (0, +∞; H s+2 ).  (4.6.43)

4.6.3. Global Existence, I. We next present a global existence result for the parabolic problem (4.6.1) when f satisfies an integrability condition as |x| → +∞, as in Theorem 4.5.3. Theorem 4.6.2. Let N ≥ 3, s > N2 + 1, and assume that aij ∈ C s (RN ), u0 ∈ H s+1 , and that f satisfies (4.5.181). There exists δ0 > 0 such that, if (4.6.44)

∇u0 s + f Cb ([0,+∞[;H s ∩Lq ) ≤ δ02 ,

the Cauchy problem (4.6.1) has a unique solution u ∈ C([0, +∞[; H s+1 ), such that ∇u ∈ Cb ([0, +∞[; H s ) and ut ∈ Cb ([0, +∞[; H s−1 ). If N ≥ 5 and 1 ≤ q < N2N ¯, then u ∈ Cb ([0, +∞[; H s+1 ). Again, the size of δ0 +4 < q depends on the coefficient of ut , which in (4.6.1) is 1, and on α0 . Proof. We proceed as in the proof of Theorem 4.5.3. We consider a local solution u ∈ Ps (τ ) of (4.6.1), as assured by Theorem 3.4.1, and extend it to its maximal time of existence Tc . We show that Tc = +∞ by means of the a priori estimate provided by Proposition 4.6.2. Let u0 and f be as in Theorem 4.6.2. Assume problem (4.6.1) has a solution u ∈ Ps (T ), for some T ∈ ]0, Tc [. There exist δ0 ∈ ]0, 1[ and K > 0, independent of T , such that if (4.6.44) holds, then for all t ∈ [0, T ], (4.6.45)

∇u(t)s ≤ δ0 ,

ut (t)s−1 ≤ K δ0 .

Consequently, Tc = +∞, and (4.6.45) holds for all t ≥ 0.

4.6. The Parabolic Problem

221

Proof. The last two claims are proven as in the corresponding parts of Proposition 4.5.7. Also, the second of (4.6.45) follows from the first, via equation (4.6.1) and the boundedness of f (t) in H s−1 . Hence, it suffices to prove the first of (4.6.45). Let δT := max0≤t≤T ∇u(t)s , and F be as in (4.5.187). By a slight modification of the proof of estimates (4.6.29) and (4.6.37), we can obtain that, if δT is sufficiently small (in particular, if δT ≤ δ0 , δ0 to be determined), then d u2s + 74 Qs (∇u) ≤ 2f, u + C f 2s + ω(δ0 ) ∇u2s , dt d ut 2s + (4.6.47) Qs (∇u) ≤ ω(δ0 ) ∇u2s + C f 2s . dt We estimate f, u as in (4.5.192); that is, (4.6.46)

f, u ≤ |f |q¯ |u|p¯ ≤ F |∇u|2 ≤ F δT ≤ δ03 .

(4.6.48)

Since also f (t)s ≤ F ≤ δ02 , taking δ0 so small that 2 ω(δ0 ) ≤ from (4.6.46) and (4.6.47) that

1 4

we deduce

d Qs (∇u) + 32 Qs (∇u) ≤ C δ03 − 2u, ut s . dt Using interpolation, we transform and estimate the last term of (4.6.49) as (4.6.49)

ut 2s +

−2u, ut s = −2u, ut  − 2∇u, ∇ut s−1 ≤ 2 |u, ut | + 2 ∇us−1 ∇ut s−1 ≤ 2 |u, ut | + ∇u2s−1 + ∇ut 2s−1

(4.6.50)

≤ 2 |u, ut | + 12 ∇u2s + C ∇u20 + ut 2s ≤ 2 |u, ut | + 12 Qs (∇u) + C ∇u20 + ut 2s . Putting (4.6.50) into (4.6.49), we obtain d Qs (∇u) + Qs (∇u) ≤ C δ03 + 2 |u, ut | + C ∇u20 . dt From equation (4.6.1), integrating by parts,

(4.6.51)

(4.6.52)

−u, ut  = −u, f + aij (∇u) ∂i ∂j u = −u, f  + aij (∇u) ∂i u, ∂j u + u, ∂j aij (∇u) ∂i u ;

thus, by (4.6.48), (4.6.53)

|u, ut | ≤ δ03 + |aij (∇u)|∞ |∇u|22 + |∂j aij (∇u) ∂i u, u| .

We estimate the last term of (4.6.53) as in (4.5.150): (4.6.54) |∂j aij (∇u) ∂i u, u| ≤ |aij  (∇u)|∞ |∇∂j u|2 |∇u|N |u|p¯ ≤ κ(∇us ) ∇u2s ∇u ≤ C δT3 ≤ C δ03 .

222

4. Global Existence

Next, we estimate |∇u(t)|2 by a variation of parameters method, as in the hyperbolic case. Again by the change of variables x → A1/2 x, A := [aij (0)], we can assume that the equation of (4.6.1) has the form ut − Δu = f + g(u) ,

(4.6.55)

where g(u) is as in (4.5.198), with aij (Du) replaced by aij (∇u). Hence, by Duhamel’s formula,  t (4.6.56) u(t) = [H(t) ∗ u0 ] + [H(t − θ) ∗ (f (θ) + g(u(θ)))] dθ ,

    0 =: v(t)

=: u ˜(t,θ)

where H is the heat kernel defined in (4.6.6). At first, by Young’s inequality (1.4.24), and (4.6.44), |∇v(t)|2 ≤ C |H(t)|1 |∇u0 |2 ≤ C ∇u0  ≤ C δ02 .

(4.6.57) Next, by (4.6.11), (4.6.58)

 |∇˜ u(t, θ)|2 ≤ C (1 + t − θ)− νq (0,1) |f (θ) + g(u(θ))|q  + |∇f (θ) + ∇g(u(θ))|2 ≤ C (1 + t − θ)− νq (0,1) (F + |g(u(θ))|q + |∇g(u(θ))|2 ) .

We estimate |g(u)|q as in (4.5.200); that is, (4.6.59)

|g(u)|q ≤ |˜ aij (∇u)|2 (|∂i ∂j u|2 + |∂i ∂j u|∞ ) ≤ κ(δT ) δT2 ≤ C δ02 ;

similarly, |∇g(u)|2 ≤ |˜ aij (∇u) ∂i ∂j ∇u|2 + |˜ aij  (∇u) ∂x2 u ∂i ∂j u|2 (4.6.60)

≤ C |∇u|∞ |∂x3 u|2 + C |∂x2 u|24 ≤ C ∇u2s ≤ C δT2 ≤ C δ02 .

Inserting (4.6.59) and (4.6.60) into (4.6.58), and then (4.6.58) and (4.6.57) into (4.6.56), we deduce that  t 2 2 (4.6.61) |∇u(t)|2 ≤ C δ0 + C (F + δ0 ) (1 + t − θ)− νq (0,1) dθ ≤ C δ02 , 0

having recalled that, by (4.5.206), νq (0, 1) > 1. By (4.6.54) and (4.6.61), it follows then from (4.6.53) that (4.6.62)

|u, ut | ≤ C δ03 + C δ04 ≤ C δ03 .

In turn, putting (4.6.62) and (4.6.61) into (4.6.51), we obtain the exponential inequality (4.6.63)

d Qs (∇u) + Qs (∇u) ≤ C1 δ03 , dt

4.6. The Parabolic Problem

223

from which we deduce that, for all t ∈ [0, T ], (4.6.64)

Qs (∇u(t)) ≤ Qs (∇u(0)) + C1 δ03 ≤ C2 δ03 ,

for suitable constant C2 independent of u and T . Clearly, (4.6.64) yields a time-independent estimate on ∇u(t)s , which is the analogous of the first of (4.5.184) in Proposition 4.5.7. We can now proceed in the same way, to reach a contradiction and conclude the proof of Proposition 4.6.2.  Finally, assume that N ≥ 5. From (4.6.56),  t (4.6.65) |u(t)|2 ≤ |v(t)|2 + |˜ u(t, θ)|2 dθ ; 0

by (4.6.11), (4.6.66)

|v(t)|2 ≤ C (1 + t)− ν2 (0,0) |u0 |2 ≤ C |u0 |2 ,

as well as, by (4.6.44) and (4.6.59), (4.6.67)

 |˜ u(t, θ)|2 ≤ C (1 + t − θ)− νq (0,0) |f (θ) + g(u(θ))|2 + |f (θ) + g(u(θ))|q ≤ C δ02 (1 + t − θ)− νq (0,0) .

The conditions q < N2N +4 and N ≥ 5 imply that νq (0, 0) > 1; hence, we deduce from (4.6.65), (4.6.66), and (4.6.67) that t → |u(t)|2 is bounded. This completes the proof of Theorem 4.6.2.  Remark. We note that if aij ∈ C s+1 (RN ), then u ∈ L2loc (0, +∞; H s+2 ). Indeed, we can first establish an estimate analogous to (4.6.63), with s replaced by s + 1, the integration of which shows that ∇u ∈ L2 (0, T ; H s+1 ) for all T > 0. Integration of the second of (4.6.45) yields that u ∈ L2 (0, T ; H s−1 ) for all T > 0; consequently, u ∈ L2loc (0, +∞; H s+2 ) as claimed.  4.6.4. Regularity for t > 0. As was the case for the linear problem, the solution of the Cauchy problem (4.6.1) acquires more regularity for t > 0, as soon as the source term f and the coefficients aij are more regular. The type of results one can prove in this regard are similar to those of Theorem 2.5.3 for the linear equation; here, we limit ourselves to the following result. Theorem 4.6.3. In addition to the assumptions of Theorem 4.6.2, assume that aij ∈ C s+1 (RN ), that u0  ≤ δ02 , and ft ∈ Cb ([0, +∞[; H s−1 ), with F5 := sup ft (t)s−1 ≤ δ02 .

(4.6.68)

t≥0

Then, ut ∈ Cb ut ∈ Cb ([0, +∞[; H s ).

([τ, +∞[; H s )

for all τ > 0. If in addition u0 ∈ H s+2 , then

224

4. Global Existence

Sketch of Proof. We first observe that, estimating f, u as in (4.6.48), by (4.6.41) and (4.6.42) we can improve (4.6.43) into d u2s+1 + ∇u2s+1 ≤ C δ03 . dt

(4.6.69)

From this, recalling (4.6.44), and also that u(t) ≤ δ02 , we obtain that, for all t ≥ 0,  t (4.6.70) u(t)2s+1 + ∇u2s+1 dθ ≤ u0 2s+1 + C δ03 t ≤ C δ03 (1 + t) . 0

Next, we differentiate the equation of (4.6.1) with respect to t, to obtain (4.6.71) utt − aij (∇u) ∂i ∂j ut = ft + (aij  (∇u) · ∇ut ) ∂i ∂j u =: ft + g1 (u) . Fix τ > 0, and set τ1 := in H s , to obtain

1 2

τ . We formally multiply (4.6.71) by 2(t − τ1 )ut

 d  (t − τ1 )ut 2s + 2(t − τ1 ) Qs (∇ut ) dt = ut 2s + 2(t − τ1 ) (R1 + R2 + R3 ) ,

(4.6.72) where (4.6.73) (4.6.74)

R1 := ft + g1 (u), ut s ,  R2 := −∂j aij (∇u) ∂xα ∂i ut , ∂x ut  , |α|≤s

(4.6.75)

R3 :=

    α β

∂xβ aij (∇u) ∂xα−β ∂i ∂j ut , ∂xα ut  .

|α|≤s 0 0, we can estimate R3 in the by now usual way; using interpolation, and (4.6.45) again, R2 + R3 ≤ κ(|∇u|∞ ) ∇us ∇ut s ut s (4.6.79)

≤ C δ0 ∇ut s ut s ≤ C δ0 ∇ut s (∇ut s + ut s−1 ) ≤ C δ0 ∇ut 2s + C K δ03 .

Consequently, if δ0 is sufficiently small, we obtain from (4.6.72) that (4.6.80)

 d  (t − τ1 )ut 2s + (t − τ1 ) ∇ut 2s ≤ C (t − τ1 ) δ03 + ut 2s . dt

From this, adding and subtracting the term (t − τ1 ) ut 20 , which is bounded by C(t − τ1 ) δ02 , we deduce the exponential inequality (4.6.81)

 d  (t − τ1 )ut 2s + (t − τ1 ) ut 2s ≤ C (t − τ1 ) δ02 + ut 2s . dt

We estimate the last term of (4.6.81) using equation (4.6.1); that is, (4.6.82)

ut s ≤ f s + aij (∇u)s˜ ∇us+1 ≤ δ02 + γ2 ∇us+1 ,

where, again, γ2 depends boundedly on δ0 . Inserting (4.6.82) into (4.6.81) yields (4.6.83)

   d  t e (t − τ1 )ut 2s ≤ C et δ02 (t − τ1 ) + δ04 + ∇u2s+1 , dt

from which, integrating on [τ1 , t], and recalling (4.6.70), (4.6.84)

et (t − τ1 )ut (t)2s ≤ C δ02 et (1 + t) ,

for all t ≥ τ1 . In particular, if t ≥ τ = 2 τ1 , (4.6.84) implies that

1 2 2 (4.6.85) ut (t)s ≤ 2 C δ0 1 + =: Kτ δ02 , τ which shows that ut ∈ L∞ (τ, +∞; H s ). By Theorem 4.6.2, we know that ut ∈ Cb ([τ, +∞[; H s−1 ); thus, by Proposition 1.7.1, ut ∈ Cw ([τ, T ]; H s ), for all T > τ . On the other hand, integration of (4.6.80) shows that ∇ut ∈ L2loc (]τ, +∞[; H s ); thus, we deduce from (4.6.72) that the function t → (t − τ1 ) ut (t)2s has a derivative which is in L1loc (τ, +∞). This function is therefore continuous on [τ, +∞[; since τ > τ1 , we deduce that the function t → ut (t)2s is continuous on [τ, +∞[ as well. Together with the weak continuity of ut into H s , this allows us to conclude that, in fact, ut ∈ Cb ([τ, +∞[; H s ), as claimed (the boundedness coming from (4.6.85)). Note that Kτ → +∞ as τ → 0; on the other hand, the weight t − τ1 was

226

4. Global Existence

introduced only to avoid the term ut (0)2s , which would appear after integration of (4.6.83) if such weight were not present. But if u0 ∈ H s+2 , then (4.6.86)

ut (0) = f (0) + aij (∇u0 )∂i ∂j u0 ∈ H s ;

thus, the factor t − τ1 is not necessary, and our estimates can be established on the whole interval [0, +∞[. This end the proof of Theorem 4.6.3.  4.6.5. Global Existence, II. As we mentioned in the introduction to this section, we can show that global existence for the parabolic problem holds, without any restriction on the size of the data u0 and f , if we assume more regularity of f . This more positive situation is due to the availability of the maximum principle; this provides a time-independent bound on the H¨older norm of local solutions, from which we can deduce an a priori bound on the solution in the space Ps (T ), for arbitrary T > 0. Theorem 4.6.4. Let N ≥ 3, and s > N2 + 1. Assume that aij ∈ C s (RN ), u0 ∈ H s+1 , and f ∈ C([0, +∞[; H s ) ∩ C 1 ([0, +∞[; H s−1 ). The Cauchy problem (4.6.1) admits a unique solution u ∈ Ps (∞), which depends continuously on u0 and f . Proof. We wish to prove that, for all T > 0, there is a unique u ∈ Ps (T ), solution of (4.6.1). Fix T > 0. The uniqueness and the continuous dependence claims are proven as in Theorem 3.4.1; in addition, the regularity claims of that theorem also hold in the present situation, with the same proof. Thus, we consider a local solution u ∈ Ps (τ ) → Vs (τ ) of (4.6.1), for some τ ∈ [0, T ], and proceed to establish an a priori bound on uVs (t) , independent of t ∈ [0, T ]. To this end, we first realize that it is sufficient to establish a time-independent bound on ∇u(t)W 1,∞ (RN ) (this is of course well known; see e.g., Taylor [157, ch. 5]). Indeed, suppose that there is M > 0 (possibly dependent on T ), such that for all t ∈ [0, T ], (4.6.87)

|∇u(t)|∞ + |∂x2 u(t)|∞ ≤ M .

Then, we go back to identity (4.6.25), but replace estimates (4.6.26) and (4.6.27) by (4.6.88)

2f, us ≤ f 2s + u2s ,

and (4.6.89)

R2,α + RG,2,α ≤ h(|∇u|∞ ) |∂x2 u|∞ ∇us us ≤ κ(M ) u2s + ∇u2s .

4.6. The Parabolic Problem

227

Inserting (4.6.88) and (4.6.89) into (4.6.25), we obtain d u2s + ∇u2s ≤ f 2s + κ(M ) u2s . dt Integration of (4.6.90) allows us to deduce, via Gronwall’s inequality, that u ∈ C([0, T ]; H s ) ∩ L2 (0, T ; H s+1 ). Proceeding as in the second part of the proof of Theorem 4.6.1, we can then deduce that ut ∈ L2 (0, T ; H s ), and ∇u ∈ C([0, T ]; H s ). Thus, u ∈ Ps (T ); since T is arbitrary, we conclude that u ∈ Ps (∞), as claimed. (4.6.90)

Thus, to prove Theorem 4.6.4, it is sufficient to establish (4.6.87); in turn, this is a consequence of classical H¨ older estimates, via the maximum principle. Referring to section 1.3 for the definitions of the H¨older spaces in time and space, for T > 0 and 0 < τ ≤ T we set Qτ := ]0, τ [ × RN , we abbreviate C [j,α] (Qτ ) := C (j+α)/2,j+α (Qτ ) ,

(4.6.91)

j = 0, 2 ,

and we denote by |·|j,α;τ the norm in these spaces. By the Sobolev imbedding (1.5.61), there is α ∈ ]0, 1[ such that u0 ∈ C 2,α (RN ); by the second part of Proposition 1.7.7, f ∈ Cb1 (QT ) → C [0,α] (QT ), with (4.6.92)

|f |0,α;T ≤ C |f |C 1 (QT ) ≤ C max (f (t)s + ft (t)s−1 ) . b

0≤t≤T

From the identity (4.6.93)

utt = ft + aij (∇u)∂i ∂j ut + aij  (∇u) · ∇ut ∂i ∂j u ,

we deduce that, since u ∈ Ps (τ ), utt ∈ L2 (0, τ ; H s−2 ); thus, by the trace theorem, ut ∈ C([0, τ ]; H s−1 ). By Proposition 1.7.7, with m = s − 1 > N2 , u ∈ Cb1 (Qτ ); in addition, since both ut , ∂x2 u ∈ C([0, τ ]; H s−1 ), and their derivatives utt , ∂x2 ut ∈ L2 (0, τ ; H s−2 ) → L2 (Qτ ), by Proposition 1.7.8 with m = s − 1 again, ut , ∂x2 u ∈ C [0,α] (Qτ ). Hence, u ∈ C [2,α] (Qτ ). The a priori estimate (4.6.87) is then a consequence of Theorem 4.6.5. Let T > 0, and assume that f ∈ Cb1 (QT ) and u0 ∈ C 2,α (RN ), for some α ∈ ]0, 1[. Assume that (4.6.1) admits a corresponding solution u ∈ C [2,α] (Qτ ), for some τ ∈ ]0, T ]. There exist CD > 0 and γ ∈ ]0, α], depending on the norms of f in Cb1 (QT ) and u0 in Cb2 (RN ), but independent of u and τ , such that (4.6.94)

|u|2,γ;τ ≤ CD (|f |0,α;T + |u0 |2,α ) .

Proof. Estimates like (4.6.94) are generally considered as well known; however, we have not been able to find in the literature an explicit proof of (4.6.94) for quasi-linear equations in the whole space RN . Thus, we reproduce the proof given in Milani [118], which is based on various results of Ladyzenskaya, Solonnikov, and Ural’tseva [86], and Krylov [83]. Note that

228

4. Global Existence

(4.6.94) reads almost exactly as the estimate reported in Krylov’s Theorem 8.9.2, which is established for a linear Cauchy problem with coefficients in C [0,α] (QT ), and where the value of the constant CD (called N there) is determined by |aij (∇u)|0,α;T . In the present situation, of course, since we are assuming that (4.6.1) does have a solution u ∈ C [2,α] (Qτ ), we can consider its equation as linear, with known coefficients auij (t, x) := aij (∇u(t, x)); however, the constant CD would then depend on the H¨older norm of ∇u, which can generally be estimated only in terms of the H¨ older norm of ∇f and the coefficients auij again; that is, the argument would run in a loop. 1) Our first step is to estimate |u| and |∇u| in Qτ , by the maximum principle. As we have remarked, we can consider the equation of (4.6.1) as linear, with known coefficients auij . These coefficients are bounded, because we are assuming that (4.6.1) has a solution u ∈ C [2,α] (Qτ ), but the bound could, in principle, depend on τ . However, we can apply Corollary 8.1.5 of Krylov [83], which yields the explicit estimate sup |u| ≤ τ sup |f | + sup |u0 | Qτ

(4.6.95)

RN

Qτ

≤ T |f |C 0 (QT ) + |u0 |C 0 (RN ) =: C0 , b

b

which is independent of τ . Next, we differentiate the equation of (4.6.1): setting ∂0 := ∂t , we see that, for 0 ≤ k ≤ N , each function v k := ∂k u satisfies the linear equation vtk − aij (∇u)∂i ∂j v k − aij  (∇u) · ∇v k ∂i ∂j u = ∂k f .

(4.6.96)

The coefficients of this equation are also bounded (including those of the lower order terms v → aij  (∇u)·(∇v) ∂i∂j u), and the same corollary yields, as in (4.6.95), the explicit estimate sup |∇u| ≤ τ sup |∇f | + sup |∇u0 | Qτ

(4.6.97)

RN

Qτ

≤ T |f |C 1 (QT ) + |u0 |C 1 (RN ) =: C1 , b

b

which is, again, independent of τ . Likewise, since ut satisfies the initial condition ut (0) = u1 := f (0, ·) + aij (∇u0 )∂i ∂j u0 ,

(4.6.98)

recalling (3.2.44), and that |∇u0 |∞ ≤ |u0 |C 1 (RN ) , we obtain that b

sup |ut | ≤ τ sup |ft | + sup |u1 | (4.6.99)

Qτ

Qτ

RN

≤ (T + 1) |f |C 1 (QT ) + a1 (∇u0 ) |u0 |C 2 (RN ) =: C2 , b

b

with C2 independent of τ . Since u ∈ and (4.6.99) also hold on Qτ .

C [2,α] (Q

τ ),

estimates (4.6.95), (4.6.97)

4.6. The Parabolic Problem

229

˜ α (∇u) (recall definition (1.3.4)). 2) Our second step is to estimate H To this end, we resort to Lemma 3.1 of Ladyzenskaya, Solonnikov, and Ural’tseva [86, ch. II, §3], which states that ∇u will satisfy a H¨older condition in t, uniformly in x, if it satisfies a H¨older condition in x, uniformly in t, and if u also satisfies a H¨older condition in t, uniformly in x. The latter is clearly implied by (4.6.99); to show the former, we consider t ∈ [0, τ ] as fixed, and u(t, ·) as solution of the quasi-linear elliptic equation (4.6.100)

− aij (∇u)∂i ∂j u = f − ut =: f˜ ,

and invoke a classical result on the H¨older continuity of the gradient of the solutions to such equations. More precisely, for fixed t ∈ [0, τ ] and arbitrary x, y ∈ RN , with x = y, we wish to estimate the ratio (4.6.101)

hβ (∇u(t); x, y) :=

|∇u(t, x) − ∇u(t, y)| , |x − y|β

for suitable β ∈ ]0, α]. If |x − y| ≥ 1, (4.6.97) yields (4.6.102)

hβ (∇u(t); x, y) ≤ |∇u(t, x) − ∇u(t, y)| ≤ 2 C1 ,

for all β ∈ ]0, 1[. If instead 0 < |x − y| < 1, we consider concentric balls Bk with center x and radii respectively equal to k = 1, k = 2 and k = 3, and choose a cut-off function ζ ∈ C0∞ (RN ), such that 0 ≤ ζ(y) ≤ 1 for all y ∈ RN , ζ(y) ≡ 1 on B1 , ζ(y) ≥ 12 in B2 , and ζ(y) ≡ 0 off B3 . In B2 , the function v := ζ u satisfies the quasi-linear elliptic equation (4.6.103) −¯ aij (y, v, ∇v)∂i ∂j v = ζ f˜ − (aij (∇u)∂i ∂j ζ) u − 2 aij (∇u)∂i ζ ∂j u =: b , in which, for y ∈ B2 , p ∈ R and q ∈ RN , the coefficients

1 1 (4.6.104) a ¯ij (y, p, q) := aij ∇ζ(y) q−p ζ(y) ζ(y)2 are of class C 1 in B2 × R × RN . Furthermore, the function y → b(t, y) is in C 0 (B2 ), and its norm in this space can be estimated in terms of C0 , C1 and C2 , because of (4.6.95), (4.6.97), and (4.6.99). Thus, we can apply Theorem 13.6 of Gilbarg and Trudinger [52, Ch. 13, §4], with Ω = B2 and Ω  = B1 , to deduce the estimate (4.6.105)

hβ (∇v(t); y, y  ) ≤ C3 d−β ,

y, y  ∈ B1 ,

y = y  ,

where d = dist(B1 , ∂B2 ) = 1, and both β ∈ ]0, 1[ and C3 depend on the fixed constants N , α0 , diam(B2 ) and, more essentially, on K(t) := |v(t, ·)|C 1(B 2 ) and B(t) := |b(t)|L∞ (B2 ) . We can of course choose β ≤ α. Again by (4.6.95) and (4.6.97), K(t) can be estimated in terms of C0 and C1 , uniformly in t ∈ [0, τ ], and independently of τ . The same is true for B(t), which, by

230

4. Global Existence

(4.6.99), is estimated also in terms of C2 . Since v = u in B1 , (4.6.105) implies that, for all y ∈ B1 \ {x}, hβ (∇u(t); x, y) ≤ C3 .

(4.6.106)

This yields the desired estimate of (4.6.101) when 0 < |x − y| < 1. In conclusion, from (4.6.102) and (4.6.106) it follows that for all x, y ∈ RN , with x = y, (4.6.107)

hβ (∇u(t); x, y) ≤ max{2C1 , C3 } =: C4 ,

where β is determined in (4.6.105). As we have observed, C4 can be estimated in terms of C0 , C1 , and C2 . Estimate (4.6.107) provides the H¨older condition in x, uniformly in t, for ∇u, which is required in Lemma 3.1 of Ladyzenskaya, Solonnikov, and Ural’tseva [86], to obtain the estimate |∇u(t, x) − ∇u(s, x)| ≤ C5 , |t − s|γ/2

(4.6.108)

for suitable γ ≤ β ≤ α. In (4.6.108), C5 is independent of x ∈ RN , and can be estimated in terms of C0 , C1 , C2 and C4 . From (4.6.107) and (4.6.108), we deduce that, for (t, x), (s, y) ∈ Qτ , with (t, x) = (s, y),

(4.6.109)

|∇u(t, x) − ∇u(s, y)| |∇u(t, x) − ∇u(t, y)| ≤ |x − y|γ (|t − s| + |x − y|2 )γ/2 +

|∇u(t, y) − ∇u(s, y)| ≤ C4 + C5 =: C6 . |t − s|γ/2

Thus, we conclude that ˜ γ (∇u) ≤ C6 ; H

(4.6.110) again, C6 is independent of τ .

3) Clearly, (4.6.97) and (4.6.110) imply that the coefficients auij satisfy the estimate (4.6.111)

|auij |0,γ;τ ≤ h(C1 ) + C6 =: C7 .

We are then in a position to apply Theorem 8.9.2 of Krylov [83] (with K = C7 of (4.6.111)), and deduce the estimate (4.6.112)

|u|2,γ;τ ≤ CK |f − u|0,γ;τ .

Note that CK can indeed be estimated in terms of the norms of f in Cb1 (QT ) and u0 in Cb2 (RN ), as claimed. By the interpolation inequality  ˜ γ (u) ≤ η H ˜ γ (ut ) + H ˜ γ (∂ 2 u) + C η −γ/2 sup |u| , (4.6.113) H η >0, x Qτ

4.6. The Parabolic Problem

231

(as per the last claim of Theorem 8.8.1 of Krylov [83, ch. 8]), and recalling (4.6.95), we obtain ˜ γ (u) |u|0,γ;τ ≤ sup |u| + H Qτ

 ˜ γ (ut ) + H ˜ γ (∂ 2 u) + Cη sup |u| ≤ η H x

(4.6.114)

Qτ

≤ η |u|2,γ;τ + Cη sup |u| . Qτ

Taking η ≤ and recalling that γ ≤ α, we deduce from (4.6.112), (4.6.114) and (4.6.95) that, for CD := 2 max{1 + Cη T, Cη } CK ,  |u|2,γ;τ ≤ C CK |f |0,γ;τ + |u0 |C 0 (RN ) b (4.6.115) ≤ CD (|f |0,α;τ + |u0 |2,α ) , 1 2 CK ,

from which (4.6.94) follows. This concludes the proof of Theorem 4.6.5 and, therefore, that of Theorem 4.6.4. 

Chapter 5

Asymptotic Behavior

5.1. Introduction 1. In this chapter we study the asymptotic behavior of the global solutions to the quasi-linear hyperbolic and parabolic problems (4.5.1) and (4.6.1), whose existence has been established in Theorems 4.5.3 and 4.6.2. We call these the hyperbolic and the parabolic solutions, and, when they appear in the same context, we denote them, respectively, by uhyp and upar . In particular, we give sufficient conditions for the convergence of these solutions to the solution of the corresponding stationary equation, that is, (5.1.1)

− aij (0, ∇v)∂i ∂j v = h

for the hyperbolic solutions and (5.1.2)

− aij (∇v)∂i ∂j v = h

for the parabolic ones. When the context is clear, we call each of these stationary solutions usta . In the homogeneous case f ≡ 0, we also want to compare the asymptotic profiles of the hyperbolic and parabolic solutions, and show that these profiles are essentially the same, in the sense that, as t → +∞, the difference uhyp (t) − upar (t) decays to 0, in a suitable sense, with a faster rate than that of both uhyp (t) and upar (t) (which turns out to be the same). This result, known as the diffusion phenomenon of hyperbolic waves, indicates that the hyperbolic problem (4.5.1) has an asymptotically parabolic structure. 2. Since we work in the whole space RN , the question of the existence of solutions to the stationary equations (5.1.1) and (5.1.2) is not trivial, and cannot be guaranteed by the classical theory of nonlinear elliptic equations in a bounded domain, with appropriate boundary conditions (as, for example, 233

234

5. Asymptotic Behavior

in Krylov [83], Ladyzenskaya and Ural’tseva [85], or Gilbarg and Trudinger [52]). On the other hand, by means of a variational technique and a fixed point argument, it is relatively straightforward to establish the existence of small, strong solutions to (5.1.1) and (5.1.2), in the spaces V m defined in part 10 of section 1.5.1. Indeed, we report from Milani and Volkmer [121] the following result concerning (5.1.1): Theorem 5.1.1. Let N ≥ 3, s > N2 + 1, m ≥ 0. Assume that aij ∈ C s+m (R1+N ), and h ∈ H s+m−2 ∩ Lq , q ∈ [1, q¯[ (¯ q = N2N +2 ). There exists 1 R0 ∈ ]0, 2 [ such that, for all R ∈ ]0, R0 ], if hs+m−2 + |h|q ≤ R2 ,

(5.1.3)

then (5.1.1) has a unique solution v ∈ V s+m , such that ∇vs+m−1 ≤ R. This solution depends continuously on h. Remark. Theorem 5.1.1 is proven in Milani and Volkmer [121] for m = 1, but the proof can be readily adapted for general m ≥ 0. In addition, a similar result can be proven for (5.1.2), assuming that aij ∈ C s+m (RN ). 

5.2. Convergence uhyp (t) → usta In this section we give sufficient conditions for the convergence, as t → +∞, of the solution to the hyperbolic Cauchy problem (4.5.1) to the solution to the stationary equation (5.1.1). We assume (3.2.14) again for simplicity; that is, α0 ≥ 1. Theorem 5.2.1. In addition to the assumptions of Theorems 4.5.3 and 5.1.1 for m = 2 (in particular, aij ∈ C s+2 (R1+N )), suppose that, as t → +∞, f (t) → h, in the sense that  +∞ (5.2.1) (f (t) − hs + |f (t) − h|q ) dt =: Λ0 < +∞ . 0

Let u and v be the solutions of (4.5.1) (with σ = 2) and (5.1.1), given respectively by Theorems 4.5.3 and 5.1.1, when δ0 and R0 are sufficiently small. Then, (5.2.2)

lim D (u(t) − v)s = 0 .

t→+∞

Proof. With abuse of notation, we write Dv := (0, ∇v). The difference z(t) := u(t) − v satisfies the equation (5.2.3)

ztt + 2 zt − aij (Du) ∂i ∂j z = f − h + g ,

where, in analogy to (3.3.33), (5.2.4)

g = g(u, v) := (aij (Du) − aij (Dv))∂i ∂j v = (Aij · Dz)∂i ∂j v ,

5.2. Convergence uhyp (t) → usta

with



(5.2.5)

1

Aij :=

235

aij  (λDu + (1 − λ)Dv) d λ .

0

Note that g ∈ L∞ (0, +∞; H s ∩ L1 ), since v ∈ V s+2 ; the L1 part of this assertion follows because, by Proposition 1.5.8, for a.a. t > 0, (5.2.6)

aij (Du(t)) − aij (Dv(t)) = (aij (Du(t)) − aij (0)) + (aij (0) − aij (Dv(t))) ∈ L2 .

1) We establish energy estimates on ∇z, similar to the ones shown for u in the proof of Theorem 4.5.3. We set (5.2.7) ˜ z (t) := ∇zt 2 + 2 ∇z, ∇zt s−1 + 2 ∇z2 + Qs−1 (∇∂x z) , N s−1 s−1 s−1 (5.2.8) ˜ z (t) := ∇zt 2 + Qs−1 (∇∂x z) E s−1 s−1 ˜ u (t) of (2.3.135)), where, in accord (compare to Nsu (t) of (4.5.141) and E s u with (4.5.138), Qs−1 (∇∂x z) = Qs−1 (a , ∇∂x z), with auij as in (3.2.49); that is, explicitly, (5.2.9)

N   Qs−1 (∇∂x z) = aij (Du) ∂i ∂xα ∂k z, ∂j ∂xα ∂k z . |α|≤s−1 k=1

We multiply (5.2.3) in V s by 2(zt + z), to obtain d ˜z ˜ z (t) = 2∇(f − h + g), ∇(zt + z)s−1 N (t) + 2 E s−1 dt s−1 N s−1     α+ek  α Aαγ +2 k (u, z), ∂x ∂k (zt + z) γ |α|=1 k=1 0 0, there is T1 ≥ 0 such that, for all t ≥ T1 , ∇Dz(t)s−1 ≤ ε .

(5.2.34)

By (5.2.1), there also is T2 ≥ 0 such that, for all t ≥ T2 ,  +∞   (5.2.35) f (θ) − h + |f (θ) − h|q dθ ≤ ε .

  t =: ζ(θ)

Fix ε ∈ ]0, 1[, and let Tε := max{T1 , T2 }, ϕε := z(Tε ), ψε := zt (Tε ); also, set χ := g + g˜. Then, for all t ≥ Tε , by the variation of parameters formulas (4.5.8) and (4.5.9), as in (4.5.203): ∇z(t) = Wd (t − Tε ) ∗ ∇(2 ϕε + ψε ) + ∂t (Wd (t − Tε ) ∗ ∇ϕε )  t + (5.2.36) ∇(Wd (t − θ) ∗ (f (θ) − h)) dθ 

Tε t

+

∇(Wd (t − θ) ∗ χ(θ)) dθ .

Tε

Let z 1 (t) := Wd (t − Tε ) ∗ (2 ϕε + ψε ). By Parseval’s formula (1.5.9), recalling (4.5.7), ∇z 1 (t)2 = F [∇z 1 (t)]2  ˆ d (t − Tε , ξ)|2 |F [∇(2 ϕε + ψε )]|2 dξ (5.2.37) = C |W  = C |h(t − Tε , ξ)|2 |F [∇(2 ϕε + ψε )]|2 dξ ; by Proposition 4.5.1, |h(t − Tε , ξ)|2 → 0 as t → +∞, for each ξ ∈ RN , and |h(t − Tε , ξ)|2 is bounded independently of t and ξ. Since ∇ϕε and ∇ψε ∈ L2 , also F [∇(2 ϕε + ψε )] ∈ L2 ; thus, by Lebesgue’s dominated convergence theorem, from (5.2.37) it follows that (5.2.38)

∇z 1 (t) → 0

as t → +∞ .

In addition, by the linear decay estimate (4.5.30) of Proposition 4.5.2, the fact that ∇ϕε ∈ L2 also implies that, as t → +∞,   (5.2.39) ∂t (Wd (t − Tε ) ∗ ∇ϕε ) ≤ O (1 + t)− 1 → 0 . We estimate the third term of (5.2.36) again by (4.5.30), with k = 0 and m = 1: recalling (5.2.35), and noting that νq (0, 1) > 1 because 1 ≤ q < q¯,

5.2. Convergence uhyp (t) → usta

we obtain

 W1 (t) :=

239

t

∇(Wd (t − θ) ∗ (f (θ) − h)) dθ  t C (1 + t − θ)− νq (0,1) ζ(θ) dθ ≤ C ε . Tε

(5.2.40) ≤

Tε

To estimate the last term of (5.2.36), we need to proceed in a slightly different manner, because, even if, as we know, g(θ) ∈ Lq for a.a. θ > 0, we do not know how to estimate |g(θ)|q in terms of ∇Dz(θ)s−1 . Thus, we 1 1 first define positive numbers λ, μ by λ1 = 1 − 4N , μ1 = 12 − 4N , and verify 2N that λ ∈ ]1, q¯[ and μ ∈ ]2, p¯[ (¯ p = N −2 ); thus, we can apply (4.5.30) again, with q = λ, k = 0 and m = 1, and obtain  t W2 (t) := ∇(Wd (t − θ) ∗ χ(θ)) dθ Tε (5.2.41)  t ≤ C (1 + t − θ)− νλ (0,1) (|χ(θ)|2 + |χ(θ)|λ ) dθ . Tε

Recalling (5.2.4), |g|2 ≤ C |Aij |∞ |Dz|p¯ |∂i ∂j v|N (5.2.42)

≤ κ(δ0 , R0 ) ∂i ∂j vs−2 ∇Dz ≤ ω(R0 ) ∇Dz ;

next, since 1 1 1 = + λ μ 2 we use interpolation to obtain

(5.2.43)

and

1 3/4 1/4 = + , μ 2 p¯

(5.2.44) |g|λ ≤ C |Aij |∞ |Dz|μ |∂i ∂j v|2 ≤ κ(δ0 , R0 ) R0 |Dz|μ 3/4

1/4

≤ ω(R0 ) |Dz|2 |Dz|p¯

≤ ω(R0 ) (δ0 + R0 )3/4 ∇Dz1/4 .

The estimate of g˜ is similar. By Proposition 1.5.8, we obtain from (5.2.33) that |˜ g |2 ≤ |aij (Du) − aij (0)|∞ |∂i ∂j z|2 ≤ C |Du|∞ |∇Dz|2 (5.2.45) ≤ C Dus−1 |∇Dz|2 ≤ C δ0 ∇Dz , and, noting that H 1 → Lμ , (5.2.46)

|˜ g |λ ≤ |aij (Du) − aij (0)|μ |∂i ∂j z|2 ≤ C |Du|μ |∇Dz|2 ≤ C Du1 |∇Dz|2 ≤ C δ0 ∇Dz .

240

5. Asymptotic Behavior

From (5.2.42), . . . , (5.2.46), and recalling (5.2.34), it follows that, for all t ≥ Tε , (5.2.47)

|χ(t)|2 + |χ(t)|λ   ≤ ω(δ0 , R0 ) ∇Dz(t) + ∇Dz(t)1/4 ≤ C ε1/4 .

As in (4.5.206), νλ (0, 1) > 1, because λ < q¯; thus, we deduce from (5.2.41) that  t 1/4 (5.2.48) W2 (t) ≤ C ε (1 + τ )− νλ dτ ≤ C ε1/4 . 0

Together with (5.2.38), (5.2.39) and (5.2.40), (5.2.48) allows us to assert, via (5.2.36), that ∇z(t) → 0 as t → +∞. 3) To prove that zt (t) → 0 as t → +∞, we multiply equation (5.2.3) in L2 by 2 zt , to obtain (5.2.49) d |zt |22 + 4 |zt |22 = 2f − h + g + aij (Du) ∂i ∂j z, zt  dt ≤ 2 |f − h|2 |zt |2 + |g + aij (Du) ∂i ∂j z|22 + |zt |22 . Recalling (5.2.24) and (5.2.34), for t ≥ Tε , (5.2.50)

|g + aij (Du) ∂i ∂j z|2 ≤ gs + |aij (Du)|∞ ∂i ∂j z ≤ C R0 ε + κ(δ0 ) ε

≤Cε;

in addition, as in (5.2.26), |zt |2 = |ut |2 ≤ δ0 ≤ 1. Consequently, by (5.2.50) we deduce from (5.2.49) that, for t ≥ Tε ,  d  2t (5.2.51) e |zt |22 ≤ 2 e2 t |f (t) − h|2 + C e2 t ε2 . dt Integrating for t ≥ Tε ,  t 2t 2 2 Tε 2 (5.2.52) e |zt (t)|2 ≤ e |zt (Tε )|2 + 2 e2 θ |f (θ) − h|2 dθ + C ε2 e2 t , Tε

from which, recalling also (5.2.35), (5.2.53)

|zt (t)|22

≤ e ≤

− 2(t−Tε )



|zt (Tε )|22

e− 2(t−Tε ) |z

2 t (Tε )|2

t

+2

|f (θ) − h|2 dθ + C ε2

Tε

+ 2 ε + C ε2 .

This shows that zt (t) → 0 as t → +∞; thus, (5.2.31) follows, and the proof of Theorem 5.2.1 is complete.  Remark. To prove (5.2.2), it would be natural to multiply (5.2.3) in H s by 2(zt + z), as in (4.5.141) of the proof of Theorem 4.5.2; however, we cannot do so, because, in contrast to the situation in that theorem, here we do not

5.3. Convergence upar (t) → usta

241

know that u(t) − v ∈ L2 , because we do not know that v ∈ L2 . Note also that Theorem 4.5.3 guarantees that u ∈ Cb ([0, +∞[; L2 ) only when N ≥ 5, and, as we have already noted in the linear case (Propositions 4.5.5 and 4.5.6), the map t → u(t) may fail to be bounded in L2 if N ≤ 4. 

5.3. Convergence upar (t) → usta In this section we give sufficient conditions for the convergence, as t → +∞, of the solution to the parabolic Cauchy problem (4.6.1) to the solution to the stationary equation (5.1.2). Theorem 5.3.1. Let N ≥ 3. In addition to the assumptions of Theorems 4.6.2 and 5.1.1 with m = 2 (in particular, aij ∈ C s+2 (RN )), suppose that (5.2.1) holds. Let u and v be the solutions of (4.6.1) and (5.1.2), given respectively by Theorems 4.6.2 and 5.1.1, when δ0 and R0 are sufficiently small. Then, u(t) → v in V s+1 as t → +∞; that is, (5.3.1)

lim ∇(u(t) − v)s = 0 .

t→+∞

Proof. We proceed as in the proof of Theorem 5.2.1. The difference z(t) := u(t) − v satisfies the equation (5.3.2)

zt − aij (∇u) ∂i ∂j z = f − h + g ,

where g ∈ L∞ (0, +∞; H s ∩ L1 ) is as in (5.2.4), but with D replaced by ∇. Similarly, we define Qs−1 (∇∂x z) as in (5.2.9), with D replaced by ∇. 1) Multiplying (5.3.2) in V s by 2z, we obtain d ∇z2s−1 + 2 Qs−1 (∇∂x z) = 2∇(f − h + g), ∇zs−1 dt N s−1      αγ α+ek Ak (u, z), ∂xα ∂k z +2 γ (5.3.3) |α|=1 k=1 0 0, we know that there is Tε ≥ 0 such that, for all t ≥ Tε , (5.3.13)

∇∂x z(t)s−1 ≤ ε

5.3. Convergence upar (t) → usta

243

(because of (5.3.11)), and (5.2.35) holds (because of (5.2.1)). Let χ := g + g˜. Then, for all t ≥ Tε , by the variation of parameters formula (4.6.56),  t z(t) = H(t − Tε ) ∗ z(Tε ) + H(t − θ) ∗ (f (θ) − h) dθ Tε  t (5.3.14) + H(t − θ) ∗ χ(θ) dθ =: z 1 (t) + z 2 (t) + z 3 (t) . Tε

By Parseval’s formula (1.5.9),  1 2 ∇z (t) = |F [H(t − Tε ) ∗ ∇z(Tε )](ξ)|2 dξ  2 ≤ C e− 2(t−Tε )|ξ| |F [∇z(Tε )](ξ)|2 dξ (5.3.15)

  =: (t,ξ)

≤ C ∇z(Tε ) . 2

Since (t, ξ) → 0 as t → +∞ for all ξ ∈ RN \ {0}, by Lebesgue’s dominated convergence theorem it follows from (5.3.15) that ∇z 1 (t) → 0 as t → +∞; that is, there is T3 ≥ Tε + 1 such that, for all t ≥ T3 , ∇z 1 (t) ≤ ε .

(5.3.16)

Next, by Young’s inequality (1.4.24), and (4.6.7), similarly to (5.2.40),  t ∇z 2 (t) ≤ C |H(t − θ)|1 |∇(f (θ) − h)|2 dθ Tε (5.3.17)  t ≤ C f (θ) − h1 dθ ≤ C ε . Tε

For the last term of (5.3.14), we proceed as in the estimate of W2 (t) in (5.2.41), just replacing D with ∇ in (5.2.42), (5.2.44), (5.2.45) and (5.2.46). By (4.6.11),  t 3 |∇z (t)|2 ≤ |∇(H(t − θ) ∗ χ(θ))|2 dθ Tε (5.3.18)  t ≤ C (1 + t − θ)− νλ (0,1) (|∇χ(θ)|2 + |χ(θ)|λ ) dθ . Tε

Acting as in (3.2.25) of Lemma 3.2.1, and recalling that χ = g + g˜, we first estimate |∇g|2 ≤ (aij (∇u) − aij (∇v)) ∂i ∂j v1 (5.3.19)

≤ aij (∇u) − aij (∇v)1 ∂i ∂j vs−1 ≤ R0 ψ(δ0 , R0 ) ∇u − ∇v1 ≤ κ(δ0 , R0 ) ∇∂x z ;

244

5. Asymptotic Behavior

likewise, (5.3.20)

|∇˜ g|2 ≤ a˜ij (∇u)s−1 ∂i ∂j z1 ≤ κ(δ0 ) ∇∂x z1 .

Thus, recalling also (5.2.44) and (5.2.46), by (5.3.13) we obtain that, for all t ≥ T3 , (5.3.21)

 |∇χ(t)|2 + |χ(t)|λ ≤ κ(δ0 , R0 ) ∇∂x z(t)1 + ∇∂x z(t)1/4 ≤ C ε1/4 .

Replacing this into (5.3.18), and recalling that νλ (0, 1) > 1, we deduce that (5.3.22)

∇z 3 (t) ≤ C ε1/4 .

Together with (5.3.16) and (5.3.17), (5.3.22) allows us to assert, via (5.3.14), that ∇z(t) → 0 as t → +∞, as claimed. This completes the proof of Theorem 5.3.1.  Remark. In contrast to the hyperbolic situation, we cannot expect the asymptotic decay of zt (t) = ut (t), unless f (t) − h also decays to 0. More precisely, by (5.3.2) (5.3.23)

f (t) − h = zt − aij (∇u)∂i ∂j z − g ;

by (5.3.11) and (5.2.24) (with D replaced by ∇), (5.3.24)

aij (∇u(t))∂i ∂j z(t) + g(t)s−1 → 0

as

t → +∞ .

Thus, we deduce from (5.3.23) that, under the assumptions of Theorem 5.3.1, (5.3.25)

zt (t)m → 0

⇐⇒

f (t) − hm → 0 ,

where 0 ≤ m ≤ s − 1. In fact, resorting to arguments similar to those used in the proof of the regularity Theorem 4.6.3, one can also find sufficient conditions on f (t) − h so that ut (t) → 0 in H s . For example, one such condition is that ft ∈ L1 ([0, +∞[; H s−1 ). 

5.4. Stability Estimates In this section we assume that the equations of the quasi-linear hyperbolic and parabolic problems (4.5.1) and (4.6.1) are homogeneous (that is, f ≡ 0), and prove some stability estimates for their solutions. More precisely, noting that (4.5.135) and (4.5.136) of Theorem 4.5.2, as well as (4.6.19) and (4.6.20) of Theorem 4.6.1 are obviously satisfied when f ≡ 0, we deduce that these solutions decay to 0 as t → +∞. In the linear case, Corollary 4.5.1 and Proposition 4.6.1 yield explicit decay rates of u and v; here, we consider the question of whether we can determine similar rates for the decay of u and v in the quasi-linear case, or whether a loss of decay can be expected. For

5.4. Stability Estimates

245

some results on this question, for equations which are small perturbations of the linear equations (4.5.4) or (4.6.4), that is, of the form (5.4.1)

utt + ut − Δ u = F (∇u, ∂x2 u) ,

(5.4.2)

ut − Δ u = F (∇u, ∂x2 u) ,

where F is sufficiently smooth and at least quadratic in a neighborhood of the origin, see, e.g., Li Ya-Chun [89] for (5.4.1), and Zheng and Chen [169]. We also mention that the stability results we obtain can be used to investigate the existence and the properties of the attractors of the semiflows generated by (4.5.1), in the phase space H s+1 × H s , and (4.6.1), in the phase space H s+1 . For a general reference on semiflows, see, e.g., Milani and Koksch [119]. 1. In general, the asymptotic decay properties of the solutions of (4.5.1) and (4.6.1), with f ≡ 0, depend in an essential way on the behavior of the coefficients aij at the origin; more precisely, on the rate at which the functions p → a ˜ij (p) := aij (p) − aij (0), p ∈ R1+N , vanish as p → 0, as measured by an estimate of the type (5.4.3)

|aij (p) − aij (0)| ≤ ω(|p|) ,

ω ∈ K0 ,

for all p in a neighborhood of 0 in R1+N . For example, quadratic vanishing of the a ˜ij ’s would be described by ω(r) = C r2 ; more generally, we may assume that, for some ς ≥ 1 and C, δ > 0, |aij (p) − aij (0)| ≤ C |p|ς ,

(5.4.4)

for all p ∈ R1+N with |p| ≤ δ. We will give a brief mention of how conditions like (5.4.4) influence the decay properties of the solutions in estimate (5.4.63) below. Note, however, that (5.4.4) is trivially satisfied at least for ς = 1, as follows from the identity  1 (5.4.5) a ˜ij (p) = aij (p) − aij (0) = aij  (λp) · p dλ ; 0

in fact, in the sequel we will only consider this less favorable situation. 2. Before proving the nonlinear decay estimates, we recall the following results from classical analysis. Lemma 5.4.1. Let α ∈ R>1 , β ∈ R>0 , and set γ := min{α, β}. There exists C > 0, depending on α and β, such that, for all t ≥ 0,  t (5.4.6) I(t) := (1 + t − θ)− α (1 + θ)− β dθ ≤ C (1 + t)− γ . 0

Proof. Following Segal [142], we split I(t) = I1 (t) + I2 (t), where  t/2 (5.4.7) I1 (t) := (1 + t − θ)− α (1 + θ)− β dθ . 0

246

5. Asymptotic Behavior

Since 1 + t − θ ≥ 1 +

(5.4.8)

if 0 ≤ θ ≤ 2t , if β = 1 we obtain from (5.4.7) that  t/2   t −α I1 (t) ≤ 1 + 2 (1 + θ)− β dθ 0   − α  1−β 1 = 1−β 1 + 2t 1 + 2t −1 . t 2

If β > 1, recalling that α ≥ γ, (5.4.9)

I1 (t) ≤

1 β−1

 − α 1 + 2t ≤ C (1 + t)− γ ,

while if 0 < β < 1, noting that α − 1 + β > β ≥ γ,  −α+1−β 1 (5.4.10) I1 (t) ≤ 1−β 1 + 2t ≤ C (1 + t)− γ . If β = 1, we modify (5.4.8) into − α    (5.4.11) I1 (t) ≤ 1 + 2t ln 1 + 2t ≤ C (1 + t)− γ , which follows because α > 1 = γ. Next, we estimate  t (5.4.12) I2 (t) := (1 + t − θ)− α (1 + θ)− β dθ . t/2

Since now 1 + θ ≥ 1 + 2t , it follows that, since α > 1 and β ≥ γ,  − β t/2  I2 (t) ≤ 1 + 2t (1 + τ )− α dτ (5.4.13) 0   1 t −β ≤ α−1 1 + 2 ≤ C (1 + t)− γ . Together with (5.4.9), (5.4.10) and (5.4.11), (5.4.13) allows us to deduce (5.4.6).  To deal with the parabolic equation (4.6.1), we need to replace Lemma 5.4.1 with Lemma 5.4.2. Let α, β ∈ ]0, 1[. There exists C > 0, depending on α and β, such that, for all t > 0,  t (5.4.14) I(t) := (t − θ)− α θ− β dθ ≤ C t1−α−β . 0

Proof. First note that the integral + in * (5.4.14) is defined, since both α and t β ∈ ]0, 1[. Fix t > 0, and for ε ∈ 0, 2 , let  t−ε (5.4.15) Iε (t) := (t − θ)− α θ− β dθ . ε

Splitting as in Lemma integral into Iε,1 (t) and Iε,2 (t), respectively * t * 5.4.1*the * t on the intervals ε, 2 and 2 , t − ε , we obtain, as in (5.4.8) and (5.4.13),

5.4. Stability Estimates

that (5.4.16) (5.4.17)

247

 1 − α 2t  1 − β Iε,2 (t) ≤ 2 t

Iε,1 (t) ≤

1 1−β



θ1−β

t/2

ε * + t/2 1−α 1 1−α τ ε

≤ C t−α+1−β , ≤ C t−β+1−α ,

with C independent of ε. Hence, (5.4.14) follows, letting ε → 0.



Lemma 5.4.3. Let α and r ∈ R>0 . There is C > 0, depending on α and r, such that for all t ≥ 0,  t (5.4.18) (1 + θ)− r eαθ dθ ≤ C (1 + t)− r eαt . 0

Consequently, if h ∈ C 1 (R>0 ) satisfies the exponential inequality h  + α h ≤ K (1 + t)− r

(5.4.19)

on an interval [T, +∞[, with K > 0 independent of t, then for all t ≥ T h(t) ≤ C1 (1 + t)− r ,

(5.4.20)

with C1 depending on α, r, T , K, and the initial condition h(T ). 't Proof. 1) Let f (t) := (1 + t)− r eαt and F (t) := 0 f (θ) dθ. Then both f (t) and F (t) → +∞ as t → +∞. We compute that

− 1 r F  (t) 1 (5.4.21) lim = ; = lim α −  t→+∞ f (t) t→+∞ 1+t α hence, by l’Hˆopital’s rule, also (5.4.22)

lim

t→+∞

F (t) 1 = . f (t) α

(t) Thus, the function t → Ff (t) , which is continuous and positive on [0, +∞[, admits a maximum value C = C(α, r) on [0, +∞[. This implies (5.4.18).

2) As for (5.4.20), we first deduce from (5.4.19) that, for t ≥ T ,  t αt αT e h(t) ≤ e h(T ) + K (1 + θ)− r eα θ dθ T (5.4.23)  t αT ≤ e h(T ) + K (1 + θ)− r eα θ dθ ; 0

hence, by (5.4.18), (5.4.24)

h(t) ≤ e− α(t−T ) h(T ) + C K (1 + t)− r .

This implies (5.4.20), with (5.4.25)

 C1 := h(T ) sup e− α(t−T ) (1 + t)r + C K . t≥T

This concludes the proof of Lemma 5.4.3.



248

5. Asymptotic Behavior

5.4.1. Hyperbolic Decay. In the dissipative hyperbolic case, we can restate Theorem 4.5.2 by claiming that for all integer s > N2 + 1, there is δ ∈ ]0, 1] such that, for all u0 ∈ H s+1 and u1 ∈ H s , with u0 s+1 + u1 s ≤ δ, the corresponding solution of the homogeneous Cauchy problem for the equation (5.4.26)

utt + ut − aij (Du) ∂i ∂j u = 0

has a unique global, bounded solution u ∈ Zs,b (∞), such that (5.4.27)

Du(t)s + utt (t)s−1 → 0

as t → +∞ .

As we have said, if (5.4.26) is linear, Corollary 4.5.1 yields explicit decay rates of u; in particular, we deduce from Matsumura’s estimates (4.5.30), with q = 2, that for k, m ∈ N, with 0 ≤ k ≤ 2 and 1 ≤ k + m ≤ s + 1, (5.4.28)

∂tk ∂xm u(t) ≤ Ck,m (1 + t)− ν2 (k,m)

for all t ≥ 0, where Ck,m is a constant that depends on u0 k+m and u1 k+m−1 . In this section we give a partial extension of this result to the rates of decay of the solution to (5.4.26). We recall that, for k, m ∈ N and q ∈ [0, 1] the exponent νq (k, m) is defined in (4.5.29). Theorem 5.4.1. Let N ≥ 3, and s ∈ N, with N2 + 1 < s ≤ N . Assume that aij ∈ C s (R1+N ), with aij  (0) = 0 (thus, (5.4.4) is satisfied only for ς = 1), and that u0 ∈ H s+1 ∩ L1 , u1 ∈ H s ∩ L1 . There exists δ ∈ ]0, 1] such that, if (5.4.29)

u0 s+1 + |u0 |1 + u1 s + |u1 |1 ≤ δ ,

the corresponding solution u of the Cauchy problem for (5.4.26), i.e. with initial values u(0) = u0 and ut (0) = u1 , is in Zs,b (∞), and satisfies the decay estimates (5.4.30)

∂xm ∇u(t) ≤ C0,m (1 + t)− ν2 (0,m+1) ,

0≤m≤s,

(5.4.31)

∂xm ut (t) ≤ C1,m (1 + t)− ν2 (1,m) ,

0≤m≤s,

(5.4.32)

∂xm−1 utt (t) ≤ C2,m (1 + t)− ν2 (2,m−1) ,

1≤m≤s.

In addition, (5.4.33)

u(t) ≤ C0,0 (1 + t)− ν1 (0,0) .

The constants Ck,m , k = 0, 1, 2, depend on δ of (5.4.29). Remarks. Estimates (5.4.30), (5.4.31) and (5.4.32) imply that the decay rates for the solution of the quasi-linear equation (5.4.26) are the same as those of the linear equation, described in (5.4.28). The upper limitation on s, that is, s ≤ N , is technical; while it seems unnatural, we ignore if it is actually necessary. However, the possible relaxing of such limitation seems to depend, at least in part, on conditions like (5.4.4) (see (5.4.63) below). 

5.4. Stability Estimates

249

Proof of Theorem 5.4.1. We follow Cherrier and Milani [28]. 1) If δ is sufficiently small, Theorem 4.5.2 implies that the Cauchy problem for (5.4.26) does have a global solution u ∈ Zs,b (∞). In addition, by (4.5.163) and (4.5.147) (with f = 0), u satisfies the global bounds (5.4.34)

sup Du(t)s ≤ R , t≥0

sup utt (t)s−1 ≤ κ(R) R , t≥0

where κ ∈ K, and R depends on δ (see (4.5.161) and (4.5.162)), and can be made as small as desired by taking δ conveniently small; that is, explicitly, (5.4.35)

∀ R1 > 0 ∃ δ0 ∈ ]0, 1[

such that

δ ≤ δ0 =⇒ R ≤ R1 .

In particular, we can assume that R ≤ 1; many of the universal constants in the sequel will depend on the number κ(1). We also note that it is sufficient to establish the decay estimates for the lowest and highest indicated values of m, because if we know that these estimates hold for two indices m1 < m2 , their validity for the intermediate values m1 < m < m2 follow by interpolation. Indeed, consider for example (5.4.30). The GagliardoNirenberg inequalities imply that (5.4.36) with α =

|∂xm ∇u(t)|2 ≤ C |∂xm2 ∇u(t)|α2 |∂xm1 ∇u(t)|21−α ≤ C (1 + t)− ν , m−m1 m2 −m1 ,

and

ν := α ν2 (0, m2 + 1) + (1 − α) ν2 (0, m1 + 1) (5.4.37) =

1 2

(1 + m1 + α(m2 − m1 )) =

m+1 2

= ν2 (0, m + 1) .

Thus, (5.4.30) holds for m as well. A similar argument holds for (5.4.31), and (5.4.32). In the sequel, we omit to record the dependence of the constants Ck,m on the indices k, m, etc.; likewise, we generally omit the explicit reference to the variables θ and t, unless necessary. As usual, we assume that α0 ≥ 1. 2) We proceed in six steps. In Step 1, we prove (5.4.30) and (5.4.31) for m = 0, and (5.4.33). In Step 2, we prove an intermediate estimate on |∂xs−k ∂tk u(t)|2 , 0 ≤ k ≤ 3 (recall that s ≥ 3 because N ≥ 3), in terms of |∂xs Du(t)|2 . In Step 3, we use this estimate, as part of an energy estimates procedure, to prove (5.4.30) for m = s; as a consequence, we deduce (5.4.31) and (5.4.32) for m ≤ s − 1. In Step 4, we use a similar energy method, to prove (5.4.31) and (5.4.32) for m = s. In Steps 5 and 6 we prove two technical estimates, which are used in Steps 3 and 4. In Steps 1 and 2, we follow, in part, a procedure similar to the one we used in the proof of Theorem 4.5.3, whereby we apply, as in (4.5.205), the linear estimates of Proposition 4.5.2 to the representation of u via Duhamel’s formula, as in (4.5.203). More precisely, using the change of variables

250

5. Asymptotic Behavior

  u ˜(t, x) = 2 u 12 t, 12 x we rewrite equation (5.4.26) in the linearized form (4.5.197); that is, (5.4.38)

utt + 2 ut − aij (0) ∂i ∂j u = g(u) := a ˜ij (Du) ∂i ∂j u .

Then, we decompose u = v + w, where, as in (4.5.203), (5.4.39) (5.4.40)

Wd (t) ∗ (2 u0 + u1 ) + ∂t (Wd (t) ∗ u0 ) ,  t w(t) := Wd (t − θ) ∗ g(u(θ)) dθ . v(t)

=

0

By the very definition of Wd (see (4.5.6)), it follows that  t (5.4.41) wt (t) = ∂t [Wd (t − θ) ∗ g(u(θ))] dθ ; 0

in addition, as we shall need in Step 4,  t (5.4.42) wtt (t) = g(u(t)) + ∂tt [Wd (t − θ) ∗ g(u(θ))] dθ , 0  t (5.4.43) wttt (t) = ∂t (g(u(t))) − 2 g(u(t)) + ∂t3 [Wd (t − θ) ∗ g(u(θ))] dθ . 0

The decomposition u = v + w shows that we can expect u to decay, at best, at a rate as fast as that of v (that is, as per the linear estimates (4.5.30)), and that, in fact, it is sufficient to show the decay estimates (5.4.30), (5.4.31) and (5.4.32) for w. Step 1. We prove (5.4.30) for m = 0, by showing that u satisfies the faster decay estimate (5.4.44)

Du(t) ≤ C (1 + t)− ν1 (0,1) = C (1 + t)−(N/4+1/2) .

To this end, we set (5.4.45)

Φ0 (t) := sup

 (1 + θ)ν1 (0,1) Du(θ)

0≤θ≤t

and prove that Φ0 is bounded. Since ν1 (0, 1) ≤ ν1 (1, 0), we obtain from (5.4.40) and (5.4.41) that  t (5.4.46) |Dw(t)|2 ≤ C (1 + t − θ)− ν1 (0,1) (|g(u)|2 + |g(u)|1 ) dθ . 0

Since a ˜ij (0) = 0, we can estimate the term a ˜ij (Du) of g(u) by Proposition 1.5.8; recalling (5.4.34), and the definition (5.4.45) of Φ0 ,

(5.4.47)

|g(u)|2 + |g(u)|1 ≤ |˜ aij (Du)|2 (|∂i ∂j u|∞ + |∂i ∂j u|2 )   ≤ κ(R) |Du|2 |∂x2 u|∞ + |∂x2 u|2 ≤ ω(R) |Du|2 ≤ ω(R) Φ0 (θ)(1 + θ)− ν1 (0,1) ,

5.4. Stability Estimates

251

with ω ∈ K0 . We replace (5.4.47) into (5.4.46); since Φ0 is non-decreasing, and ν1 (0, 1) > 1 because N ≥ 3, Lemma 5.4.1 implies that  t |Dw(t)|2 ≤ ω(R) Φ0 (t) (1 + t − θ)− ν1 (0,1) (1 + θ)− ν1 (0,1) dθ (5.4.48) 0 ≤ ω(R) Φ0 (t) (1 + t)− ν1 (0,1) . Since u0 and u1 ∈ L1 , v satisfies the linear estimate |Dv(t)|2 ≤ CL (1 + t)− ν1 (0,1) ;

(5.4.49)

hence, we conclude from (5.4.49) and (5.4.48) that (5.4.50)

(1 + t)ν1 (0,1) |Du(t)|2 ≤ CL + ω(R) Φ0 (t) .

Choosing R so small that ω(R) ≤ 12 (which, by (5.4.35), can be done by choosing δ sufficiently small), and recalling that Φ0 is non-decreasing, we deduce from (5.4.50) that Φ0 (t) ≤ 2 CL .

(5.4.51)

Thus, Φ0 is bounded; then, (5.4.44) and, consequently, (5.4.30) for m = 0 follow. We can then proceed to prove (5.4.31) for m = 0, and (5.4.33). Indeed, from (5.4.41), (5.4.47) and (5.4.51),  t |wt (t)|2 ≤ C (1 + t − θ)− ν1 (1,0) (|g(u)|2 + |g(u)|1 ) dθ 0  t ≤ C ω(R) (1 + t − θ)− ν1 (1,0) Φ0 (θ)(1 + θ)− ν1 (0,1) dθ (5.4.52) ≤ C (1 + t)

0 − ν1 (0,1)

≤ C (1 + t)− ν2 (1,0) .

Since |vt (t)|2 satisfies a similar estimate, (5.4.31) for m = 0 follows. In the same way, we derive from (5.4.40) and (4.5.31) that  t (5.4.53) |w(t)|2 ≤ C (1 + t − θ)− ν1 (0,0) (|g(u)|2 + |g(u)|1 ) dθ . 0

By (5.4.47), (5.4.51) and Lemma 5.4.1, we then deduce that (5.4.54)

|w(t)|2 ≤ 2 CL C ω(R) (1 + t)− ν1 (0,0) ,

from which, since v(t) satisfies a similar estimate, (5.4.33) follows. Step 2. 1) We set (5.4.55)



2  ν2 (0,s+1) s Φs (t) := max 1, sup (1 + θ) ∂x Du(θ) 0≤θ≤t

(note that Φs is continuous), and prove the intermediate estimates   (5.4.56) |∂xs−k ∂tk u(t)|2 ≤ Ck + ωk (R)(Φs (t))2 (1 + t)− ν2 (k,s−k)

252

5. Asymptotic Behavior

for 0 ≤ k ≤ 2, and ωk ∈ K0 , and |∂xs−3 ∂t3 u(t)|2 ≤ C3 (Φs (t))7 (1 + t)− ν2 (3,s−3) .

(5.4.57)

In (5.4.56) and (5.4.57), the constants Ck are independent of R; in fact, they depend on the constants Ck,s−k,2 of (4.5.64), and are, therefore, of the type ω(δ), with ω ∈ K0 and δ as in (5.4.29). For convenience, in each of the estimates that follow, we use the same letters λ, μ and ν to denote various exponents, that are different in different estimates; since the context is clear, this should lead to no confusion. Likewise, for r ∈ R>0 we write Φrs (t) instead of (Φs (t))r , and for m ∈ N>0 , we write ∂xm g(u), ∂tm g(u), instead of ∂xm (g(u)) and ∂tm (g(u)). From (5.4.40), (5.4.41), (5.4.42), and (5.4.43), we see that, for 0 ≤ k ≤ 3,

(5.4.58)

|∂xs−k ∂tk w(t)|2 ≤ |∂xs−k Γk (u)|2  t   +C (1 + t − θ)− ν2 (k,s−k) |∂xs−1 g(u)|2 + |g(u)|2 dθ ,   0 =: Ik (t,θ)

with Γ0 (u) = Γ1 (u) = 0, Γ2 (u) = g(u), and Γ3 (u) = ∂t g(u) − 2 g(u). Recalling (5.4.44), and that Φs is non-decreasing, we can modify the estimate of |g(u)|2 in (5.4.47) into |g(u(θ))|2 ≤ |˜ aij (Du)|2 |∂i ∂j u|∞ ≤ κ(R) |Du|2 |∂xs ∇u|λ2 |∇u|21−λ

(5.4.59)

≤ ω(R) Φsλ−ε (t) (1 + θ)− ν+ε(s+1)/2 , with λ = (5.4.60)

N +2 2s

∈

+1

* , 1 , ε ∈ ]0, λ[ to be determined, and s

ν :=

N 4

+

1 2



(2 − λ) +

s+1 2

λ=

3(N +2) 4

−

N (N +2) 8s

.

The assumption N2 + 1 < s ≤ N and the limitation k ≤ 3 imply that ε ν > s+3 2 ≥ ν2 (k, s − k). Thus, we can choose ε > 0 such that ν − 2 (s + 1) ≥ ν2 (k, s − k), and deduce from (5.4.59) that (5.4.61)

|g(u(θ))|2 ≤ ω(R) Φsλ−ε (θ) (1 + θ)− ν2 (k,s−k) .

Since Φs (t) ≥ 1, we conclude that (5.4.62)

|g(u(θ))|2 ≤ ω(R) Φs (θ) (1 + θ)− ν2 (k,s−k) .

5.4. Stability Estimates

253

We remark in passing that if (5.4.4) were to hold for some ς > 1, then, using interpolation, (5.4.59) and (5.4.61) could be replaced by |g(u)|2 ≤ C |Du|ς2 ς |∂x2 u|∞ ≤ C |∂xs Du|2λ+η |Du|21−λ+ς−η

(5.4.63)

≤ C Φsλ+η (t) (1 + t)− νς , with η =

N 2s

(ς − 1) and νς

(5.4.64)

 N 1 = (λ + η) s+1 2 + 4 + 2 (2 − λ + ς − η − 1)     = ν + η 2s − N4 + (ς − 1) N4 + 12  2 = ν + 12 (ς − 1) N + 1 − N4s .

Since νς > ν if ς > 1 (as we see from the second line of (5.4.64)), it follows that we could consider values of s higher than N . 2) Going back to (5.4.58), by Leibniz’ formula we decompose    α β (5.4.65) ∂xs−1 g(u) = a ˜ij (Du) ∂xα−β ∂i ∂j u . β ∂

x   |α|=s−1 β≤α

For suitable p, q ≥ 1, such that we estimate

1 p

=: Λαβ

+ 1q = 12 , as in the proof of Theorem 1.5.4, |β|

(5.4.66)

s−|β|

|Λαβ |2 ≤ κ(R) |∂x Du|p |∂x

∇u|q

2−(λ+μ)

≤ C |∂xs Du|2λ+μ |Du|2

,

where p, q, λ and μ are related by (5.4.67)

1 |β| 1 s = + −λ , p N 2 N

1 s − |β| 1 s = + −μ . q N 2 N

N Thus, λ + μ = 1 + 2s ∈ ]1, 2[. Recalling (5.4.30) and (5.4.31) for m = 0, we 1 obtain from (5.4.66), and step 1) of this proof, that, for ε = s+1 ,

(5.4.68) with (5.4.69)

|Λαβ |2 ≤ C |∂xs Du|ε2 Φsλ+μ−ε (t) (1 + θ)− ν ,  ν := 1 +

N 2s

−

1 s+1



s+1 2

+

N 4

+

1 2

 1−

N 2s



.

N Then, λ + μ − ε ≤ 1 + 2s ≤ 2, and ν ≥ s+3 2 ≥ ν2 (k, s − k) (because of the N assumption 2 + 1 < s ≤ N ); consequently, from (5.4.65),

(5.4.70)

|∂xs−1 g(u(θ))|2 ≤ ω(R) Φ2s (θ) (1 + θ)− ν2 (k,s−k) .

254

5. Asymptotic Behavior

Together with (5.4.62), (5.4.70) implies, via Lemma 5.4.1, that  t (5.4.71) Ik (t, θ) dθ ≤ ω(R) Φ2s (t) (1 + t)− ν2 (k,s−k) . 0

When k = 2, we estimate |∂xs−2 Γ2 (u)|2 = |∂xs−2 g(u)|2 by interpolation from (5.4.70) and (5.4.62); recalling that Φs (t) ≥ 1, (5.4.72)

|∂xs−2 Γ2 (u)|2 ≤ ω(R) Φ2s (t) (1 + θ)− ν2 (k,s−k) .

Inserting (5.4.71) and (5.4.72) into (5.4.58), we conclude that, if 0 ≤ k ≤ 2, (5.4.73)

|∂xs−k ∂tk w(t)|2 ≤ ω(R) Φ2s (t) (1 + t)− ν2 (k,s−k) .

Adding this to the analogous estimate for |∂xs−k ∂tk v(t)|2 , (5.4.56) follows. When k = 3, we still need to estimate (5.4.74)

  ∂xs−3 ∂t g(u) = ∂xs−3 a ˜ij (Du)∂i ∂j ut + a ˜ij  (Du) · Dut ∂i ∂j u =: S1 + S2 .

By Leibniz’ formula, (5.4.75)

S1 =

   α β ˜ij (Du) ∂xα−β ∂i ∂j ut ; β ∂x a |α|=s−3 β≤α

thus, by the Gagliardo-Nirenberg inequalities,  |S1 |2 ≤ C |∂x|β| a ˜ij (Du)|p |∂xs−1−|β| ut |q |β|≤s−3

≤ κ(R) |∂xs Du|λ2 |Du|21−λ |∂xs ut |μ2 |ut |21−μ

(5.4.76)

2−(λ+μ)

≤ C |∂xs Du|2λ+μ |Du|2

with p, q ∈ [2, +∞] and λ, μ ∈ [0, 1] related by (5.4.77)

1 |β| 1 s = + −λ , p N 2 N

Then, λ + μ = 1 + (5.4.44), (5.4.78)

N −2 2s

,

+

1 q

= 12 , and

1 s − 1 − |β| 1 s = + −μ . q N 2 N

∈ ]1, 2[, so that we obtain from (5.4.76), using

|S1 |2 ≤ ω(R) Φs (t) (1 + t)− ν ≤ C Φs (t) (1 + t)− ν2 (3,s−3) ,

with (5.4.79)

1 p

ν :=

s+1 2

 + 1−

N −2 2s

 N 4

+

1 2



≥

s+3 2

= ν2 (3, s − 3) ,

as is straightforward to verify. Likewise,     α β γ (5.4.80) S2 = a ˜ij  (Du) ∂xβ−γ Dut ∂xα−β ∂i ∂j u . β γ ∂

x   |α|=s−3 β≤α

γ≤β

=: S2αβγ

When γ > 0, again by the Gagliardo-Nirenberg inequalities, (5.4.81)

|S2αβγ |2 ≤ κ(R) |∂x|γ| Du|p |∂x|β|−|γ| Dut |q |∂xs−2−|β| ∇u|r ,

5.4. Stability Estimates

with

1 p

+

(5.4.82)

1 q

+

1 r

255

= 12 , |∂x|γ| Du|p ≤ C |∂xs−1 Du|λ2 |Du|21−λ ,

(5.4.83)

|∂x|β|−|γ| Dut |q ≤ C |∂xs−1 Dut |μ2 |Dut |21−μ ,

(5.4.84)

|∂xs−2−|β| ∇u|r ≤ C |∂xs−1 ∇u|σ2 |∇u|21−σ ,

and (5.4.85)

1 p

=

|γ| 1 s−1 + −λ , N 2 N

(5.4.86)

1 q

=

|β| − |γ| 1 s−1 + −μ , N 2 N

1 s − 2 − |β| 1 s−1 = + −σ . r N 2 N In (5.4.82) and (5.4.84), we use (5.4.56) for k = 0 and k = 1 to estimate   (5.4.88) |∂xs−1 ∇u(t)|2 ≤ |∂xs−1 Du(t)|2 ≤ C + ω(R)Φ2s (t) (1 + t)− ν2 (0,s) , (5.4.87)

for suitable C > 0 and ω ∈ K0 . In (5.4.83), using equation (5.4.26) we first estimate (5.4.89) |Dut |2 ≤ |∇ut |2 + |utt |2 ≤ |∇ut |2 + |ut |2 + |aij (Du)|∞ |∂i ∂j u|2 1/s

1−1/s

≤ C |∂xs ut |2 |ut |2

1/s

1−1/s

+ |ut |2 + κ(R) |∂xs ∇u|2 |∇u|2

.

Recalling (5.4.44), we deduce from (5.4.89) that −α |Dut (t)|2 ≤ C Φ1/s + C (1 + t)− ν1 (0,1) , s (t) (1 + t)       1 N 1 s+1 1 N 1 with α = s+1 ; and, since Φ + 1 − + (t) ≥ 1 and ≥ + s 2s s 4 2 2s s 4 2 , we finally obtain that

(5.4.90)

(5.4.91)

|Dut (t)|2 ≤ C Φs (t) (1 + t)− ν1 (0,1) .

Similarly, (5.4.92) |∂xs−1 Dut |2 ≤ |∂xs−1 ∇ut |2 + |∂xs−1 utt |2 ≤ |∂xs ut |2 + |∂xs−1 (aij (Du) ∂i ∂j u − ut )|2 ≤ |∂xs ut |2 + |∂xs−1 ut |2 + |∂xs−1 g(u)|2 + |∂xs−1 (aij (0)∂i ∂j u)|2 .

256

5. Asymptotic Behavior

By (5.4.56) and (5.4.70), we can proceed with |∂xs−1 Dut |2 ≤ (1 + |aij (0)|) |∂xs Du|2 + |∂xs−1 ut |2 + |∂xs−1 g(u)|2 ≤ C Φs (t)(1 + t)− ν2 (0,s+1)   + C1 + ω1 (R)Φ2s (t) (1 + t)− ν2 (1,s−1)

(5.4.93)

+ (C + ω(R) Φs (t))(1 + t)− ν2 (3,s−3) ≤ C Φ2s (t) (1 + t)− ν2 (0,s+1) . Consequently, replacing (5.4.88), (5.4.91) and (5.4.93) into (5.4.82), (5.4.83), (5.4.84), and then into (5.4.81), we obtain that (5.4.94)

|S2αβγ |2 ≤ C Φs2(λ+μ+σ)+1−μ (t) (1 + t)− ν ,

where (5.4.95)

ν := 2s (λ + μ + σ) + 12 μ +

N 4

+

1 2



(3 − (λ + μ + σ)) .

−1 We compute that λ+μ+σ = 1+ Ns−1 ∈ [2, 3[, again because thus, since also N ≥ 3,    3N −1 s N 1 3 ν ≥ 1 + Ns−1 2 − 4 − 2 + 4 + 2  N  N N −1 (5.4.96) ≥ s+3 + 2 − 12 2 + 2(s−1) s − 2 + 1

≥

s+3 2

N 2

+1 < s ≤ N ;

= ν2 (3, s − 3) .

Consequently, we deduce from (5.4.94) that, if γ > 0, |S2αβγ |2 ≤ C Φ7s (t) (1 + t)− ν2 (3,s−3) .

(5.4.97)

If γ = 0, we replace (5.4.81) with (5.4.98)

|S2αβ0 |2 ≤ κ(R) |∂x|β| Dut |q |∂xs−2−|β| ∇u|r ,

with, now, 1q + 1r = 12 . Proceeding as in (5.4.83) and (5.4.84), and recalling (5.4.91) and (5.4.93), we obtain that (5.4.99)

|S2αβ0 |2 ≤ C |∂xs−1 Dut |μ2 |Dut |21−μ |∂xs−1 ∇u|σ2 |∇u|21−σ 2(μ+σ)+1−μ

≤ C Φs where μ + σ = 1 +

N −2 2(s−1)

ν := (5.4.100)

(t) (1 + t)− ν ,

∈ ]1, 2[, and, now,  N s 1 1 2 (μ + σ) + 4 + 2 (2 − (μ + σ)) + 2 μ

≥

 1+

≥

s 2

+

1 2

N −2 2(s−1)

+

N 4

 +

s 2

−

1 2

N −2 4(s−1)

− 

N 4

s−



+

N 2

N 2

+1

 −1 .

5.4. Stability Estimates

257

If N ≥ 4, we proceed with ν ≥ (5.4.101)

s 2

+

3 2

= ν2 (3, s − 3), and deduce that

|S2αβ0 |2 ≤ C Φ5s (t) (1 + t)− ν2 (3,s−3) .

If N = 3, the conditions 32 + 1 < s ≤ 3 imply that s = 3 as well; thus, recalling (5.4.80), the sum S2 reduces to the single term S2000 . Recalling (5.4.91) and (5.4.93), as well as (5.4.44) and the definition (5.4.55) of Φs , and that ν2 (3, 0) = 3, we estimate |S2000 |2 ≤ |˜ aij (Du)|∞ |Dut |∞ |∂i ∂j u|2 ≤ κ(R) |Dut |∞ |∂x2 u|2 (5.4.102)

3/4

1/4

1/3

2/3

≤ C |∂x2 Dut |2 |Dut |2 |∂x3 ∇u|2 |∇u|2 ≤ C Φp3 (t) (1 + t)− q ,

where p = 2 · 34 + 13 · 14 + 13 ≤ 2, and q = 2 · 34 + 54 · 14 + 2 · 13 + 54 · 23 ≥ 3. Thus, we conclude from (5.4.102) that (5.4.103)

|S2000 |2 ≤ C Φ2s (t) (1 + t)− ν2 (3,s−3) ,

s=3.

Recalling (5.4.74), we deduce from (5.4.97), (5.4.101) and (5.4.103), together with (5.4.78), that (5.4.104)

|∂xs−3 ∂t g(u)|2 ≤ C Φ7s (t) (1 + t)− ν2 (3,s−3) .

Estimating |∂xs−3 g(u)|2 again by interpolation, as we did to obtain (5.4.72), we finally conclude that Γ3 (u) (defined after (5.4.58)) satisfies the estimate (5.4.105)

|∂xs−3 Γ3 (u)|2 ≤ C Φ7s (θ) (1 + θ)− ν2 (3,s−3) .

Inserting (5.4.105) and (5.4.71) into (5.4.58) (for k = 3), and adding the corresponding linear estimate for ∂xs−3 ∂t3 v, we finally conclude the proof of (5.4.57). Step 3. 1) We show that the intermediate estimate (5.4.56) for k = 1, i.e., (5.4.106)

|∂xs−1 ut (t)|2 ≤ (C1 + ω1 (R)Φ2s (t))(1 + t)− ν2 (1,s−1) ,

allows us to prove, by means of an energy estimates argument, that Φs is bounded. In analogy to (4.5.183), and recalling the definition (2.3.13) of Q0 , we set (5.4.107)

P˜ u (t) := |∂xs ut (t)|22 + ∂xs u(t), ∂xs ut (t) + Q0 (au(t) , ∇∂xs u(t)) .

258

5. Asymptotic Behavior

We differentiate equation (5.4.26) α times with respect to x, |α| = s, and multiply the resulting equations in L2 by 2∂xα ut + ∂xα u. Summing the resulting identities for |α| = s, we obtain d ˜u ˜u P +P dt  = (R0,α + R1,α + R2,α + RG,1,α + RG,2,α )

(5.4.108)

|α|=s

=: Λ0 , where the terms R0,α , . . . , RG,2,α , are defined as in (3.3.89), (3.3.90), (3.3.97), (3.3.91) and (3.3.98). In Step 5, we shall prove that there is ω ∈ K0 such that Λ0 ≤ ω(R) |∂xs Du(t)|22 + ω(R)Φ3s (t) (1 + t)− 2ν2 (0,s+1) .

(5.4.109)

Assuming this for the moment, we deduce from (5.4.108) that (5.4.110)

 d  t ˜u ≤ et ω(R) |∂xs Du(t)|22 + Φ3s (t) (1 + t)− 2ν2 (0,s+1) e P dt   ≤ ω(R) et Φ2s (t) + Φ3s (t) (1 + t)− 2ν2 (0,s+1) .

Since Φs ≥ 1, integration of (5.4.110) yields



e P˜ u (t) ≤ P˜ u (0) + ω(R) Φ3s (t) t

(5.4.111)

t

eθ (1 + θ)− 2ν2 (0,s+1) dθ .

0

Recalling that, by (5.4.29), |P˜ u (0)| ≤ C δ 2 ≤ C, by Lemma 5.4.3 we obtain from (5.4.111) that   (5.4.112) P˜ u (t) ≤ C + ω(R)Φ3s (t) (1 + t)− 2ν2 (0,s+1) =: Z0 (t) . Recalling (5.4.107), we further deduce from (5.4.112) that |∂xs Du(t)|22 ≤ Z0 (t) − ∂xs u(t), ∂xs ut (t) (5.4.113)

≤ Z0 (t) + ∂xs+1 u(t), ∂xs−1 ut (t) ≤ Z0 (t) + 12 |∂xs ∇u(t)|22 + 12 |∂xs−1 ut (t)|22 ,

from which (5.4.114)

|∂xs Du(t)|22 ≤ 2 Z0 (t) + |∂xs−1 ut (t)|22 .

By (5.4.106), noting that ν2 (1, s − 1) = ν2 (0, s + 1), and recalling the definition of Z0 in (5.4.112), we obtain that (with different C and ω)   (5.4.115) |∂xs Du(t)|22 ≤ C + ω(R) Φ4s (t) (1 + t)− 2ν2 (0,s+1) ,

5.4. Stability Estimates

259

from which Φs (t) ≤ Cs + ωs (R) Φ2s (t) ,

(5.4.116)

for suitable Cs ≥ 1 independent of R, and ωs ∈ K0 . It is then immediate to show that, if R is sufficiently small, there is C2 > Cs such that, for all t ≥ 0, Φs (t) ≤ C2 .

(5.4.117)

Indeed, assuming otherwise, for all R > 0 and C > Cs there would be tC > 0 such that Φs (tC ) > C. Take C = 2 Cs ; since, by (5.4.29), Φs (0) = |∂xs Du(0)|2 ≤ δ < Cs , by the continuity of Φs there is t1 ∈ ]0, t2 Cs [ such that (5.4.118)

∀ t ∈ [0, t1 ] ,

Φs (t) ≤ Φs (t1 ) = 2 Cs .

Then, from (5.4.116) for t = t1 , (5.4.119)

2 Cs ≤ Cs + ωs (R) (2 Cs )2 ,

i.e., 1 ≤ 4 ωs (R) Cs ,

which leads to a contradiction for small R. Consequently, (5.4.117) holds; that is, under the reservation that (5.4.109) holds, Φs is bounded. This implies (5.4.30) for m = s, with C0,s := C2 . 2) Inserting (5.4.117) into (5.4.56) and (5.4.57), we deduce that (5.4.120)

|∂xs−k ∂tk u(t)|2 ≤ C (1 + t)− ν2 (k,s−k) ,

0≤k ≤3;

in particular, this implies (5.4.31) and (5.4.32) for m = s − 1. We now prove (5.4.32) for m = 1. From (5.4.42) we derive that |wtt (t)|2 ≤ |g(u(t))|2 

(5.4.121)

t

+C

(1 + t − θ)− ν2 (2,0) (|∇g(u)|2 + |g(u)|2 ) dθ .

0

Since s + 1 ≥ 4 because N ≥ 3, it follows that (5.4.62) with k = 1, and by (5.4.117), (5.4.122)

s+1 2

≥ 2 = ν2 (2, 0); hence, by

|g(u(θ))|2 ≤ ω(1) C2 (1 + θ)− ν2 (1,s−1) ≤ C (1 + θ)− ν2 (2,0) .

Next, we use interpolation between (5.4.70) with k = 1 and (5.4.122), to deduce, again via (5.4.117), that 1/(s−1)

(5.4.123)

|∇g(u(θ))|2 ≤ C |∂xs−1 g(u(θ))|2

1−1/(s−1)

|g(u(θ))|2

≤ C (1 + θ)− ν2 (1,s−1) ≤ C (1 + θ)− ν2 (2,0) .

Thus, we conclude from (5.4.121), via Lemma 5.4.1, that (5.4.124)

|wtt (t)|2 ≤ C (1 + t)− ν2 (2,0) ;

together with the analogous estimate for |vtt (t)|2 , (5.4.124) implies (5.4.32) for m = 1.

260

5. Asymptotic Behavior

Step 4. We use an energy method, similar to the one we used in Step 3, to prove (5.4.31) and (5.4.32) for m = s. Let k = 1, 2. We generalize (5.4.107) into (5.4.125) P˜ku := |∂xs−k ∂tk ut |22 + ∂xs−k ∂tk u, ∂xs−k ∂tk ut  + Q0 (au , ∇∂xs−k ∂tk u) ; we also set (compare to (5.4.55))  (5.4.126)



Ψk (t) := max 1, sup 0≤θ≤t

(1 + θ)ν2 (k,s+1−k) ∂xs−k ∂tk Du(θ)

2

(thus, Ψ0 = Φs ). We differentiate equation (5.4.26) first k times with respect to t, to obtain ∂tk utt + ∂tk ut − aij (Du) ∂i ∂j ∂tk u (5.4.127) =

k  k 

∂t a ˜ij (Du) ∂tk− ∂i ∂j u =: Dk (u) ;

=1

then, we differentiate (5.4.127) α times with respect to x, |α| = s − k, and multiply the resulting equations in L2 by 2∂xα ∂tk ut + ∂xα ∂tk u. Summing the resulting identities for |α| = s − k, we obtain (5.4.128)

 

d ˜u ˜u P + Pk = dt k

(k)

(k)

(k)

(k)

(k)



R0,α + R1,α + R2,α + RG,1,α + RG,2,α

|α|=s−k

+



∂xα Dk (u), ∂xα ∂tk (2 ut + u) =: Λk + Mk ,

|α|=s−k (k)

(k)

where the terms R0,α , . . . , RG,2,α , are defined in analogy to (3.3.89), (3.3.90), (3.3.97), (3.3.91) and (3.3.98); that is, (k)

(5.4.129)

R0,α := ∂t (aij (Du))∂i ∂xα ∂tk u, ∂j ∂xα ∂tk u ,

(5.4.130)

R1,α + R2,α := − ∂j (aij (Du))∂i ∂xα ∂tk u, ∂xα ∂tk (2ut + u) ,

(k)

(k)

and (5.4.131) (k)

(k)

RG,1,α + RG,2,α :=

   α β α−β ∂i ∂j ∂tk u, ∂xα ∂tk (2ut + u) . β ∂x aij (Du) ∂x 0 0 is implied by the inequality 1 (5.4.155) 1+ ≤λ+μ. 2(s + 1) From (5.4.148) and (5.4.149), we compute that N k−2 k−1 (5.4.156) λ+μ=1+ − + μ, 2s s s so that (5.4.155) follows. With the choice of ε in (5.4.154), and recalling that N ≥ 3, (5.4.153) yields that  1 ν ≥ (s + 1) 1 + 2(s+1) + N2 (5.4.157) ≥ s + 3 ≥ s + 1 + k = ν2 (k, s + 1 − k) .

264

5. Asymptotic Behavior

Since also (5.4.158)

2(λ−ε) 1−μ

=2+

1 s+1

< 3, we finally conclude from (5.4.152) that

RG,1,α ≤ ω(R) |∂xs−k ∂tk Du|22 + ω(R) Φ3s (t) (1 + t)− ν2 (k,s+1−k) , (k)

as desired in (5.4.143). Next, as in (5.4.146) and (5.4.147), (5.4.159)

(k)

RG,2,α ≤ κ(R)

s−k 

|∂xb Du|p |∂xs+2−k−b ∂tk u|q |∂xs−k ∂tk u|

b=1

2N N −2

,

for p, q ∈ ]2, +∞[ such that 1p + 1q = 12 + N1 , determined as per Corollary b s−b 1.5.1, with r = N2N −2 > 2, s1 = s−b and s2 = b−1, considering ∂x Du ∈ H and ∂xs+2−k−b ∂tk u ∈ H b−1 . In particular, we find that q = 2 if b = 1, and that, if min{s − b, b − 1} > N2 , we can choose either p = 2 and q = N , or p = N and q = 2. From (5.4.159), we proceed as in (5.4.152), that is (5.4.160) (k)

RG,2,α ≤ ω(R) |∂xs Du|2λ−ε |Du|21−λ |∂tk u|21−μ |∂xs−k ∂tk Du|1+μ 2 ≤ ω(R) |∂xs−k ∂tk Du|22 + ω(R) Φs2(λ−ε)/(1−μ) (t) (1 + t)− ν , with the same λ and μ of (5.4.148) and (5.4.149), and the same ν and ε of (5.4.153) and (5.4.154). The only difference is that, now, (5.4.156) has to be replaced by (5.4.161)

λ+μ=1+

N 2s

−

k−1 s

+

k−1 s

μ,

and (5.4.155) is implied by the inequality 1 k−1 N + (1 − μ) ≤ , 2(s + 1) s 2s

(5.4.162)

which does hold. Thus, from (5.4.160), (5.4.163)

RG,2,α ≤ ω(R) |∂xs−k ∂tk Du|22 + ω(R) Φ3s (t) (1 + t)− ν2 (k,s+1−k) , (k)

again as desired in (5.4.143). In conclusion, inserting (5.4.144), (5.4.145), (5.4.146), (5.4.158) and (5.4.163) into the definition of Λk in (5.4.128), we see that (5.4.143) follows. In particular, Λ0 satisfies (5.4.109); thus, Step 3 is now complete, and (5.4.117) can be assumed to hold. As we have seen in part 2 of Step 3, this implies the validity of (5.4.120), and of (5.4.32) for m = 1; as a consequence, (5.4.30) holds for m ≤ s; in addition, (5.4.31) and (5.4.32) hold at least for m ≤ s − 1. Step 6. We prove estimate (5.4.133). 1) Let k = 1. Recalling the definition of Mk in (5.4.128), and of D1 (u) in (5.4.127), we split  (5.4.164) M1 = ∂xα D1 (u), ∂xα (2 utt + ut ) =: M11 + M12 . |α|=s−1

5.4. Stability Estimates

265

At first, (5.4.165) M11 ≤ C |∂xs−1 D1 (u)|2 |∂xs−1 utt |2 ≤ C |∂xs−1 D1 (u)|2 |∂xs−1 ∂t1 Du|2 . The estimate of ∂xs−1 D1 (u) is similar to that of the term S2 of (5.4.74), to which we refer. As in (5.4.80),        αβγ α β (5.4.166) ∂xs−1 D1 (u) = . β γ S2 |α|=s−1 β≤α

γ≤β

When γ > 0, the analogous of (5.4.81) is (5.4.167)

|S2αβγ |2 ≤ κ(R) |∂x|γ| Du|p |∂x|β|−|γ| Dut |q |∂xs−|β| ∇u|r ,

and we replace estimates (5.4.82), (5.4.83), (5.4.84) with (5.4.168)

|∂x|γ| Du|p ≤ C |∂xs Du|λ2 |Du|21−λ ,

(5.4.169)

|∂x|β|−|γ| Dut |q ≤ C |∂xs−1 Dut |μ2 |Dut |21−μ ,

(5.4.170)

|∂xs−|β| ∇u|r ≤ C |∂xs ∇u|σ2 |∇u|21−σ ,

where, now, (5.4.171)

1 p

=

|γ| 1 s + −λ , N 2 N

(5.4.172)

1 q

=

|β| − |γ| 1 s−1 + −μ , N 2 N

1 s − |β| 1 s = + −σ . r N 2 N By (5.4.117), (5.4.93), and (5.4.91), (5.4.173)

(5.4.174)

|∂xs Du(t)|2 ≤ C (1 + t)−

s+1 2

,

(5.4.175)

|∂xs−1 Dut (t)|2 ≤ C (1 + t)−

s+1 2

,

(5.4.176)

−

−

1 2

|Dut (t)|2 ≤ C (1 + t)

N 4

;

in addition, in (5.4.168) we decompose (5.4.177)

|∂xs Du|λ2 = |∂xs Du|ε2 |∂xs Du|2λ−ε ≤ Rε |∂xs Du|2λ−ε ,

for ε ∈ ]0, λ[ to be determined. Hence, we deduce from (5.4.167) that (5.4.178)

|S2αβγ |2 ≤ ω(R) (1 + t)− ν+ε (s+1)/2 ,

where (5.4.179)

ν=

s+1 2

(λ + μ + σ) +

N 4

+

1 2



(3 − (λ + μ + σ)) .

From (5.4.171), (5.4.172) and (5.4.173), we compute that N μ (5.4.180) λ+μ+σ =1+ + ∈ ]1, 3] . s s

266

5. Asymptotic Behavior

This, together with the assumptions N2 +1 < s ≤ N , allows us to deduce that s+1 s+2 ν > s+2 2 . Thus, we can choose ε ∈ ]0, λ[ such that ν −ε 2 ≥ 2 = ν2 (1, s), and conclude from (5.4.178) that (5.4.181)

|S2αβγ |2 ≤ ω(R) (1 + t)− ν2 (1,s) .

The estimate of S2αβ0 is similar: replacing (5.4.167) with (5.4.182) 1 q

|S2αβ0 |2 ≤ κ(R) |∂x|β| Dut |q |∂xs−|β| ∇u|r ,

+ 1r = 12 , and proceeding as in (5.4.169) and (5.4.170) (compare to (5.4.98)

and the subsequent estimates), we obtain that S2αβ0 satisfies the same estimate (5.4.181). Then, by (5.4.182) and (5.4.181), it follows from (5.4.166) and (5.4.165) that (5.4.183)

M11 ≤ ω(R) |∂xs−1 ∂t1 Du|22 + C (1 + t)−2 ν2 (1,s+1−1) ,

as desired in (5.4.133) when k = 1. To estimate M12 , we proceed as in (5.4.146) and (5.4.159); that is, M12 ≤ C |∂xs−1 D1 (u)| (5.4.184)

≤ C |∂xs−1 D1 (u)|

2N N +2 2N N +2

|∂xs−1 ut |

2N N −2

|∂xs−1 ∇ut |2 .

We can then follow the same steps as in the estimate of S2αβγ above, except that, now, the indices p, q and r in (5.4.167) satisfy p1 + 1q + 1r = 12 + N1 , and, in (5.4.180), we find that λ + μ + σ = 1 + Ns − 1−μ s , as we deduce from (5.4.171), (5.4.172) and (5.4.173). However, we can still verify that ν > s+2 2 ; thus, we conclude that, as in (5.4.181), (5.4.185)

|S2αβγ |

2N N +2

≤ ω(R) (1 + t)− ν2 (1,s) .

Consequently, we deduce from (5.4.184) that (5.4.186)

M12 ≤ ω(R) |∂xs−1 ∂t1 Du|22 + C (1 + t)−2 ν2 (1,s+1−1) .

Together with (5.4.183), (5.4.186) implies (5.4.133) for k = 1. 2) When k = 2, recalling the definition (5.4.128) of M2 , we start with M2 ≤ C |∂xs−2 D2 (u)|2 |∂xs−2 ∂t2 ut |2 (5.4.187)

+ C |∂xs−2 D2 (u)| 2N |∂xs−2 ∂t2 u| 2N N +2 N −2  ≤ C |∂xs−2 D2 (u)|2 + |∂xs−2 D2 (u)| 2N |∂xs−2 ∂t2 Du|2 , N +2

5.4. Stability Estimates

267

with D2 (u) defined in (5.4.127); that is, D2 (u)

=

aij  (Du) · Dut ∂i ∂j ut + aij  (Du)(Dut , Dut ) ∂i ∂j u + aij  (Du) · Dutt ∂i ∂j u

(5.4.188)

=: D21 (u) + D22 (u) + D23 (u) . The estimate of (5.4.187) proceeds along lines similar to those we followed in the estimate of M11 and M12 . Here, we limit ourselves to the most difficult step; that is, the estimate of the term M := |∂xs−2 D23 (u)|

(5.4.189)

2N N +2

|∂xs−2 ∂t2 Du|2 ,

that would appear in (5.4.187). By Leibniz’ formula, as usual, (5.4.190) ∂xs−2 D23 (u) =

   α β



β |α|=s−2 β≤α

γ γ≤β

∂ γ aij  (Du) · ∂xβ−γ Dutt ∂xα−β ∂i ∂j u .

x   αβγ =: S23

2a) When γ > 0, we choose p, q and r ≥ 2 such that and estimate

1 p

+ 1q + 1r =

1 2

+ N1 ,

(5.4.191) αβγ |S23 |

2N N +2

≤ κ(R) |∂x|γ| Du|p |∂x|β|−|γ| Dutt |q |∂xs−1−|β| ∇u|r ≤ C |∂xs Du|2λ+σ |Du|22−λ−σ |∂xs−2 Dutt |μ2 |Dutt |21−μ ≤ ω(R) |∂xs Du|2λ+σ−ε |Du|22−λ−σ |Dutt |21−μ |∂xs−2 ∂t2 Du|μ2 ,

with (5.4.192)

1 p

=

|γ| 1 s + −λ , N 2 N

(5.4.193)

1 q

=

|β| − |γ| 1 s−2 + −μ , N 2 N

1 s − 1 − |β| 1 s = + −σ , r N 2 N and ε ∈ ]0, λ[. As in (5.4.89), (5.4.194)

(5.4.195)

|Dutt |2 ≤ |∇utt |2 + |uttt |2 ;

by (5.4.32) for m = 1 and (5.4.120) for k = 2 (which do hold, as we noted at the end of Step 5), 1

(5.4.196)

1 1− s−2

|∇utt |2 ≤ C |∂xs−2 utt |2s−2 |utt |2

≤ C (1 + t)− 2 . 5

268

5. Asymptotic Behavior

In analogy to (5.4.151), using (5.4.32) for m = 1 and (5.4.31) for m = 2, (5.4.197) |uttt |2 ≤ |utt |2 + |aij (Du)|∞ |∂i ∂j ut |2 + |aij  (Du)|∞ |Dut |p |∂i ∂j u|q ≤ C (1 + t)− 2 + κ(R) (1 + t)− 2 + κ(R) |∂xs−1 Dut |λ2 |Dut |21−λ |∂xs−1 ∂x2 u|μ2 |∂x2 u|21−μ ≤ C (1 + t)− 2 + C (1 + t)− ν , where (5.4.198)

1 1 1 + = , p q 2

1 1 s−1 = −λ , p 2 N

1 1 s−1 = −μ , q 2 N

N so that λ + μ = 2(s−1) and, recalling (5.4.175) and (5.4.176), as well as (5.4.30) for m = 1,  N 1 ν = s+1 2 (λ + μ) + (1 − λ) 4 + 2 + 1 − μ  N N 3 1 = s−1 2 (λ + μ) + 4 + 2 − λ 4 − 2 (5.4.199)   = N4 + N4 + 32 − λ N4 − 12   = N4 + 2 + N4 − 12 (1 − λ) ≥ N4 + 2 > 2 .

Thus, we deduce from (5.4.197), (5.4.196) and (5.4.195) that |Dutt (t)|2 ≤ C (1 + t)− 2 .

(5.4.200)

We insert (5.4.200) into (5.4.191), and then into (5.4.189), and proceed as in the estimate of M1 : using (5.4.44) and (5.4.174), we obtain that αβγ |S23 |

(5.4.201)

2N N +2

|∂xs−2 ∂t2 Du|2

≤ ω(R) |∂xs Du|2λ+σ−ε |Du|22−λ−σ |Dutt |21−μ |∂xs−2 ∂t2 Du|1+μ 2 ≤ ω(R) |∂xs−2 ∂t2 Du|22 + C (1 + t)− ρ1 +ε (s+1)/(1−μ) ,

where (5.4.202)

ρ1 := (s + 1) λ+σ 1−μ +

N 2

+1



2−λ−σ 1−μ

+4.

We wish to show that ρ1 > s + 3, which is equivalent to     (5.4.203) (λ + μ + σ) s − N2 + N + 3 > s − μ N2 − 1 . From (5.4.192), (5.4.193) and (5.4.194) we compute that (5.4.204)

λ+μ+σ =1+

N −2 s

+

2μ s

≥1+

N −2 s

.

5.4. Stability Estimates

269

Thus, (5.4.203) is implied by the inequality    (5.4.205) 1 + N s−2 s − N2 + N + 3 > s , which holds beacuse N2 + 1 < s ≤ N . We can therefore choose ε ∈ ]0, λ[ so small that ρ1 − ε(s+1) 1−μ ≥ s + 3; and, since s + 3 = 2 ν2 (2, s − 1), we conclude from (5.4.201) that (5.4.206) αβγ |S23 |

2N N +2

|∂xs−2 ∂t2 Du|2 ≤ ω(R) |∂xs−2 ∂t2 Du|22 + C (1 + t)− 2 ν2 (2,s−1) .

2b) When γ = 0, we replace (5.4.191) with αβ0 |S23 |

(5.4.207)

2N N +2

|β|

s−1−|β|

≤ κ(R) |∂x Dutt |q |∂x

∇u|r

≤ C |∂xs Du|σ2 |Du|21−σ |∂xs−2 Dutt |μ2 |Dutt |21−μ ,

with 1q + 1r = 12 , and μ, σ as in (5.4.192), (5.4.193) and (5.4.194) (with γ = 0). Correspondingly, (5.4.201) becomes αβ0 |S23 |

2N N +2

|∂xs−2 ∂t2 Du|2

≤ ω(R) |∂xs Du|σ−ε |Du|21−σ |Dutt |21−μ |∂xs−2 ∂t2 Du|1+μ 2 2

(5.4.208)

≤ ω(R) |∂xs−2 ∂t2 Du|22 + C (1 + t)− ρ2 +ε (s+1)/(1−μ) , where, now, σ ρ2 := (s + 1) 1−μ +

(5.4.209)

N 2

+1

 1−σ 1−μ

+4.

We verify again that ρ2 > s + 3; hence, we deduce from (5.4.208) that, if ε is sufficiently small, (5.4.210) αβ0 |S23 |

2N N +2

|∂xs−2 ∂t2 Du|2 ≤ ω(R) |∂xs−2 ∂t2 Du|22 + C (1 + t)− 2 ν2 (2,s−1) .

Together with (5.4.206), (5.4.210) implies, via (5.4.189), that (5.4.211)

M ≤ ω(R) |∂xs−2 ∂t2 Du|22 + C (1 + t)− 2 ν2 (2,s−1) ,

as desired in (5.4.133) for k = 2. Reasoning in a similar way for the other terms at the right side of (5.4.187), we can conclude the proof of (5.4.133) for k = 2. Thus, Step 6 is complete, and this ends the proof of Theorem 5.4.1.  Remarks. 1) Comparing (5.4.32) for m = s − 1 with (5.4.30) for m = s and (5.4.31) for m = s − 1, we see that, in L2 , ∂xs−1 ut (t) and ∂xs−1 ∂i ∂j u(t) s−1 decay with the same rate r = s+1 2 , while ∂x utt (t) decays at the faster rate r +1 = s+3 2 . This motivates the conjecture that the asymptotic profile of the

270

5. Asymptotic Behavior

solution of (5.4.26) should coincide with that of a solution of the parabolic equation (3.1.7). We shall indeed prove such kind of result in section 5.5. 2) If we assume that, in addition to the assumptions of Theorem 5.4.1, aij ∈ C s+1 (R1+N ), u0 ∈ H s+2 and u1 ∈ H s+1 , then, following an energy method analogous to the one used in Steps 3 and 4, we can prove the decay estimate (5.4.212)

∂xs+2 u(t) ≤ C (1 + t)− ν2 (0,s+2) .

The key point in this argument is the analog of the integration by parts step in (5.4.137), which would now be (5.4.213) 1 s+1 |∂ ∇u|22 + C (1 + t)−(s+2) , 4 x in the last term of which we used (5.4.31) for m = s. −∂xs+1 u, ∂xs+1 ut  = ∂xs+2 u, ∂xs ut  ≤



5.4.2. Parabolic Decay. As in the hyperbolic case, we can restate Theorem 4.6.1 by claiming that for all integer s > N2 + 1, there is δ ∈ ]0, 1] such that, for all u0 ∈ H s+1 , with u0 s+1 ≤ δ, the corresponding solution of the homogeneous Cauchy problem for the equation ut − aij (∇u) ∂i ∂j u = 0

(5.4.214)

has a unique global solution u ∈ Ps (∞), with ut ∈ Cb ([0, +∞[; H s−1 ) by Theorem 4.6.2, such that (4.6.19) and (4.6.20) hold; that is, (5.4.215)

lim (∇u(t)s + ut (t)s−1 ) = 0 .

t→+∞

Again, if (5.4.214) is linear, Proposition 4.6.1 yields explicit decay rates of u; in fact, we deduce from estimates (4.6.11), with q = 2, that, for k = 0, 1, 0 ≤ 2k + m ≤ s + 1, and all t ≥ 0,  (5.4.216) ∂tk ∂xm u(t) ≤ Ck,m (1 + t)− ν2 (k,m) ∂x2k+m u0  + u0  . In this section we give a partial extension of this result to the rates of decay of the solution to (5.4.214). In analogy to Theorem 5.4.1, we claim: Theorem 5.4.2. Let N ≥ 3, and s ∈ N, with s > N2 + 1. Assume that aij ∈ C s (RN ), with aij  (0) = 0 (thus, (5.4.4) is satisfied only for ς = 1), and that u0 ∈ H s+1 . Let ν2 (k, m) be defined as in (4.5.29). There exists δ ∈ ]0, 1] such that, if u0 s+1 ≤ δ, the corresponding solution of the Cauchy problem for (5.4.214) satisfies the decay estimates (5.4.217) (5.4.218)

∂xm ∇u(t) ≤ C0,m (1 + t)− ν2 (0,m+1) ∂xm ut (t)

≤ C1,m (1 + t)

− ν2 (1,m)

for

0≤m≤s−1,

for

0≤m≤s−2.

5.4. Stability Estimates

271

If u0 ∈ L1 , with sufficiently small norm, or if aij ∈ C s+1 (RN ), (5.4.217) also holds for m = s, (5.4.218) holds for m = s − 1, and u(t) ≤ C0,0 (1 + t)− ν1 (0,0) .

(5.4.219)

The constants Ck,m , k = 0, 1, depend on k, m, N , and δ. Proof. The proof of Theorem 5.4.2 follows essentially the same lines of that of Theorem 5.4.1. Here too, if δ is sufficiently small, Theorem 4.6.1 implies that the Cauchy problem for (5.4.214) does have a global, bounded solution u ∈ Ps,b (∞); in addition, by (4.6.31) and (4.6.23), u satisfies the global bound  +∞ 2 (5.4.220) sup u(t)s+1 + Du2s dθ ≤ R2 , t≥0

0

where R depends on δ, and can be made as small as desired by taking δ conveniently small (as in (5.4.35)); in particular, we can assume that R ≤ 1. We also note that (5.4.218) follows from (5.4.217); more precisely, if (5.4.217) holds for 1 ≤ m ≤ s, then (5.4.218) holds for 0 ≤ m ≤ s − 1. This is a consequence of the identity (5.4.221)

∂xα ut = ∂xα (aij (∇u) ∂i ∂j u) ,

|α| = m ,

whose right side we estimate by means of the Sobolev product estimates and the chain rule, using (5.4.217) and keeping in mind the already noted identity ν2 (1, m) = ν2 (0, m + 2). To prove (5.4.217), as in (5.4.38) we rewrite (5.4.214) in the linearized form (5.4.222)

ut − aij (0) ∂i ∂j u = h(u) := a ˜ij (∇u) ∂i ∂j u ,

and note that, by the linear estimates (4.6.10) and (4.6.11), it is sufficient to estimate the difference w = u − v, where, with notation analogous to the one in (4.6.56), v(t) := H(t) ∗ u0 , and  t (5.4.223) w(t) := H(t − θ) ∗ h(u(θ)) dθ . 0

In analogy with (5.4.55), for 0 ≤ m ≤ s and t > 0 we define  2  (5.4.224) Φm (t) := max 1, sup (1 + θ)ν2 (0,m+1) ∂xm ∇u(θ) , 0≤θ≤t

and seek to establish time-independent bounds on Φ0 (t) and Φs (t) (the other cases following by interpolation, as in (5.4.36)). As in the proof of Theorem 5.4.1, we occasionally abbreviate Φm for Φm (t) or Φm (θ). 1) We set (5.4.225)

Φ(t) := Φ0 (t) + Φs−1 (t) ,

 := 2 −

1 s−1

∈ ]1, 2[ .

272

5. Asymptotic Behavior

By the linear estimates (4.6.11), we deduce from (5.4.223) that  t (5.4.226) |∇w(t)|2 ≤ C (1 + t − θ)− ν1 (0,1) (|h(u)|1 + |∇h(u)|2 ) dθ . 0

By the H¨older and Gagliardo-Nirenberg inequalities, and (5.4.220), (5.4.227) |h(u)|1 ≤ |˜ aij (∇u)|2 |∂i ∂j u|2 ≤ κ(R) |∇u|2 |∂x2 u|2 1

1 1− s−1

≤ C |∇u|2 |∂xs u|2s−1 |∇u|2

≤ ω(R) |∇u|2

≤ ω(R) Φ0 (θ) (1 + θ)− /2 ≤ ω(R) Φ0 (θ) (1 + θ)− 1/2 , with ω ∈ K0 . Likewise, with some abuse of notation, |∇h(u)|2

(5.4.228)

≤

|˜ aij  (∇u) ∂x ∇u ∂i ∂j u|2 + |˜ aij (∇u) ∂i ∂j ∇u|2

=: H1 + H2 . By (5.4.220), setting η :=

N +4 4(s−1)

∈ ]0, 1[, 2(1−η)

H1 ≤ κ(R) |∂x2 u|24 ≤ C |∂xs u|22 η |∇u|2 ≤ C Rη Φηs−1 (θ) (1 + θ)− η s/2 Φ0

2(1−η)

(5.4.229)

2(1−η)

≤ ω(R) Φηs−1 (θ) Φ0

(θ) (1 + θ)−(1−η)

(θ) (1 + θ)− ρ ,

with ρ := η 2s + 1 − η. Similarly, for suitable p and q ≥ 1 such that 1p + 1q = 12 , H2 ≤ |˜ aij (∇u)|p |∂i ∂j ∇u|q (5.4.230)

≤ κ(R) |∇u|p |∂x3 u|q ≤ C |∂xs u|2λ+μ |∇u|22−λ−μ ,

1 2 1 s−1 with λ, μ ∈ [0, 1] related to p and q by 1p = 12 − λ s−1 N , q = N + 2 −μ N . N +4 Noting that λ + μ = 2(s−1) = 2 η, we deduce from the first line of (5.4.229) that H2 satisfies the same estimate (5.4.229) as H1 ; thus, recalling (5.4.228), 2(1−η)

|∇ h(u(θ))|2 ≤ ω(R) Φηs−1 (θ) Φ0

(5.4.231)

(θ) (1 + θ)− ρ .

Inserting (5.4.227) and (5.4.231) into (5.4.226), by Lemma 5.4.1 we obtain that (5.4.232)



t

(1 + t − θ)− ν1 (0,1) Φ0 (θ) (1 + θ)− 1/2 dθ 0  t 2(1−η) + ω(R) (1 + t − θ)− ν1 (0,1) Φηs−1 (θ) Φ0 (θ) (1 + θ)− ρ dθ

|∇w(t)|2 ≤ ω(R)

≤

0  ω(R) Φ0 (t) (1 +

t)− 1/2 + ω(R) Φηs−1 (t) Φ0

2(1−η)

(t) (1 + t)− σ ,

5.4. Stability Estimates

273

where σ := min{ν1 (0, 1), ρ} ≥ 12 . Thus, since v satisfies the linear estimate |∇v(t)|2 ≤ CL (1 + t)− 1/2 ,

(5.4.233)

for suitable CL > 0 depending on u0 , we conclude that (5.4.234)

2(1−η)

Φ0 (t) ≤ CL + ω(R) Φ0 (t) + ω(R) Φηs−1 (t) Φ0

(t) .

1 Noting that 2(1 − η) < 2 − s−1 = , by H¨older’s inequality we further obtain from (5.4.234) that

(5.4.235) with r =

Φ0 (t) ≤ CL + ω(R) Φ0 (t) + ω(R) Φrs−1 (t) =: Ψ(t) , η −2(1−η)

≤ .

2) We now use an energy method to obtain an intermediate estimate on Φs−1 (t). We differentiate equation (5.4.214) α times with respect to x, |α| = s, and multiply the resulting equation in L2 by 2(1 + t)n ∂xα u, with n ≥ s. Summing in α, we obtain that, for all t ≥ 0, (5.4.236)  d  (1 + t)n |∂xs u|22 + 2(1 + t)n Q0 (au , ∇∂xs u) dt = n (1 + t)n−1 |∂xs u|22 + 2 (1 + t)n



(R2,α + RG,2,α ) ,

|α|=s

where, as in (4.5.140), R2,α and RG,2,α , are defined in (3.3.97) and (3.3.98), but with the coefficients auij of (3.2.49) replaced by auij (t, x) = aij (∇u(t, x)). For θ ∈ ]0, 1[ to be chosen later, R2,α ≤ C |aij  (∇u)|∞ |∂j ∇u|∞ |∇∂xs u|2 |∂xs u|2 (5.4.237)

1−θ+θ ≤ κ(R) |∂x2 u|∞ |∇∂xs u|2 |∂xs u|21−θ+θ

 θ ≤ ω(R) |∇∂xs u|2 |∂x2 u|∞ |∂xs u|2 , having recalled (5.4.220). By the Gagliardo-Nirenberg inequalities, we can proceed with (5.4.238)

1+θ(σ1 +σ2 )

R2,α ≤ ω(R) |∂xs ∇u|2

θ(2−σ1 −σ2 )

|∇u|2

,

+2 with σ1 = N2s and σ2 = s−1 s . We now choose θ so that θ(σ1 + σ2 ) = 1; since 1 < σ1 + σ2 < 2, it follows that 12 < θ < 1, and θ(2 − σ1 − σ2 ) = 2 θ − 1 > 0. Hence, we obtain from (5.4.238) that

(5.4.239)

R2,α ≤ ω(R) |∇∂xs u|22 .

274

5. Asymptotic Behavior

Likewise, for suitable p, q, r ≥ 1 such that p1 + 1q + 1r = 1, and keeping in mind that β > 0,  RG,2,α ≤ C |∂x|β| aij (∇u)|p |∂xs−|β|+1 ∇u|q |∂xs u|r 0 4 C4 . Now, by (5.4.220), Φ(0) ≤ 2 R ≤ 2 C4 ; hence, since Φ is continuous, there would be t1 ∈ ]0, t2 [ such that Φ(t1 ) = 4 C4 . But then, (5.4.253) would imply the contradiction (5.4.255)

4 C4 = Φ(t1 ) ≤ C4 + ω(R) (4 C4 ) ≤ 2 C4 .

Thus, Φ is bounded; therefore, so are Φ0 and Φs−1 ; that is, (5.4.217) holds for m = 0 and m = s − 1. As we have remarked above, this also implies that (5.4.218) holds for 0 ≤ m ≤ s − 2. For future reference, we note that if u0 ∈ L1 , then the decay estimate (5.4.217) for m = 0 can be improved into (5.4.256)

|∇u(t)|2 ≤ C1 (1 + t)− /2 .

Indeed, if u0 ∈ L1 , then v satisfies the linear estimate (5.4.257)

|∇v(t)|2 ≤ CL (1 + t)− ν1 (0,1) ;

on the other hand, since Φ0 and Φs−1 are bounded, we deduce from the third line of (5.4.227) and from (5.4.231) that, since ρ ≥ 2 , (5.4.258)

|h(u)|1 + |∇h(u)|2 ≤ C (1 + t)− min{ρ,/2} = C (1 + t)− /2 .

276

5. Asymptotic Behavior

Since also ν1 (0, 1) ≥ 2 , we deduce from (5.4.226) that |∇w(t)|2 ≤ C (1 + t)− min{ν1 (0,1),/2} = C (1 + t)− /2 .

(5.4.259)

Together with (5.4.257), (5.4.259) yields (5.4.256). 3) We now prove (5.4.217) for m = s, under the additional assumption that either u0 ∈ L1 , or aij ∈ C s+1 (RN ). In the latter case, we can repeat the argument of the second part of this proof, differentiating once more the equation, i.e., with |α| = s + 1 in (5.4.236). This leads to an estimate analogous to (5.4.250), with s replaced by s + 1 and n > s + 1, which implies (5.4.217) for m = s. If instead u0 ∈ L1 , we first note that, using the improved estimate (5.4.256) of |∇u|2 in (5.4.248) leads to modifying the latter into (5.4.260) n (1 + t)n−1 |∂xs u|22 ≤ C (1 + t)n σ2 |∇∂xs u|22 σ2 (1 + t)(n−)(1−σ2 )−1 1 (1 + t)n |∇∂xs u|22 + C (1 + t)n−−s . 2 In turn, (5.4.260) allows us to modify (5.4.249) into  d  (5.4.261) (1 + t)n |∂xs u|22 + 12 (1 + t)n |∇∂xs u|22 ≤ C (1 + t)n−−s , dt from which, choosing n = s and integrating, we obtain that  t  C (5.4.262) (1 + t)1− − 1 . (1 + θ)s |∇∂xs u|22 dθ ≤ 2 |∂xs u(0)|22 + 1− ≤

0

We now note that 1 −  < 0, because s ≥ 3; hence, we finally conclude from (5.4.262) that  t (5.4.263) (1 + θ)s |∇∂xs u|22 dθ ≤ C1 . 0

We now multiply the differentiated equations in L2 by 2 (1 + t)s+1 ∂xα ut , |α| = s. As in (4.6.33) of the proof of Theorem 4.6.1, we obtain  d  2 (1 + t)s+1 |∂xs ut |22 + (1 + t)s+1 Q0 (∇∂xs u) dt (5.4.264)

= (s + 1) (1 + t)s Q0 (∇∂xs u) + (1 + t)s+1



(R0,α + R1,α + RG,1,α ) .

|α|=s

We modify the estimates (4.6.35) and (4.6.36) of the last terms of (5.4.264) as follows. At first, we note that, from equation (5.4.214) itself, (5.4.265)

|ut |2 ≤ |aij (∇u)|∞ |∂i ∂j u|2 ≤ κ(R) R = ω(R) .

5.4. Stability Estimates

Next, for γ1 :=

N +2 2s ,

277

γ2 :=

4 2−γ1 ,

and γ3 :=

2(s+1) 2−γ1 ,

and recalling (5.4.265),

R0,α ≤ h(|∇u|∞ ) |∇ut |∞ |∇∂xs u|22 1 ≤ κ(R) |∂xs ut |γ21 |ut |1−γ |∇∂xs u|22 2

(5.4.266)

4 2−γ1

≤ ω(R) |∇∂xs u|2

+ 13 |∂xs ut |22

≤ ω(R) Φγs 2 (t) (1 + t)

−

2(s+1) 2−γ1

+ 13 |∂xs ut |22

2) ≤ ω(R) Φ2(1+γ (t) (1 + t)− γ3 + 13 |∂xs ut |22 , s

having noted that γ2 ≤ 2(1 + γ1 ). Likewise, since (1 + γ1 )(s + 1) ≥ γ3 , R1,α + RG,1,α ≤ h(|∇u|∞ ) |∇∂x u|∞ |∇∂xs u|2 |∂xs ut |2 1 1 ≤ κ(R) |∇u|1−γ |∇∂xs u|1+γ |∂xs ut |2 2 2

(5.4.267)

2(1+γ1 )

≤ ω(R) |∇∂xs u|2

+ 23 |∂xs ut |22

1) ≤ ω(R) Φ2(1+γ (t) (1 + t)−(1+γ1 )(s+1) + 23 |∂xs ut |22 s 1) ≤ ω(R) Φ2(1+γ (t) (1 + t)− γ3 + 23 |∂xs ut |22 . s

Inserting (5.4.266) and (5.4.267) into (5.4.264), we obtain (1 + t)s+1 |∂xs ut |22 + (5.4.268)

 d  (1 + t)s+1 Q0 (∇∂xs u) dt

≤ (s + 1) (1 + t)s Q0 (∇∂xs u) 1) + ω(R) Φ2(1+γ (t) (1 + t)s+1−γ3 , s

from which, integrating,  t (1 + θ)s+1 |∂xs ut |s2 dθ + (1 + t)s+1 Q0 (∇∂xs u(t)) 0  t ≤ Q0 (∇∂xs u(0)) + (s + 1) (1 + θ)s Q0 (∇∂xs u) dθ (5.4.269) 0  t 1) + ω(R) Φ2(1+γ (t) (1 + θ)s+1−γ3 dθ . s 0

Recalling (5.4.263), and noting that γ3 −(s+1) > 1, we deduce from (5.4.269) that, in particular, (5.4.270)

1) (1 + t)s+1 |∇∂xs u(t)|s2 ≤ C2 + ω(R) Φ2(1+γ (t) , s

278

5. Asymptotic Behavior

with C2 depending also on C1 . In turn, we deduce from (5.4.270) that (5.4.271)

1 Φs (t) ≤ C + ω(R) Φ1+γ (t) , s

an inequality qualitatively similar to (5.4.253). As in (5.4.254), we deduce that, if R is sufficiently small, Φs is bounded; hence, (5.4.217) for m = s follows. Consequently, (5.4.218) for m = s − 1 also follows. 4) Finally, (5.4.219) is proven as (5.4.33) in Theorem 5.2.9. This concludes the proof of Theorem 5.4.2.  Remark. In contrast to the hyperbolic case, we are not able to establish estimates for the higher order derivatives of ut , similar to those of Theorem 5.4.1. This is due to the qualitative difference between the linear parabolic estimates of Proposition 4.6.1 and the hyperbolic ones of Corollary 4.5.1, which do not allow us to estimate the higher order derivatives of the difference wt = ut − vt via Duhamel’s formula. Indeed, if we try to exploit the minimal regularity in the initial value, reflected in the presence of only the term |w0 |q in (4.6.10), we encounter the problem of the integrability of the function t → t− νq (k,m) at t = 0, which requires q to be close to 2, k = 0,   2N and m ≤ 1 (more precisely, q ∈ N +4 , 2 if m = 0, and q ∈ N2N +2 , 2 if m = 1). On the other hand, if we try to remedy this by using (4.6.11), we need the corresponding higher regularity of the initial value. This is also reflected in the fact that the solution of the hyperbolic equation satisfies ut ∈ Cb ([0, +∞[; H s ), while the solution of the parabolic equation satisfies ut ∈ Cb ([0, +∞[; H s−1 ) only. Thus, a full extension of the results of Theorem 5.4.1 to the parabolic case is not to be expected. On the other hand, we point out that Theorem 5.4.2 holds for all s > N2 + 1, and not just for N  2 + 1 < s ≤ N , as we had to assume in Theorem 5.4.1.

5.5. The Diffusion Phenomenon 1. Consider a smooth solution u to the linear, dissipative homogeneous equation (5.5.1)

utt + 2 ut − Δ u = 0 .

The decay estimates (4.5.30) of Proposition 4.5.2 imply that, as t → +∞, ut (t) and Δu(t) decay, in a specified norm, with the same rate, while utt (t) decays, in the same norm, with a faster rate. For example,     (5.5.2) ut (t) ≤ O (1 + t)− 1 , ∂i ∂j u(t) ≤ O (1 + t)− 1 , while (5.5.3)

  utt (t) ≤ O (1 + t)− 2 .

5.5. The Diffusion Phenomenon

279

As we have seen in Theorem 5.4.1, the same is true for solutions of the quasilinear dissipative equation (5.4.26). This is due to the obvious observation that the decay rate νq (k, m) defined in (4.5.29) satisfies the identities (5.5.4)

νq (k + 1, m) = νq (k, m + 2) = νq (k + 2, m) − 1 .

The fact that utt decays with a faster rate than ut and Δ u motivates the conjecture that, as t → +∞, the asymptotic profile of u tends to coincide with that of a solution of the heat equation (5.5.5)

vt − Δ v = 0 ;

likewise, as we have already remarked, solutions of the quasi-linear equation (5.4.26) should asymptotically converge to a solution of (5.4.214). In this section we show that this result, known as the diffusion phenomenon of hyperbolic waves, does hold for the linear equation (5.5.1), in the sense that u(t) converges, in a norm to be specified, to the solution of (5.5.5) whose initial value is v(0) = u(0) + ut (0). We remark, in passing, that, as shown in Proposition 4.6.1, the solution of (5.5.5) satisfies the same decay estimates as the solution of (5.5.1). We then proceed to generalize this result, showing that the diffusion phenomenon also holds, with the same decay rates, for small solutions of the homogeneous quasi-linear hyperbolic and parabolic equations (5.5.6)

utt + ut − aij (∇u) ∂i ∂j u = 0 ,

(5.5.7)

vt − aij (∇v) ∂i ∂j v = 0 ,

under suitable assumptions on the coefficients aij . Thus, also the solutions of the quasi-linear hyperbolic equation (5.5.6) have an asymptotically parabolic profile. Note that, in (5.5.6), we assume that the aij ’s depend only on ∇u, as opposed to Du; note also that, since f ≡ 0, by (4.5.135) and (4.5.136) of Theorem 4.5.2 (respectively, (4.6.19) and (4.6.20) of Theorem 4.6.1), it follows that ∇u(t), ut (t) and utt (t) (respectively, ∇v(t) and vt (t)) do vanish, in the specified norms, as t → +∞. 2. The diffusion phenomenon was originally observed by Hsiao and Liu [59], for the system of hyperbolic conservation laws with damping  vt − ux = 0 , (5.5.8) ut + (p(v))x = −α u , where α > 0, with smooth initial data u(0) = u0 , v(0) = v0 that are asymptotically constant, in the sense that (5.5.9)

(u0 (x), v0 (x)) → (u± , v± )

in R2

as x → ±∞. In (5.5.8) and (5.5.9), it is assumed that p > 0 and p  < 0 in R>0 , and v0 , v± > 0. Hsiao and Liu [59] showed that, if v+ = v− , the

280

5. Asymptotic Behavior

solution (u, v) behaves asymptotically like the diffusion wave (¯ u, v¯), solution of the parabolic system ⎧ α v¯t = − (p(¯ v ))xx , ⎪ ⎪ ⎨ (p(¯ v ))x = − α u ¯, (5.5.10) ⎪ ⎪ ⎩ v¯(0, x) → v± as x → ±∞ . The same problem was also considered by Li Ta-Tsien [92], who obtained better estimates than those of Hsiao and Liu [59]. Note that for special initial data (u0 , v0 ) satisfying  ∞ (5.5.11) lim (u0 (x), v0 (x)) = (0, v¯0 ) , (v0 (x) − v¯0 ) dx = 0 , |x|→+∞

−∞

for some v¯0 > 0, system (5.5.8) is reduced to the quasi-linear dissipative hyperbolic Cauchy problem ⎧ ⎪ ⎪ vtt + α vt − (p(v¯0 + vx ) − p(v¯0 ))x = 0 , ⎪ ⎪  x ⎨ (5.5.12) v(0, x) = (v0 (y) − v¯0 ) dy , ⎪ ⎪ −∞ ⎪ ⎪ ⎩ v (0, x) = u (x) . t 0 System (5.5.8) was also considered by Nishihara in [127], with an improvement of the estimates of Hsiao and Liu [59]. In [128], Nishihara also considered the equivalent second order formulation in one space dimension  utt + α ut − (a(ux ))x = 0 , (5.5.13) u(0, x) = u0 (x) , ut (0, x) = u1 (x) , and proved that solutions of (5.5.13) behave asymptotically as those of the linear parabolic problem  α vt − a (0) vxx = 0 , (5.5.14) v(0, x) = u0 (x) + α1 u1 (x) , in the sense that, while |u(t)|∞ and |v(t)|∞ decay as O(t− 1/2 ) as t → +∞, the faster decay   (5.5.15) |u(t) − v(t)|∞ = O t− 1 holds for the difference u − v. In addition, analogous faster decay rates hold for the differences ux − vx and ut − vt ; namely,    (5.5.16) |ux (t) − vx (t)|∞ = O t− 3/2 , |ut (t) − vt (t)|∞ = O t− 2 . A first result on the diffusion phenomenon for quasi-linear equation in higher dimensions was given in Yang and Milani [164], for the equations in

5.5. The Diffusion Phenomenon

281

divergence form (5.5.17)

utt + ut − div (a(∇u)∇u) = 0 ,

(5.5.18)

vt − div (a(∇v)∇v) = 0 ,

with a a smooth function satisfying a(y) = 1 + O (|y|α ) as |y| → 0, for some α ∈ N>0 . More precisely, small, smooth solutions to (5.5.17), (5.5.18) are shown to satisfy the decay estimate  (5.5.19) |u(t) − v(t)|∞ = O t−(N +1)/2 , which generalizes (5.5.15) for N > 1. Since both |u(t)|∞ and |v(t)|∞ decay like t− N/2 , (5.5.19) implies that the diffusion phenomenon does hold for (5.5.17). Other results on the diffusion phenomenon, for different kinds of equations (e.g., semi-linear), can be found in, e.g., Hsiao and Liu [60], Hosono and Ogawa [58], Cavazzoni [22], and the literature quoted therein. In particular, Ikehata and Nishihara [64] and Chill and Haraux [30] give a generalization of these results to dissipative equations in an abstract setting. Finally, Yamazaki [163], and Wirth [162], have investigated the wave equation with time-dependent dissipation. 5.5.1. The Linear Case. In this section we describe the diffusion phenomenon for the linear equation (5.5.1), assuming, for simplicity, that the initial values u0 , u1 and v0 for (5.5.1) and (5.5.5) are in C0∞ (RN ). We take q = 1 in (4.5.29), consider arbitrary k, m ∈ N, and set ν(k, m) := ν1 (k + 1, m) = ν1 (k, m + 2). Recalling the decomposition (4.5.8) (with the coefficient 2 of u0 deleted, because of the different coefficient of ut in (5.5.1)), we immediately deduce from Proposition 4.5.2 that, for |α| = m, as t → +∞,  k α k α − ν(k,m) (5.5.20) ∂t ∂x ut (t) and ∂t ∂x Δ u(t) = O (1 + t) , while (5.5.21)

 ∂tk ∂xα utt (t) = O (1 + t)−ν(k,m)−1 .

By Proposition 4.6.1, the same is true for the solution of (5.5.5); that is,  (5.5.22) ∂tk ∂xα vt (t) and ∂tk ∂xα Δ v(t) = O (1 + t)− ν(k,m) , while (5.5.23)

 ∂tk ∂xα vtt (t) = O (1 + t)−ν(k,m)−1 .

We are now ready to describe the diffusion phenomenon for the linear equation (5.5.1).

282

5. Asymptotic Behavior

Theorem 5.5.1. Let u0 , u1 ∈ C0∞ (RN ), and let u be the corresponding solution of (5.5.1). Let v0 := u0 +u1 , and let v be the corresponding solution of (5.5.5). For all k, m ∈ N, and α ∈ NN , with |α| = m, there is Ck,m > 0 such that, for all t ≥ 1, ∂tk ∂xα (u(t) − v(t)) ≤ Ck,m t−ν1 (k,m)−1 .

(5.5.24)

Consequently, the diffusion phenomenon holds for the linear hyperbolic equation (5.5.1). Remark. In particular, (5.5.24) implies that, in L2 , ∂tk ∂xα (ut − vt ) and ∂tk ∂xα (Δu − Δv) decay with the same rate, which is the same decay rate of ∂tk ∂xα utt .  Proof. We follow Yang and Milani [164]; for an alternative proof, based on Fourier transform techniques, see Volkmer [161]. The difference z = u − v satisfies the Cauchy problem  zt − Δ z = − utt , (5.5.25) z(0) = − u1 . By Duhamel’s formula, as, e.g., in (4.6.56), problem (5.5.25) can be solved exactly, with 

t

z(t) = − H(t) ∗ u1 −

(5.5.26)

H(t − θ) ∗ utt (θ) dθ .

0

Splitting the integral on [0, t] into two parts, we obtain 

t/2

z(t) = − H(t) ∗ u1 − 

(5.5.27) −

H(t − θ) ∗ utt (θ) dθ

0 t

H(t − θ) ∗ utt (θ) dθ .

t/2

In the middle term of the right side of (5.5.27), we integrate by parts to obtain (5.5.28)



t/2

−

H(t − θ) ∗ utt (θ) dθ

0

= −H

t 2

∗ ut

t 2



t/2

+ H(t) ∗ ut (0) − 0

Ht (t − θ) ∗ ut (θ) dθ .

5.5. The Diffusion Phenomenon

283

Replacing this into (5.5.27), we conclude that  t/2     z(t) = − H 2t ∗ ut 2t − Ht (t − θ) ∗ ut (θ) dθ 0  t − (5.5.29) H(t − θ) ∗ utt (θ) dθ t/2

=: − H

t 2

∗ ut

t 2

− I1 (t) − I2 (t) .

We note that it is at this step that the special form of the initial value for v0 is needed, for the cancellation of the term with ut (0) (see the remark at the end of this proof). In the estimates that follow, we make extensive use of Young’s inequality (1.4.24), and the constants will depend on the norms of u0 and u1 in, respectively, H m+k ∩ L1 and H m+k−1 ∩ L1 . 1) Recalling (4.6.7) and (4.5.64), we start with        ∂xα [H 2t ∗ ut 2t ] ≤ ∂xα H 2t 1 (5.5.30) ≤ C t−(m/2) (1 + t)−(N/4+1)

&  t & & ut & 2

  = O t−ν1 (0,m)−1 .

To estimate I1 , we integrate again by parts in t, and decompose  t/2 t t (5.5.31) I1 (t) = Ht 2 ∗ u 2 − Ht (t) ∗ u0 + Htt (t − θ) ∗ u(θ) dθ .

0   =: I3 (t)

Recalling (4.6.7) and (4.5.64), and proceeding as before,        &  & ∂xα [Ht 2t ∗ u 2t ] ≤ ∂xα Ht 2t 1 &u 2t & (5.5.32)  − N/4   ≤ C t−(1+m/2) 1 + 2t ≤ O t−ν1 (0,m)−1 . In the same way, using (4.6.9), ∂xα [Ht (t) ∗ u0 ] ≤ |∂xα Ht (t)|2 |u0 |1 (5.5.33)

 ≤ C t−(N/4+m/2+1) |u0 |1 ≤ O t−ν1 (0,m)−1 .

To estimate I3 , we resort to the semigroup property of the heat kernel (5.5.34)

H(t, ·) ∗ H(s, ·) = C H(t + s, ·) ,

for suitable C > 0; (5.5.34) is easily proven via the identity (5.5.35)

ˆ y) = C1 e−t |y|2 , H(t,

with C depending on C1 . Equation (5.5.34) allows us to decompose     (5.5.36) H(t − θ) = C H 14 t ∗ H 3t 4 −θ ;

284

5. Asymptotic Behavior

by the associative property of the convolution product (recall (1.1.10)), and by Parseval’s formula (1.5.9), we obtain that  t/2 t   α α ∂x I3 (t) ≤ C |∂x Htt 4 |1 |H 3t − θ ∗ u(θ)|2 dθ 4 0 (5.5.37)  t/2   −(2+m/2) ˆ 3t − θ u ≤ Ct |H ˆ(θ)|2 dθ . 4

0

Keeping (5.5.35) in mind, the integrand in (5.5.37) can be estimated by   3t  2 2 ˆ |H 4 − θ u ˆ(θ)|2 ≤ C e− 2(3t/4−θ) |y| |ˆ u(θ, y)|2 dy  ∞ 2 2 ≤ C |ˆ u(θ)|∞ (5.5.38) rN −1 e− 2(3t/4−θ) r dr 0

≤ C

|ˆ u(θ)|2∞

3 t−θ 4

− N/2 ,

having recalled (4.5.45). By (4.5.8) and (4.5.10),   |ˆ u(θ)|∞ ≤ C |h|L∞ (Q) |2ˆ u0 + u ˆ1 |∞ + |ht |L∞ (Q) |ˆ u0 | ∞ (5.5.39) ≤ C (|u0 |1 + |u1 |1 ) . Inserting this in (5.5.37), we obtain that (5.5.40)

∂xα I3 (t)

≤ Ct ≤ C

−(2+m/2)



t/2 

0 −(1+m/2+N/4) t

3 4

t−θ

− N/4

dθ

= C t− ν1 (0,m)−1 .

We estimate I2 (t) using (4.5.64):  t α ∂x I2 (t) ≤ |H(t − θ)|1 |∂xα utt (θ)|2 dθ t/2



(5.5.41)

≤ C

t

(1 + θ)−(N/4)−2−m/2 dθ

t/2

≤ C (1 + t)− ν1 (0,m)−1 . Together with (5.5.30), (5.5.32), (5.5.33) and (5.5.40), (5.5.41) allows us then to conclude from (5.5.29) that, for t ≥ 1, (5.5.42)

∂xα z(t) ≤ C t− ν1 (0,m)−1 ;

that is, (5.5.24) holds for k = 0. 2) We can then proceed by induction on k. Assume that (5.5.24) holds for some k ≥ 0. Then, from the equation of (5.5.25), (5.5.43)

∂tk+1 ∂xα z(t) ≤ ∂tk ∂xα Δz(t) + ∂tk ∂xα utt (t) .

5.5. The Diffusion Phenomenon

285

Thus, by the induction assumption on k and the decay estimate (4.5.64) of Corollary 4.5.1 for utt , (5.5.44)

∂tk+1 ∂xα z(t) ≤ C (1 + t)− ν1 (k,m+2)−1 + C (1 + t)− ν1 (k+2,m) ≤ C (1 + t)− ν1 (k+1,m)−1 ,

as desired in (5.5.24) for k + 1. This ends the proof of Theorem 5.5.1. Since the decay rate in (5.5.24) is higher than those in (5.5.20), . . . , (5.5.23), it does follow that the asymptotic profile of u converges to that of v.  Remarks. 1) A natural question concerning Theorem 5.5.1 is whether the condition v0 = u0 +u1 is actually necessary. A partial answer can be found in Volkmer [161], where it is shown, by means of Fourier transform techniques, that the decay estimate |u(t) − v(t)|2 = O(t− N/4−1/2 )

(5.5.45) holds, if and only if (5.5.46)

 (u0 (x) + u1 (x) − v0 (x)) dx = 0 .

In addition, the decay estimate corresponding to (5.5.42) for α = 0, that is, (5.5.47)

|u(t) − v(t)|2 = O(t− N/4−1 ) ,

holds, if and only if (5.5.46) holds and, for all j = 1, . . . , N ,  (5.5.48) xj (u0 (x) + u1 (x) − v0 (x)) dx = 0 . This result can be extended to higher order derivatives of the difference u−v, and is related to the conditions on the optimality of the decay estimates for u described in Theorem 5.5.1 (as we have mentioned, analogous results hold for v, as shown by Volkmer in [161]). 2) Estimates of the difference u − v can also be given in a more general Lp − Lq setting; that is, the decay of the difference u(t) − v(t) can be estimated, in appropriate Lp -norms, in terms of an appropriate Lq -norm of the difference of the initial data. For example, see Marcati and Nishihara [109] for the one-dimensional case N = 1, Nishihara [129] for the case N = 3, and Narazaki [124] for the cases N ≤ 5. In these works, some applications of the linear estimates to semi-linear equations are also discussed.  5.5.2. The Quasi-Linear Case. In this section we present a result on the diffusion phenomenon for the quasi-linear equation (5.5.6), which generalizes the one given in Yang and

286

5. Asymptotic Behavior

Milani [164]. Given N ≥ 3, we assume that N2 + 1 < s ≤ N , and the higher regularity conditions aij ∈ C s+1 (RN ), u0 and u1 ∈ H s+2 ∩ L1 , with δ := u0 s+2 + |u0 |1 + u1 s+1 + |u1 |1

(5.5.49)

so small that, by Theorems 4.5.2 and 5.4.1, the hyperbolic equation (5.5.6) with initial data u0 and u1 admits a unique solution u ∈ Zs+1,b (∞), satisfying the decay estimates (5.4.30), (5.4.31), (5.4.32), (5.4.33) and (5.4.212). We let v0 := u0 + u1 , and assume that δ of (5.5.49) is also so small that, by Theorems 4.6.1 and 5.4.2, the parabolic equation (5.5.7) with initial value v0 admits a unique solution v ∈ Ps+1,b (∞), satisfying the decay estimates (5.4.217) and (5.4.218), with s replaced by s + 1, and (5.4.219). We claim: Theorem 5.5.2. Let N ≥ 3. Under the above stated assumptions, the difference u − v satisfies, for t ≥ 1, the decay estimates (5.5.50)

∂xm (u(t) − v(t)) ≤ C t− ν2 (0,m)−3/8 ,

1≤m≤s+1,

(5.5.51)

∂xm (ut (t) − vt (t)) ≤ C t− ν2 (1,m)−3/8 ,

0≤m≤s−1,

where C depends on N , s, m, and δ. Consequently, the diffusion phenomenon holds for the quasi-linear hyperbolic equation (5.5.6). Sketch of Proof. 1) Let z := u − v. From the linearized equations (5.5.52)

utt + ut − aij (0) ∂i ∂j u = a ˜ij (∇u) ∂i ∂j u ,

(5.5.53)

vt − aij (0) ∂i ∂j v = a ˜ij (∇v) ∂i ∂j v ,

we derive that (5.5.54)

zt − aij (0) ∂i ∂j z = − utt + F (u, v) ,

where (5.5.55)

F (u, v) := a ˜ij (∇u) ∂i ∂j u − a ˜ij (∇v) ∂i ∂j v .

Since z(0) = − u1 , we can, as in (5.5.26), represent the solution of (5.5.54) as  t z(t) = − H(t) ∗ u1 − H(t − θ) ∗ utt (θ) dθ 0  t + (5.5.56) H(t − θ) ∗ F (u(θ), v(θ)) dθ 0

=: − zlin (t) + znln (t) .

5.5. The Diffusion Phenomenon

287

As in (5.5.29) of the proof of Theorem 5.5.1, we decompose  t/2 t t zlin (t) = H 2 ∗ ut 2 + Ht (t − θ) ∗ ut (θ) dθ 0  t + (5.5.57) H(t − θ) ∗ utt (θ) dθ t/2

=: z1 (t) + z2 (t) + z3 (t) . 2) We first assume that 1 ≤ m ≤ s. Acting as in (5.5.30), by (4.6.7) and (5.4.31) it follows that     |∂xm z1 (t)|2 ≤ |∂xm H 2t |1 |ut 2t |2 (5.5.58) ≤ C t− m/2 (1 + t)− 1 ≤ C t− ν2 (0,m)−1 . Likewise, recalling also (5.4.44),  t/2 m |∂x z2 (t)|2 ≤ |∂xm Ht (t − θ)|1 |ut (θ)|2 dθ 0



t/2

≤ C

(5.5.59)

(t − θ)−1−m/2 (1 + θ)−(N +2)/4 dθ

0

≤ C

 t −1−m/2 2



t/2

(1 + θ)−(N +2)/4 dθ

0

≤ C t−1−m/2 = C t− ν2 (0,m)−1 , having noted that N4 + 12 > 1. Next, in a similar way, if 1 ≤ m ≤ s we obtain, by (4.6.7) and (5.4.32),  t m |∂x z3 (t)|2 ≤ |∇H(t − θ)|1 |∂xm−1 utt (θ)|2 dθ t/2



≤ C

(5.5.60)

t

(t − θ)− 1/2 (1 + θ)−(2+(m−1)/2) dθ

t/2

≤ C t− ν2 (2,m−1)+1/2 = C t− ν2 (0,m)−1 . If instead m = 0,



t

|z3 (t)|2 ≤ (5.5.61)

|H(t − θ)|1 |utt (θ)|2 dθ

t/2



≤ C

t

(1 + θ)− 2 dθ ≤ C t− ν2 (0,0)−1 .

t/2

In conclusion, from (5.5.58), (5.5.56), (5.5.60) and (5.5.61) we obtain that, for N ≥ 3, (5.5.62)

|∂xm zlin (t)|2 ≤ C t− ν2 (0,m)−1 .

288

5. Asymptotic Behavior

3) To estimate znln (t), we again split  t/2 znln (t) = H(t − θ) ∗ F (u, v) dθ 0 (5.5.63)  t + H(t − θ) ∗ F (u, v) dθ =: z4 (t) + z5 (t) . t/2

Using (4.6.9), we start with  m |∂x z4 (t)|2 ≤

t/2 0

(5.5.64)



|∂xm H(t − θ)|2 |F (u, v)|1 dθ

t/2

≤ C

(t − θ)−(N/4+m/2) |F (u, v)|1 dθ

0

≤ C

 t −(N/4+m/2)



2

t/2

|F (u, v)|1 dθ .

0

Recalling (5.5.55) and (5.4.30), (5.5.65)

|˜ aij (∇u) ∂i ∂j u|1 ≤ C |∇u|2 |∂i ∂j u|2 ≤ C (1 + t)− 3/2 ,

and analogously for the term in v, via (5.4.217). Consequently, we obtain from (5.5.64) that (5.5.66)

|∂xm z4 (t)|2 ≤ C t−(N/4+m/2) ≤ C t− ν2 (0,m)−3/4 .

Next,



t

|∂xm z5 (t)|2 ≤

|∇H(t − θ)|1 |∂xm−1 F (u, v)|2 dθ

t/2



(5.5.67)

≤ C

t

(t − θ)− 1/2 |∂xm−1 F (u, v)|2 dθ .

t/2

Using Leibniz’ formula to expand ∂xm−1 F (u, v) and proceeding as usual, since m ≤ s we arrive at the estimate (5.5.68)

m |∂xm−1 (˜ aij (∇u) ∂i ∂j u)|2 ≤ C |∂xs ∇u|λ2 m |∇u|2−λ , 2

m−1 F (u, v) where λm := N +2m 2s ; an analogous estimate holds for the term of ∂x containing v. Again by (5.4.30), and (5.4.217) with m = s,

(5.5.69)

|∂xm−1 F (u, v)|2 ≤ C (1 + θ)− rm ,

with (5.5.70)

rm := λm

s+1 1 − 2 2

+1=

N + 2m +1. 4

Consequently, from (5.5.67), (5.5.71)

|∂xm z5 (t)|2 ≤ C (1 + t)−(N +2m)/4−1/2 ≤ C (1 + t)− ν2 (0,m)−5/4 .

5.5. The Diffusion Phenomenon

289

Together with (5.5.62) and (5.5.66), (5.5.71) implies (5.5.50) for 1 ≤ m ≤ s (in fact, with the better decay rate ν2 (0, m) + 34 ), via (5.5.56). 4) If instead m = s + 1, we use the interpolation inequality 1/2

1/2

|∂xs+1 (u − v)|2 ≤ C |∂xs+2 (u − v)|2 |∂xs (u − v)|2

(5.5.72)

,

together with the estimates |∂xs+2 u(t)|2 + |∂xs+2 v(t)|2 ≤ C (1 + t)−ν2 (0,s+2) ,

(5.5.73)

which follow from (5.4.212) for u, and, for v, from (5.4.217) of Theorem 5.4.2, with s replaced by s+1. It is at this step that we require the higher regularity assumptions on the coefficients and the initial values; in particular, the assumptions aij ∈ C s+1 (RN ) and u0 + u1 ∈ H s+2 ∩ L1 allow us to invoke the second part of Theorem 5.4.2, and deduce that ∂xs+2 v satisfies (5.5.73). Thus, from (5.5.72), (5.5.73) and (5.5.50) for m = s (with the exponent 34 instead of 38 , as seen at the end of part 3 of this proof), it follows that |∂xs+1 (u(t) − v(t))|2 ≤ C (1 + t)− ν2 (0,s+1)−3/8 .

(5.5.74)

5) To prove (5.5.51), we compute from (5.5.54) that, for 0 ≤ |α| = m ≤ s − 1, ∂xα zt = aij (0) ∂i ∂j ∂xα z − ∂xα utt + ∂xα F (u, v) .

(5.5.75) By (5.5.50), (5.5.76)

|aij (0) ∂i ∂j ∂xα z|2 ≤ C t− ν2 (0,m+2)−3/8 = C t− ν2 (1,m)−3/8 .

Next, recalling (5.4.32), (5.5.77)

|∂xm utt (t)|2 ≤ C (1 + t)− ν2 (2,m) = C (1 + t)− ν2 (1,m)−1 .

Then, we write F (u, v) = (˜ aij (∇u) − a ˜ij (∇v)) ∂i ∂j u (5.5.78)

+a ˜ij (∇v) ∂i ∂j (u − v) =: F1 (u, v) + F2 (u, v) .

We expand both ∂xm F1 (u, v) and ∂xm F2 (u, v) by means of Leibniz’ formula, and estimate each term of the resulting sums by the H¨older and GagliardoNirenberg inequalities, in the usual way. We only consider the most significant estimate, that is, that of the term (5.5.79)

Am := |∂xm (˜ aij (∇u) − a ˜ij (∇v))|2 |∂i ∂j u|∞ ,

coming from F1 (u, v), for which we proceed as follows. At first, by (5.4.30) and (5.4.44), (5.5.80)

|∂i ∂j u|∞ ≤ C |∂xs ∇u|μ2 |∇u|21−μ ≤ C (1 + t)− ρ ,

290

5. Asymptotic Behavior

 +2 with μ = N2s and ρ := 2s − with some abuse of notation,

N 4



1 2

μ+

N 4.

+

Then, by Leibniz’ formula,

∂xm (˜ aij (∇u) − a ˜ij (∇v)) (5.5.81)

=

 m  m r

1 0

r=0

∂xr aij  (λ∇u + (1 − λ)∇v) · ∂xm−r ∇z dλ .

  =: Amr

If r = 0, by (5.5.50) with m = s and, as before, the exponent 3 8,

3 4

instead of

|Am0 |2 ≤ |aij  (λ∇u + (1 − λ)∇v)|∞ |∂xm ∇z|2 ≤ κ(R) |∂xm ∇z|2

(5.5.82)

C (1 + t)− ν2 (0,m+1)−3/4

≤

= C (1 + t)− ν2 (1,m)−1/4 , with R defined at the beginning of the proofs of Theorems 5.4.1 and 5.4.2. If 0 < r ≤ m ≤ s − 1, by Proposition 1.5.8 we can proceed with |Amr |2 ≤ κ(R) (|∂xr ∇u|pr + |∂xr ∇v|pr ) |∂xm−r ∇z|qr ,

(5.5.83)

with pr and qr ≥ 2 such that p1r + inequalities, (5.4.30) and (5.4.44),

=

1 2.

By the Gagliardo-Nirenberg

r |∂xr ∇u|pr ≤ C |∂xs ∇u|λ2 r |∇u|1−λ ≤ C (1 + t)− αr , 2

(5.5.84)

with λr ∈ [0, 1] defined by (5.5.85)

1 qr

αr =

s+1 2

1 pr

λr +

=

r N

4

+

N

+  1 2

1 2

− λr

s N,

and  (1 − λr ) = 2s −

N 4



1 2 1 L )

λr +

Analogously, by (5.4.217) with m = s (recall that u0 + u1 ∈

+

N 4

.

and m = 0,

r |∂xr ∇v|pr ≤ C |∂xs ∇v|λ2 r |∇v|1−λ ≤ C (1 + t)− βr , 2

(5.5.86) with (5.5.87)

βr =

s+1 2

λr + 12 (1 − λr ) =

s 2

λr +

1 2

≤ αr .

Likewise, by (5.5.50), r |∂xm−r ∇z|qr ≤ C |∂xs ∇z|μ2 r |∇z|1−μ ≤ C (1 + t)− γr , 2

(5.5.88)

with μr ∈ [0, 1] defined by (5.5.89) Since λr + μr = (5.5.90)

N s

=

m−r N

+

1 2

− μr

s N,

and

1 s 1 γr = s+1 2 μr + 2 (1 − μr ) = 2 μr + 2 .  m 1  2m+N N + 2 = 2s , we deduce from (5.5.83) that

|Amr |2 ≤ C (1 + t)−(βr +γr ) ≤ C (1 + t)− ν2 (1,m)−3/4 .

Thus, noting that ρ ≥ that (5.5.91)

1 qr

N 4

+ 12 , we deduce from (5.5.82), (5.5.90) and (5.5.80)

Am ≤ C (1 + t)− ν2 (1,m)−1/4−ρ ≤ C (1 + t)− ν2 (1,m)−3/2 .

5.5. The Diffusion Phenomenon

291

From this, it follows that (5.5.92)

|∂xm F (u, v)|2 ≤ C t− ν2 (1,m)−3/2 ;

in turn, (5.5.76), (5.5.77) and (5.5.92) yield (5.5.51). This concludes the proof of Theorem 5.5.2. 

Chapter 6

Singular Convergence

6.1. Introduction 1. In this chapter we consider again the Cauchy problem for the quasi-linear equation (0.0.2), when ε > 0, σ = 1 (for simplicity), and aij = aij (∇u), as in section 5.5; that is (compare to (5.5.6)), (6.1.1)

ε utt + ut − aij (∇u) ∂i ∂j u = f ε ,

with initial data (6.1.2)

u(0) = uε0 ,

ut (0) = uε1 .

Our goal is to study the dependence of the solutions, and their life-span, on the parameter ε, which we consider as a measure of the hyperbolicity of the equation. In particular, when ε is small we wish to study the relations, if any, of the solutions to (6.1.1)+(6.1.2) to those of the limit problem formally corresponding to (6.1.1) when ε = 0; namely, of the Cauchy problem consisting of the parabolic equation (6.1.3)

ut − aij (∇u) ∂i ∂j u = f ,

together with the initial value (6.1.4)

u(0) = u0 .

More precisely, we can either consider (6.1.1)+(6.1.2) as a perturbation, for small ε, of (6.1.3)+(6.1.4), or, if f ε → f and uε0 → u0 when ε → 0 (of course, in a sense to be specified), we can study the possible convergence of the solutions u = uε of (6.1.1)+(6.1.2) to a solution u = u0 of (6.1.3)+(6.1.4). This is an example of a convergence process in which the equation changes type, from hyperbolic to parabolic; in addition, in the limit Cauchy problem (6.1.3)+(6.1.4), there is no initial condition on ut . The corresponding loss of 293

294

6. Singular Convergence

initial condition when the problem changes type as ε → 0 gives rise to a socalled initial layer, which describes a loss of regularity in the time derivatives of the solutions to (6.1.1) at t = 0. Because of this, the convergence process uε → u0 is sometimes referred to as a singular perturbation problem. 2. As mentioned in the preface to this book, singular perturbation results of this type are particularly interesting in applications, where equation (6.1.3) is often considered simpler and easier to study (e.g., from a numerical point of view). In these cases, it becomes important to be able to control, in terms of ε, the error uε − u0 introduced in this approximation process. For example, in Milani [114] we presented some results along these lines for the quasi-linear Maxwell’s equations in ferro-magnetic media, where the parameter ε is a measure of the displacement currents, which are typically small and habitually neglected in simulations. Another application is the model for the heat equation proposed by Cattaneo in [21], where he argued that the conduction of heat in a nonlinear homogeneous medium should be modeled by the perturbed equation (6.1.5)

ε utt + ut − (σ(ux ))x = f ,

which is of type (6.1.1). In (6.1.5), ε is a measure of the thermal relaxation properties of the medium, and σ is a C 2 function satisfying σ  (r) r > 0 for all r ∈ R \ {0}. Typically, the thermal relaxation is quite small, but not negligible, and its explicit presence in (6.1.5) allows us to correct the inconsistencies of instant propagation with infinite speed of the heat flow, that one is forced to deduce from the standard model of the heat equation. A similar model for the heat equation with delay was proposed by Li Ta-Tsien in [92]. To briefly describe this model, we recall that, in the standard model for heat conduction, the basic equations relating the temperature u = u(t, x) and the flux q = q(t, x) are  ut + qx = f , (6.1.6) q + k ux = 0 , the second of which expresses Fourier’s law. If there is a delay τ > 0, the second of (6.1.6) is replaced by (6.1.7)

q(t + τ, x) = −k ux (t, x) .

Formally approximating the left side of (6.1.7) by means of Taylor’s expansion, and neglecting higher order terms, we obtain (6.1.8)

q(t + τ, x) = q(t, x) + τ qt (t, x) = −k ux (t, x) ,

from which (6.1.9)

qx (t, x) + τ qxt (t, x) = −k uxx (t, x) .

6.2. An Example from ODEs

295

Replacing this in the first of (6.1.6), differentiated with respect to t, we obtain the heat equation with delay τ utt + ut − k uxx = f + τ ft ,

(6.1.10)

which is again of type (6.1.1). An analogous model can be obtained for nonlinear heat conduction with delay, that is, when, in the second equation of (6.1.6), k is a function of ux In both cases, one is interested in small values of the delay parameter τ . A similar analysis can also be carried out for more general reaction-diffusion processes with delays. 3. In the remainder of this chapter, we keep the same basic assumptions on the data and the coefficients of equations (6.1.1) and (6.1.3), that we had in the previous chapters. Specifically, we fix an integer s > N2 + 1, and maintain the standing assumptions that, at least, uε0 ∈ H s+1 ,

(6.1.11)

uε1 ∈ H s ,

f ε ∈ L2 (0, T ; H s ) ,

for some T ∈ ]0, +∞] and each ε > 0, as well as that aij ∈ C s (RN ), aij = aji , and the uniformly strong ellipticity condition (3.2.1) holds, with α0 ≥ 1. We will not repeat these assumptions explicitly in the statement of our results; rather, we will only list any additional assumptions that may be needed.

6.2. An Example from ODEs 1. In this section we try to describe the main ideas of the singular perturbation problems we wish to study for (6.1.1), by means of a simple example from ODEs. Given ε ∈ ]0, 1[, consider the two Cauchy problems  ε h  + 2 h  + h = 0 , (6.2.1) h(0) = a , h  (0) = b (loosely corresponding to (6.1.1)+(6.1.2)), and  2p + p = 0 , (6.2.2) p(0) = a (corresponding to (6.1.3)+(6.1.4)). Their explicit solutions are, respectively,    1−R  ε ε hε (t) = Cε exp − 1+R (6.2.3) ε t + Dε exp − ε t , p(t) = a e− t/2 , √ where Rε := 1 − ε, and (6.2.4)

(6.2.5)

Cε :=

1 2Rε

((Rε − 1) a − ε b) ,

Dε :=

1 2Rε

((Rε + 1) a + ε b) .

Noting that, as ε → 0, (6.2.6)

Rε → 1 ,

1−Rε ε

→

1 2

,

Cε → 0 ,

Dε → a ,

296

6. Singular Convergence

we easily see from (6.2.3) that, keeping t ≥ 0 fixed, and letting ε → 0, hε (t) → a e− t/2 = p(t) .

(6.2.7)

That is, the solution to (6.2.1) converges, pointwise in t ≥ 0, to the solution to (6.2.2). The same is true for the first derivatives, as long as t = 0. Indeed, we compute from (6.2.3) that     ε ε (hε )  (t) = Cε − 1+R exp − 1+R ε ε t (6.2.8)     ε ε + Dε − 1−R exp − 1−R ε ε t ; thus, as ε → 0 we see that, for each t > 0 fixed, (6.2.9)

(hε )  (t) → − a2 e− t/2 = p  (t) ,

as claimed. On the other hand, for t = 0, (6.2.10)

hε  (0) = b = p  (0) = − 12 a

(unless, of course, a + 2b = 0; this would correspond to the situation which, in [82], Kreiss calls a “preparation” of the initial data). We summarize these findings in the following diagrams. For the solutions, hε (t)

−→

↓

(6.2.11)

hε (0) = a

p(t) ↓

=

p(0) = a

−→

p  (t)

and for their derivatives, (hε )  (t) (6.2.12)

↓

↓

(hε )  (0) = b

=

p  (0) = − 12 a

where the top horizontal arrows mean convergence for fixed t > 0 as ε → 0, the left vertical arrows mean convergence for fixed ε > 0 as t → 0, and the right vertical arrows mean convergence as t → 0. 2. The last line of (6.2.12) reflects the mentioned initial layer, related to the singular convergence process; that is, the part of the solution hε which is sensible to the loss of the initial condition on (hε )  (0). More precisely, (6.2.3) can be read as (6.2.13)

hε (t) = hεil (t) + hεex (t) ,

6.2. An Example from ODEs

where

297

(6.2.14)

  ε hεil (t) := Cε exp − 1+R ε t ,

(6.2.15)

  ε hεex (t) := Dε exp − 1−R ε t .

Then, for t positive and ε small, hεil ≈ 0 and hεex (t) ≈ p(t). The terms hεil represents the initial layer, in the sense that, as ε → 0 and t > 0,   ε ε (6.2.16) hεil  (t) = − Cε 1+R exp − 1+R ε ε t → 0 exponentially in t, while if t = 0, ε hεil  (0) = − 1+R ε Cε →

(6.2.17) In contrast, note that (6.2.18)

hεex  (t) = − Dε

1−Rε ε

1 2

a+b.

  − t/2 1 ε exp − 1−R = p  (t) ε t → − 2 ae

for all t ≥ 0. 3. In fact, we can say a lot more about the convergence of hε (t) to p(t) and (hε )  (t) to p  (t). Proposition 6.2.1. As ε → 0, (6.2.19)

hε (t) → p(t)

uniformly on

[0, +∞[ ,

and for all τ > 0, (6.2.20)

(hε )  (t) → p  (t)

uniformly on

[τ, +∞[ .

If a + 2b = 0, (6.2.20) also holds for τ = 0. Proof. Without loss of generality, we assume that ε ≤ 34 , which implies that 1 ≤1+ε 0, and lim f (t) < 0, there is a unique t0 > 0, depending on ε, such that f (t) ≥ 0

t→+∞

if and only if 0 ≤ t ≤ t0 . For t ≥ t0 ,

D4 (t) ≤ |a| e−t/2 |f (t)| = − f (t) |a| e−t/2 ≤ 2 |a| e−t/2 .  Given arbitrary η > 0, let Tη := 2 ln 2|a| : then, (6.2.26) implies that η

(6.2.26)

D4 (t) ≤ η

(6.2.27)

if

t ≥ Tη .

For t0 ≤ t ≤ Tη , (6.2.28)

  D4 (t) ≤ 2|a| 2Rε − (1 + Rε ) e− αε Tη =: γ(ε, Tη ) .

+ + Since αε → 0 as ε → 0, and Tη is independent of ε, there exists ε1 ∈ 0, 12 such that for all ε ≤ ε1 , (6.2.29)

D4 (t) ≤ γ(ε, Tη ) ≤ η

if

t0 ≤ t ≤ Tη .

Finally, if 0 ≤ t ≤ t0 , (6.2.30)

D4 (t) ≤ 2|a| ((1 + Rε ) e− αε t − 2 Rε ) ≤ 2|a| (1 − Rε ) ≤ 2|a| ε .

Together with (6.2.24), (6.2.27), and (6.2.29), (6.2.30) implies (6.2.19). 2) To prove (6.2.20), recalling (6.2.8) we compute that   a ε (hε )  (t) − p  (t) = 2R exp − 1+R ε t ε (6.2.31)

+

b (1+Rε ) 2Rε

  ε exp − 1+R ε t +

−

a 2Rε

 −1+R  1 − t/2 ε t + 2 ae . ε

exp

Writing the last line of (6.2.31) as  − t/2   a ε (6.2.32) − exp −1+R t + 2 e ε

b (Rε −1) 2Rε

a Rε −1 2 Rε

exp

exp

 −1+R  ε t ε

 −1+R

ε

ε

 t ,

6.2. An Example from ODEs

299

and recalling (6.2.21) and (6.2.22), (6.2.33)

  ε |(hε )  (t) − p  (t)| ≤ |a| exp − 1+R ε t   ε + 2 |b| exp − 1+R t + |b| ε ε +

1 2

  ε −1| ε |a| |RR exp − 1+R ε t ε

+

1 2

     − t/2  ε |a|  exp −1+R t − e  ε

=: D5 (t) + D6 (t) + D7 (t) + D8 (t) + D9 (t) . For arbitrary τ > 0 and all t ≥ τ , (6.2.34) D5 (t) + D6 (t) + D7 (t) + D8 (t) ≤ (|a| + 2 |b|) e− τ /ε + |b| ε + |a| ε . + + Given arbitrary η > 0, there exists then ε2 ∈ 0, 12 (dependent of τ ), such that, for all ε ≤ ε2 , (6.2.35) D5 (t) + D6 (t) + D7 (t) + D8 (t) ≤ η

if

t≥τ.

Next, recalling (6.2.33) and the definition of αε ,   (6.2.36) D9 (t) = 12 |a| e− t/2 e− αε t − 1 = 12 |a| e− t/2 (1 − e− αε t ) .  Let Tη := 2 ln |a| 2η : then, (6.2.37)

D9 (t) ≤

1 2

|a| e− t/2 ≤ η

if

t ≥ Tη .

If instead 0 ≤ t ≤ Tη , (6.2.38)

D9 (t) ≤

1 2

|a| (1 − e− αε Tη ) ;

and since Tη is independent of ε, recalling that αε → 0 as ε → 0, we deduce that there exists ε3 ≤ ε2 such that, for all ε ≤ ε3 , (6.2.39)

D9 (t) ≤ η

if

0 ≤ t ≤ Tη .

In conclusion, (6.2.20) follows from (6.2.33), (6.2.35), (6.2.38), and (6.2.39). 3) If a = −2b, the first two terms at the right side of (6.2.31) can be written as   ε (6.2.40) γ(t) = 2Rb ε (−1 + Rε ) exp − 1+R t ; ε thus, recalling (6.2.22), the estimate of D5 (t) and D6 (t) can be replaced by (6.2.41)

|γ(t)| ≤ |b| ε .

300

6. Singular Convergence

Since the estimates of D7 (t) and D8 (t) are independent of τ , we conclude that (6.2.20) also holds for τ = 0, if a + 2b = 0. This ends the proof of Proposition 6.2.1.  4. When a + 2b = 0, we can still recover some kind of uniform convergence of the derivatives on all of [0, +∞[, by introducing a correction term to offset the loss of the initial condition on the derivatives, described by the condition b = − 12 a. More precisely, we claim: Proposition 6.2.2. Assume that a + 2b = 0, and, for ε > 0, define ε (a + 2b)(1 − e− 2t/ε ) ,

(6.2.42)

θε (t) :=

(6.2.43)

Rε (t) := hε (t) − p(t) − θε (t) .

1 4

Then, as ε → 0, both Rε (t) → 0 and (Rε )  (t) → 0 uniformly on [0, +∞[. Proof. The uniform decay of Rε (t) follows immediately from (6.2.19) and (6.2.42). As for the decay of (Rε )  (t), keeping the same notations of the proof of Proposition 6.2.1, we recall that D7 (t) and D8 (t) vanish uniformly for t ≥ 0 as ε → 0. Hence, recalling (6.2.31) and the definition of θε , we need only to show that the term (6.2.44) γ1 (t) :=

a 2Rε

  ε exp − 1+R ε t +

b(1+Rε ) 2Rε

  ε exp − 1+R ε t −

1 2

(a + 2b) e− 2t/ε

vanishes uniformly in [0, +∞[. Recalling (6.2.21),    1−R    ε |γ1 (t)| ≤ |a|  exp ε t − Rε     1−R    ε +|b| (1 + Rε ) exp (6.2.45) ε t − 2 Rε  =: D9 (t) + D10 (t) . By (6.2.22), for 0 ≤ t ≤ τ we can proceed with D9 (t) ≤ |a| (eτ − 1 + 1 − Rε ) ≤ |a| (eτ − 1 + ε) . * + Given η > 0, there exist τ1 > 0 and ε4 ∈ 0, 12 , such that, for all ε ≤ ε4 ,

(6.2.46)

(6.2.47)

eτ 1 − 1 ≤

η , 2(|a| + |b|)

ε≤

η ; 2(|a| + |b|)

hence, we deduce from (6.2.46) that (6.2.48)

D9 (t) ≤ η

if

0 ≤ t ≤ τ1 .

6.3. Uniformly Local and Global Existence

301

Likewise, if ε ≤ ε4 and 0 ≤ t ≤ τ1 , D10 (t) ≤ |b| ((1 + Rε ) eτ1 − 2 Rε ) = |b| ((1 + Rε )(eτ1 − 1) + 1 − Rε ) (6.2.49)

≤ |b| (2(eτ1 − 1) + ε) ≤ |b|

η η + |b| ≤ |a| + |b| 2(|a| + |b|)

3 2

η.

Together with (6.2.48) and (6.2.20) with τ replaced by τ1 , this allows us to conclude the proof of Proposition 6.2.2.  Remark. If a+2b = 0, (6.2.43) implies that Rε (t) = hε (t)−p(t), so that the conclusions of Proposition 6.2.2 are a restatement of those of Proposition 6.2.1. To motivate the choice (6.2.42) of the corrector θε , we note that, if we pose hε (t) = p(t) + θε (t), from (6.2.1) and (6.2.2) we deduce 0 = ε(hε ) + 2 (hε )  + hε (6.2.50)

= ε(p  + (θε )  ) + 2 (p  + (θε )  ) + (p + θε ) = ε p  + (ε (θε )  + 2 (θε )  ) + θε .

Neglecting the term ε p  , which is O(ε) uniformly in t ≥ 0, we are then led to look for a function t → θε (t), such that also θε (t) = O(ε), uniformly in t, and satisfies the ODE (6.2.51)

ε (θε )  + 2(θε )  = 0 .

To this ODE, we attach the initial conditions (6.2.52) (6.2.53)

θε (0) = hε (0) − p(0) = 0 , (θε )  (0) = (hε )  (0) − p  (0) = b +

a 2

.

It is then immediate to verify that the solution to the Cauchy problem (6.2.51)+(6.2.52)+(6.2.53) is precisely the function θε defined in (6.2.42); clearly, this function is such that, as desired, θε (t) = O(ε), uniformly in t. 

6.3. Uniformly Local and Global Existence In this section we briefly adapt the results of Chapters 3 and 4, to prove local and global existence results for the Cauchy problem (6.1.1)+(6.1.2). In particular, we are interested in how the local time of existence determined in Theorem 3.2.1 and the smallness of the data required in the global existence Theorem 4.5.3 depend on ε. When such quantities can be determined independently of ε, we call the corresponding result, respectively, uniformly local or global.

302

6. Singular Convergence

1. We start with a uniformly local existence result, analogous to that of Theorem 3.2.1. Theorem 6.3.1. Let the assumptions on the coefficients and the data stated in part 3 of section 6.1 hold. Then, for all ε > 0 there is τε ∈ ]0, T ], and a unique uε ∈ Ys (τε ), solution of (6.1.1)+(6.1.2). If in addition there is M > 0 such that

1/2  T ε ε 2 ε 2 ε 2 (6.3.1) I0 := ε u1 s + u0 s+1 + 4 f s dt ≤M 0

for all ε > 0, then (6.3.2)

inf τε =: τ > 0 ,

ε>0

and each uε satisfies, in [0, τ ], the estimate (6.3.3)

Esε (t)

:=

ε uεt (t)2s

+ u

ε

(t)2s+1



t

+ 0

uεt 2s dθ ≤ M12 ,

with M1 depending on M and τ , but independent of ε. In general, M1 ≥ M . Proof. Arguing as in the beginning of the proof of Theorem 4.5.2, we can adapt the proof of the local existence Theorem 3.2.1 to show the local existence claim, recalling that, as remarked after the statement of Theorem 3.2.1, the weaker assumption on f ε in (6.1.11) is sufficient to determine a solution of (6.1.1) in Ys (τε ). In addition, recalling identity (4.5.139) and estimating its right side as in (4.5.149), (4.5.151) and (part of) (4.5.153), we deduce that, on [0, τε ], uε satisfies the estimate  t

ε ε 2 ε ε (6.3.4) Es (t) ≤ (I0 ) exp C (ut s + k(∇u s )) dθ , 0

for suitable k ∈ K. If we now acted as in the proof of Theorem 3.2.1, we would proceed from (6.3.4) with   (6.3.5)

Esε (t) ≤ (I0ε )2 exp C sup (uεt (θ)s + k(∇uε (θ)s )) t

.

0≤θ≤t

Per se, estimate (6.3.5) is not sufficient to allow us to show (6.3.2) and (6.3.3). To see this, consider, e.g., an interval [0, tε ] ⊆ [0, τε ] where (6.3.6)

Esε (t) ≤ 4 Esε (0) ≤ 4 (I0ε )2 .

Then, from (6.3.5) we would deduce that, for t ∈ [0, tε ],   (6.3.7) Esε (t) ≤ (I0ε )2 exp C √1ε 2 I0ε + k(2 I0ε ) t . At this point, even if the uniform bound on I0ε assumed in (6.3.1) allows us to proceed from (6.3.7) with (6.3.8)

Esε (t) ≤ M 2 eC β(ε) t ,

6.3. Uniformly Local and Global Existence

303

where β(ε) := √1ε 2 M + k(2M ), the fact that β(ε) → +∞ as ε → 0 prevents us from obtaining (6.3.2). However, in the present situation, the ε-uniform bound on uεt in L2 (0, τε ; H s ) implied by the definition of Esε , does allow us to proceed from (6.3.4) with   

1/2 t (6.3.9) Esε (t) ≤ (I0ε )2 exp C t uεt 2s dθ + k(∇uε s ) t , 0

instead of (6.3.5). In turn, (6.3.9) allows us to replace (6.3.8) with   √ √ √ (6.3.10) Esε (t) ≤ M 2 exp C M + k(M ) t t ≤ M 2 eh(M,T ) t , the right side of which is independent of ε. Thus, local solutions uε ∈ Ys (τε ) can be extended to a common interval [0, τ ], with τ depending only on M , and uε ∈ Ys (τ ). We explicitly note that the possibility of obtaining (6.3.4), which then allows us to deduce the ε-uniform estimate (6.3.10) via (6.3.9), depends in a crucial way on the fact that the coefficients aij in (6.1.1) depend only on ∇u, and not on ut . When 0 < ε ≤ 1, this is essential, because, in the a priori estimates that lead to (6.3.9), we could control, independently √ of ε, the norms of uεt in L2 (0, τε ; H s ), and of ε uεt (0) in H s . If the aij ’s depended also on ut , we would also need to control the norm of uεtt at least in L2 (0, τε ; H s−1 ) (as we see, e.g., from the terms R0,α of (4.4.2)), or of ε uεtt (0) in H s−1 . However, uεtt can only be computed from equation (6.1.1) itself, and, unless we assume additional conditions of the data, this yields at most √ an ε-independent control of ε uεtt L2 (0,τε ;H s−1 ) or ε uεtt (0)s−1 . Alternatively, we can argue as in the proof of Theorem 4.5.2. Let Tcε denote the life-span of uε . If (6.3.2) did not hold, then, as ε → 0, Tcε → 0 at least along a sequence (εn )n≥1 . It follows that, for each n ≥ 1, there is tn ∈ ]τεn , Tcεn [, such that for all t ∈ [0, tn ], (6.3.11)

Esεn (t) ≤ 4 M 2 = Esεn (tn ) .

But then, (6.3.10) for ε = εn and t = tn would imply that (6.3.12)

4 M 2 ≤ M 2 eh(M,T )

√ tn

.

Since tn is supposed√ to vanish as n → +∞, it is possible to choose n large enough that eh(M,T ) tn ≤ 2: then, for such n, (6.3.12) yields a contradiction. Consequently, (6.3.2) holds. Finally, (6.3.3) follows from (6.3.10), letting t = τ at its right side. We remark that (6.3.2) means that the ε-uniform bound of (6.3.1) is sufficient to imply a uniformly local existence result for problem (6.1.1)+(6.1.2).  2. We now present a uniformly global existence result, analogous to the global existence result of Theorem 4.5.3.

304

6. Singular Convergence

Theorem 6.3.2. Let N ≥ 3, and assume that, as in (4.5.181), (6.3.13)

f ε ∈ Cb ([0, +∞[; H s ∩ Lq ) ,

1 ≤ q < q¯ :=

2N N +2

.

There exists δ0 > 0, such that, if √ (6.3.14) ε uε1 s + uε0 s+1 + f ε Cb ([0,+∞[;H s ∩Lq ) ≤ δ02 for all ε > 0, then the Cauchy problem (6.1.1)+(6.1.2) has a unique solution uε ∈ C([0, +∞[; H s+1 ), which is such that (6.3.15)

Duε ∈ Cb ([0, +∞[; H s ) ∩ Cb1 ([0, +∞[; H s−1 ) .

If N ≥ 5, uε ∈ Cb ([0, +∞[; L2 ) as well, and uε ∈ Zs,b (∞). Remark. An alternative uniformly global existence result would also hold, analogous to that of Theorem 4.5.2. As in Theorems 4.5.2 and 4.5.3, the value of δ0 in (6.3.14) depends on the ellipticity constant α0 , and on the coefficient σ of the dissipation term ut in (6.1.1); here, α0 ≥ 1 and σ = 1.  Proof. The proof of Theorem 6.3.2 follows the same steps of the proof of Theorem 4.5.3, with a natural rescaling of the norms, due to the presence of the parameter ε. More precisely, in analogy to Proposition 4.5.7 we assume that (6.1.1)+(6.1.2) has a local solution uε ∈ Ys (T ), for some T ∈ ]0, Tcε [ independent of ε (which we can do, by of the second claim of Theorem 6.3.1), and proceed to establish the a priori estimates √ (6.3.16) ε uεt s + ∇uε s ≤ δ0 , ε uεtt s−1 ≤ K1 δ0 , which play the role of (4.5.184). To this end, instead of the norm (Nsu )1/2 and the quadratic form Psu of (4.5.141) and (4.5.183), we introduce the ε-weighted norm (Nsε )1/2 and the quadratic form Psε , defined by (6.3.17) ⎧ ⎨ ε uεt 2s + εuεt , uε s + Qs (∇uε ) , if 0 < ε ≤ 1 , ε ε Ps (u ) := ⎩ ε uε 2 + uε , uε  + Q (∇uε ) , if ε ≥ 1 , s s t s t and (6.3.18)

Nsε (uε ) :=

⎧ ⎨ Psε (uε ) + 12 uε 2s ,

if

00 of solutions to the Cauchy problem (6.1.1)+(6.1.2), when ε → 0. Some results on this process are given in Theorems 6.4.1 and 6.4.2 below. As we have mentioned, the loss of the initial condition on uεt introduces a degree of singularity, at t = 0, in the convergence process; Theorem 6.4.3 gives some results on the corresponding initial layer. Singular convergence results of this type are well known in the linear case; for example, see Zlamal [170, 171], and Lions [100, ch. VI]. In alternative, one can solve each of the Cauchy problems (6.1.1)+(6.1.2) and (6.1.3)+(6.1.4) separately, and then consider the question of comparing the corresponding solutions, in the sense of being able to estimate their difference, in a suitable norm, in terms of ε. For example, this would provide an estimate of the error one introduces, if, instead of the full equation (6.1.1), one considers only the simpler equation (6.1.3). We present a result of this type in Theorem 6.4.4 below. 6.4.1. Singular Convergence. In this section we present two results on the convergence of sequences of solutions uε of (6.1.1)+(6.1.2) to a solution of the parabolic Cauchy problem (6.1.3)+(6.1.4). We assume that (6.3.1) holds, and, without loss of generality, that ε ∈ ]0, 1]. Then, there are a vanishing sequence (εn )n≥1 , and corresponding functions u0 ∈ H s+1 , u1 ∈ H s , and f ∈ L2 (0, T ; H s ), such that, as εn → 0, (6.4.1) (6.4.2) (6.4.3)

√

uε0n → u0 εn uε1n → εn f

u1

→f

in

H s+1

in

s

in

H

weak , weak ,

2

L (0, T ; H s )

weak .

For simplicity, in the sequel we omit the dependence of εn on n, and simply say “as ε → 0”, to mean that ε → 0 along the sequence (εn )n≥0 . Because of (6.3.3), there also is u0 ∈ L∞ (0, τ ; H s+1 ), with u0t ∈ L2 (0, τ ; H s ), such that, as ε → 0, (6.4.4)

uε → u0

in

L∞ (0, τ ; H s+1 )

weak∗ ,

(6.4.5)

uεt → u0t

in

L2 (0, τ ; H s ) weak .

306

6. Singular Convergence

It is then natural to expect that the limit function u0 should be a solution, in some sense, of the limit Cauchy problem (6.1.3)+(6.1.4). 1. As a first result in this direction, we claim: Theorem 6.4.1. Let N ≥ 3, s ∈ N, with s > N2 + 1, and assume that aij ∈ C s (RN ). Let u0 ∈ H s+1 , f ∈ L2 (0, T ; H s ), and u0 ∈ L∞ (0, τ ; H s+1 ) be the weak limits appearing in (6.4.1), (6.4.3) and (6.4.4). 1) If uε0 → u0 in L2 and f ε → f in L2 (0, T ; L2 ), then u0 ∈ C([0, τ ]; H s+1 ), and solves the parabolic Cauchy problem (6.1.3)+(6.1.4), where (6.1.3) holds at least in H s , for a.a. in t. In addition, uε → u0 in C([0, τ ]; H s+1−η ), η ∈ ]0, s + 1]. √ 2) If also ε uε1 → 0 in L2 (hence, u1 = 0 in (6.4.2)), then uεt → u0t in L2 (0, τ ; H s−η ), η ∈ ]0, s]. 3) If also the set (ftε )0 N2 + 1, and assume that aij ∈ C s+1 (RN ), u0 ∈ H s+1 , u1 , and f ∈ C([0, T ]; H s ) ∩ C 1 ([0, T ]; H s−1 ), for some T > 0. There is ε0 ∈ ]0, 1], depending on u0 , u1 , f and T , such that, if ε ∈ ]0, ε0 ], then the Cauchy problem (6.5.1) admits a unique solution u = uε ∈ Zs (T ). Proof. 1) Noting that the data of (6.5.1) satisfy the requirements of the uniformly local existence Theorem 6.3.1, there are τ ∈ ]0, T ], independent of ε, and a unique uε ∈ Ys (τ ), solution of (6.5.1). Since f ∈ C([0, T ]; H s−1 ), uε ∈ Zs (τ ) as well. We wish to show that we can extend uε to the whole interval [0, T ], with uε ∈ Zs (T ). To achieve this, we again follow Milani [115] (see also Milani [118]), and use a perturbation technique, based on the second global existence result for the parabolic Cauchy problem  vt − aij (∇v) ∂i ∂j v = g , (6.5.2) v(0) = v0 , presented in Theorem 4.6.4 (which explains our more restrictive assumptions on f ), and the comparison estimates of Theorem 6.4.4. Roughly speaking, this method consists in writing the local solution of (6.5.1) as uε = v+z, and show that, if ε is small, we can determine time-independent bounds for a suitable norm of z. Since v is defined on all of [0, T ], these bounds translate into time-independent bounds for uε , which can therefore be extended to all of [0, T ]. 2) To implement this method, we consider (6.5.2) with g = f and v0 = u0 , and resort to Theorem 4.6.4 to determine a solution v ∈ Ps (T ). Since aij ∈ C s+1 (RN ), we know that v ∈ P˜s (T ). As in the proof of Theorem 6.4.4, we consider (v δ )δ>0 ⊂ D([0, T ]; H s+2 ) such that (6.4.118) and (6.4.119) hold. Then, each difference z := uε − v δ is at least in Ys (τ ), and has the same life-span Tε of uε (see section 4.2). We now note that the fact that the coefficients aij of (6.1.1) depend only on ∇uε and not on uεt allows us to deduce, when retracing the arguments of the proof of the global existence results of section 4.4, that in order to guarantee almost global existence for (6.5.1), it is sufficient to establish a time-independent bound on ∇uε (t)s (as opposed to a bound on Duε (t)s ). Thus, we are led to consider the semi-norm w → max0≤t≤T [w(t)]s on Ys (T ), defined by  t 2 2 (6.5.3) [w(t)]s := ∇w(t)s + wt 2s dθ , w ∈ Ys (T ) , t ∈ [0, T ] . 0

Arguing by contradiction, assume now that Tε ≤ T , for ε arbitrarily small. Recalling the characterization (4.2.5) of the life-span, then, (6.5.4)

lim [z(t)]2s = +∞ .

t→Tε−

328

6. Singular Convergence

On the other hand, by (6.4.119), (6.5.5)

[z(0)]2s = ∇u0 − ∇v δ (0)2s = ∇v0 − ∇v δ (0)2s ≤ ω(δ) ;

thus, taking δ0 ∈ ]0, 1] such that ω(δ) ≤ such that, for all t ∈ [0, t1 ],

1 2

if δ ∈ ]0, δ0 ], we can find t1 ∈ ]0, Tε [

0 ≤ [z(t)]2s ≤ [z(t1 )]2s = 1 .

(6.5.6) Then, for t ∈ [0, t1 ], (6.5.7)

[uε (t)]s ≤ [v δ (t)]s + [z(t)]s ≤ V + 1 ,

where, now, V := 1 + vP˜s (T ) , and we can carry out the same estimates of the proof of Theorem 6.4.4, on the interval [0, t1 ], with M1 replaced by V +1 and M = M (V ) determined accordingly, and f ε = f = g, uε0 = u0 = v0 , and uε1 = u1 . In particular, we arrive at estimate (6.4.117), which now reads simply (6.5.8)

[uε (t) − v δ (t)]2s = [z(t)]2s ≤ C∗ ε u1 2s + α2 ,

+  + with C∗ = h(V, M (V ), V +1, T ). We now take, e.g., α = 12 , and ε ∈ 0, ε 12 (as per Theorem 6.4.4), such that C∗ ε u1 2s ≤ 14 . Then, (6.5.8) for t = t1 yields [z(t1 )]2s ≤ 12 , which is a contradiction. This ends the proof of Theorem 6.5.1.  Remarks. 1) The explicit dependence of ε0 on T , given by (6.4.145) with α = 12 , that is, 2  1    , (6.5.9) ε0 (T ) = min 1, 8 ψ(V, M (V ), M1 , T ) ζ δ 12 is such that (6.5.10)

lim inf ε0 (T ) = 0 , T →+∞

because ψ(V, M (V ), M1 , T ) → +∞ as T → +∞. Thus, as far as we can tell, the question of the asymptotic behavior of uε , as t → +∞, remains open. 2) When ε ≤ ε0 , estimate (6.3.3) of Theorem 6.3.1, as well as Theorems 6.4.1 and 6.4.2 on the singular convergence uε → v, obviously hold, with τ replaced by T . 3) An alternative, direct proof of the almost global existence result of Theorem 6.5.1, based on the Moser-Nash iteration technique, can be found in Michael [113].  2. We next consider the case of ε large. Theorem 6.5.2. Let N ≥ 3, s > N2 + 1, aij ∈ C s (RN ), and (6.1.11) hold. There is ε1 ≥ 1, depending on u0 , u1 , f and T , such that, if ε ≥ ε1 , then the Cauchy problem (6.5.1) admits a unique solution u = uε ∈ Ys (T ).

6.5. Almost Global Existence

329

Proof. We assume for simplicity that u1 = 0, the proof when u1 = 0 being analogous. By the local existence Theorem 6.3.1, there is τε ∈ ]0, T ], and a unique uε ∈ Ys (τε ), solution of (6.5.1). However, the uniform lower bound (6.3.2) need not hold, because, instead of condition (6.3.1), in accord with (6.1.11) we only assume that there is Ms ≥ 1 such that, for all ε ≥ 1,  T (6.5.11) ε u1 2s + Qs (∇u0 ) + f 2s dt ≤ ε Ms2 , 0

where Qs (∇u) is defined as in (4.5.138), but with the coefficients auij of (3.2.49) replaced by auij = aij (∇u). In analogy with (6.3.3), with some abuse of notation, given w ∈ Ys (T ) and t ∈ [0, T ], we now set  t ε 2 (6.5.12) Es (w, t) := ε wt (t)s + Qs (∇w(t)) + 2 wt 2s dθ . 0

Let s0 :=

 N2 

(6.5.13)

+ 2, and define J1 := u0 s0 + 2 T u1 s0 ,

J2 := u1 s0 h(J1 ) ,

where h is the function appearing in (6.5.23) below, and (6.5.14)

J3 := e(1+J2 ) T .

These quantities depend only on T and the initial data u0 and u1 ; in particular, they are independent of t and ε. Finally, let Tε denote again the life-span of uε . Theorem 6.5.2 is a consequence of the following a priori estimate: Proposition 6.5.1. Let Ms be as in (6.5.11), and J3 as in (6.5.14). There is ε1 ≥ 1 with the property that, for all ε ≥ ε1 , if τ ∈ ]0, T ] is such that the Cauchy problem (6.5.1) has a solution uε ∈ Ys (τ ), then for and all t ∈ [0, τ ], (6.5.15)

Esε (uε , t) ≤ ε Ms2 J32 .

As a consequence, if ε ≥ ε1 , uε can be extended to all of [0, T ], with uε ∈ Ys (T ). Proof. Arguing again by contradiction, suppose that for all ε1 ≥ 1 there are ε ≥ ε1 and τ ∈ ]0, T ] such that (6.5.1) has a solution uε ∈ Ys (τ ), but there is t2 ∈ ]0, τ ] such that (6.5.16)

Esε (uε , t2 ) > ε Ms2 J32 .

By (6.5.11), (6.5.17)

Esε (uε , 0) = ε u1 2s + Qs (∇u0 ) ≤ ε Ms2 < ε Ms2 J32 ;

therefore, if Esε (uε , t2 ) > 4 ε Ms2 J32 , there is t1 ∈ ]0, t2 [ such that for all t ∈ [0, t1 ], (6.5.18)

Esε (uε , t) ≤ 4 ε Ms2 J32 = Esε (uε , t1 ) ;

330

6. Singular Convergence

we set t0 := t1 . If Esε (uε , t2 ) ≤ 4 ε Ms2 J32 , again by (6.5.17) there is α ∈ ]0, 3] such that for all t ∈ [0, t2 ], (6.5.19)

Esε (uε , t) ≤ (1 + α) ε Ms2 J32 = Esε (uε , t2 ) ;

we set t0 := t2 . Since α ≤ 3, in either case we conclude that for all t ∈ [0, t0 ], (6.5.20)

Esε (uε , t) ≤ γ ε Ms2 J32 = Esε (uε , t0 ) ,

γ ∈ ]1, 4] .

1) We assume first that s ≥ s0 + 1, and proceed with the following (formal) a priori estimates. Multiplying the equation of (6.5.1) in H s by 2 uεt we obtain that for all t ∈ [0, t0 ],  d ε ε (6.5.21) (R0,α + R1,α + RG,1,α ) , Es (u , t) = 2f, uεt s + dt |α|≤s

where R0,α , R1,α and RG,1,α are defined as in (3.3.89), (3.3.90) and (3.3.91), with u and Du replaced by uε and ∇uε . At first, recalling that ε ≥ 1, (6.5.22)

2f, uεt s ≤ f 2s + uεt 2s ≤ f 2s + ε uεt 2s .

Next, as in (4.4.4), (6.5.23)

R0,α ≤ h(|∇uε |∞ ) |∇uεt |∞ ∇uε 2s .

By (6.5.20), ε uεt (θ)2s ≤ 4 ε Ms2 J32 for all θ ∈ [0, t0 ]; thus,

(6.5.24)

|∇uε (t)|∞ ≤ ∇uε (t)s0 −1 ≤ uε (t)s0  t ε ≤ u (0)s0 + uεt s dθ 0

≤ u0 s0 + 2 Ms J3 T ; the same estimate also shows that (6.5.25)

uε (t)s ≤ u0 s + 2 Ms J3 T =: J4 .

Likewise, using the equation of (6.5.1) to express uεtt , and since s0 ≤ s − 1, |∇uεt (t)|∞ ≤ ∇uεt (t)s0 −1 ≤ uεt (t)s0  t ε ≤ ut (0)s0 + uεtt s−1 dθ 0  1 t ≤ u1 s0 + (6.5.26) f − uεt + aij (∇uε ) ∂i ∂j uε s−1 dθ ε 0

1/2  T  1 1 t ε ≤ u1 s0 + f 2s−1 dθ + ut s−1 dθ T ε ε 0 0  1 t + h(|∇uε |∞ ) ∇uε s−1 ∇uε s dθ . ε 0

6.5. Almost Global Existence

331

Thus, by (6.5.25), and (6.5.20) again,

(6.5.27)

|∇uεt (t)|∞

≤ u1 s0 +

1 ε

1 + ε

 T

T

1/2 f 2s−1 dθ

0

√ 2 Ms J3 T + 1ε h(J4 ) J4 (2 ε Ms J3 ) T .

Since J3 , J4 and Ms are independent of ε, it is possible to find ε2 ≥ 1 such that, if ε ≥ ε2 , (6.5.28)

|∇uεt (t)|∞ ≤ 2 u1 s0

for all t ∈ [0, t0 ]. Then, if ε ≥ ε2 , (6.5.24) can be improved into  t |∇uε (t)|∞ ≤ u0 s0 + |∇uεt |∞ dθ (6.5.29) 0 ≤ u0 s0 + 2 u1 s0 T = J1 . Inserting this and (6.5.28) into (6.5.23), and recalling the second of (6.5.13), we finally deduce that (6.5.30)

R0,α ≤ h(J1 ) (2 u1 s0 ) ∇uε 2s = 2 J2 ∇uε 2s .

Next, proceeding as in (4.4.5) for R1,α , and as in (4.4.7) for RG,1,α , we obtain that R1,α + RG,1,α ≤ h(|∇uε |∞ ) |∂x2 uε |∞ ∇uε s uεt s . + * +2 By the Gagliardo-Nirenberg inequality, with λ := N2s ∈ 1s , 1 , and by (6.5.20) and (6.5.25), (6.5.31)

(6.5.32)

|∂x2 uε (t)|∞ ≤ C |∂xs ∇uε (t)|λ2 |∇uε (t)|21−λ √ ≤ C (2 ε Ms J3 )λ J41−λ =: ελ/2 J5 .

Inserting this into (6.5.31), and recalling (6.5.29), we obtain that (6.5.33)

R1,α + RG,1,α ≤ h(J1 ) J5 ελ/2 ∇uε s uεt s   ≤ h(J1 ) J5 ε−(1−λ)/2 ε uεt 2s + ∇uε 2s .

Since 1 − λ > 0, and J5 is independent of ε, it is possible to find ε1 ≥ ε2 such that, if ε ≥ ε1 , then for all t ∈ [0, t0 ], (6.5.34)

R1,α + RG,1,α ≤ ε uεt 2s + ∇uε 2s .

In conclusion, putting (6.5.22), (6.5.30) and (6.5.34) into (6.5.21), we deduce that, for ε ≥ ε1 , (6.5.35)

  d ε ε Es (u , t) ≤ f 2s + (2 J2 + 2) ε uεt 2s + ∇uε 2s . dt

332

6. Singular Convergence

From this, by Gronwall’s inequality, recalling (6.5.11), (6.5.14), and (3.2.14), we conclude that, for all t ∈ [0, t0 ], (6.5.36)

Esε (uε , t)



≤

Esε (uε , 0) +

≤

ε Ms2 e2(1+J2 ) T

T

f 2s dθ

e2(1+J2 ) t

0

= ε Ms2 J32 .

For t = t0 , (6.5.36) contradicts (6.5.20). Thus, Proposition 6.5.1 holds, if s ≥ s0 + 1. 2) Assume now that s = s0 . For δ > 0, consider the Friedrichs’ mollifiers ρδ in the space variables, and set, as per (1.4.7), uε,δ (t) := ρδ ∗ uε (t). Then, in (6.5.23), we replace estimate (6.5.26) of |∇uεt (t)|∞ as follows. At first, for δ to be determined, (6.5.37)

|∇uεt (t)|∞

≤

ε,δ |∇uεt (t) − ∇uε,δ t (t)|∞ + |∇ut (t)|∞

=: A1 (t) + A2 (t) . As in (1.5.36), for x ∈ RN ,

(6.5.38)

|∇uεt (t, x) − ∇uε,δ t (t, x)|  ≤ ρ(z) |∇uεt (t, x − δz) − ∇uεt (t, x)| dz ; |z|≤1

since ∇uεt ∈ C([0, t0 ]; H s−1 ), acting as in (1.7.75) of Proposition 1.7.7, and recalling (6.5.20), we obtain that, for suitable α ∈ ]0, 1[ and all z ∈ RN with |z| ≤ 1, (6.5.39)

|∇uεt (t, x − δz) − ∇uεt (t, x)| ≤ C ∇uεt (t)s−1 | δz|α ≤ C (2 Ms J3 ) δ α .

Inserting this into (6.5.38), we deduce that, for all δ > 0, A1 (t) ≤ 2 C Ms J3 δ α .

(6.5.40)

To estimate A2 , we note that (6.5.41)

ε,δ δ δ ε ε ε,δ ε uε,δ . tt = f + ρ ∗ (aij (∇u ) ∂i ∂j u ) − ut =: g

Thus, as in (6.5.26), (6.5.42)

ε,δ ε,δ |∇uε,δ t (t)|∞ ≤ ut (t)s0 ≤ ut (0)s0 +

1 ε



t 0

g ε,δ s0 dθ .

6.5. Almost Global Existence

333

Now, uεt (0) = u1 ∈ H s = H s0 , and f (t), uεt (t) ∈ H s = H s0 for almost all (respectively, all) t ∈ [0, t0 ]. Thus, by (1.5.34),  t 1 uε,δ (0) + f δ − uε,δ s0 t t s0 dθ ε 0 (6.5.43)  t 1 ≤ u1 s0 + ε f − uεt s0 dt . 0

On the other hand, aij by (1.5.39), (6.5.44)

(∇uε ) ∂

i ∂j

uε

∈ C([0, t0 ]; H s0 −1 ) only; thus, in general,

ρδ ∗ (aij (∇uε ) ∂i ∂j uε )s0 ≤ C

1 δ

aij (∇uε ) ∂i ∂j uε s0 −1 .

In conclusion, from (6.5.42), (6.5.43) and (6.5.44), recalling that ε ≥ 1 it follows that, as in (6.5.27), if δ ≤ 1,  T

1/2 2 1 A2 (t) ≤ u1 s0 + ε T f s0 dt + 2ε Ms J3 T 0

+

(6.5.45)

C εδ

√ h(J4 ) J4 (2 ε Ms J3 ) T

≤ u1 s0 +

√1 εδ

J6 ,

for suitable J6 depending on J3 , J4 , T and Ms , but independent of ε. Thus, from (6.5.37), (6.5.40) and (6.5.45) we obtain (6.5.46)

|∇uεt (t)|∞ ≤ 2 C Ms J3 δ α + u1 s0 +

√1 εδ

J6 .

At this point, we first fix δ = δ0 ∈ ]0, 1] such that 2 C Ms J3 δ0α ≤ 12 u1 s0 , and then determine ε1 ≥ 1, such that √ε1δ J6 ≤ 12 u1 s0 if ε ≥ ε1 . With 0 these choices, (6.5.46) implies that ∇uεt satisfies the same estimate (6.5.28), and the rest of the proof can proceed in the same way. 3) Since τ ∈ ]0, T ] is arbitrary, (6.5.15) is an a priori bound on Esε (uε , ·) on [0, T ]; hence, Tε > T , as claimed. This concludes the proof of Proposition 6.5.1 and, therefore, that of Theorem 6.5.2. Here too, we remark that, when ε ≥ ε1 , estimate (6.3.3) of Theorem 6.3.1 holds, with τ replaced by T .  3. As a consequence of Theorems 6.5.1 and 6.5.2, the life-span Tε of the solution of the Cauchy problem (6.5.1), with fixed data u0 , u1 and f (assumed defined on [0, +∞[), is such that (6.5.47)

lim Tε = +∞ ,

ε→0

lim Tε = +∞ .

ε→+∞

To see this, assume the contrary, that is, that there is M > 0 such that Tε ≤ M for arbitrarily small (or large) ε. Taking then T = M +1, Theorems 6.5.1 and 6.5.2 guarantee that, if ε is sufficiently small (or large), (6.5.1) has a solution uε ∈ Ys (T ). Thus, we reach the contradiction Tε > T > M ≥ Tε .

334

6. Singular Convergence

From this, it follows that the map ε → Tε would have a graph roughly similar to the one pictured in Figure 1.

t 6 T

ε0

0

ε1

-

ε

Figure 1. The life-span of uε , I.

In fact, we conjecture that, at least when f ≡ 0, the solution of (6.5.1) should be in Zs,b (∞), or at least in Zs (∞), for all ε sufficiently small; in this case, the graph of the map ε → Tε would be qualitatively similar to that of Figure 2. However, we are not able to prove or disprove this conjecture.

t 6 T

0

ε∗ ε0

ε1

Figure 2. The life-span of uε , II.

-

ε

Chapter 7

Maxwell and von Karman Equations

In this chapter we present two applications of the local and global existence results we have presented in the previous chapters, respectively, to the complete system of Maxwell’s equations in a ferromagnetic medium, and to a hyperbolic and a parabolic version of a highly nonlinear system of von Karman type on R2m , m ≥ 2.

7.1. Maxwell’s Equations In this section we describe how the complete system of Maxwell’s equations in a ferromagnetic medium can be formally transformed into a quasi-linear equation of the type (0.0.2). With the exception of Theorem 4.6.4 (because the maximum principle does not necessarily hold for systems), all other results of the previous chapters would apply to this equation, so that we can deduce the existence of local or global solutions to the corresponding Cauchy problems, study the asymptotic behavior of global solutions corresponding to small initial values, and consider a singular perturbation problem related to the fact that, in ferromagnetic media, displacement currents are usually negligible in comparison to the eddy currents (also known as Foucault’s currents). 7.1.1. The Equations. 1. Physical Principles. As is well known (see, e.g., Duvaut and Lions [44, ch. 7]), Maxwell’s equations describe the evolution of the electromagnetic fields and inductions in a physical medium. Ferromagnetic media are 335

336

7. Maxwell and von Karman Equations

characterized by a nonlinear dependence between the magnetic field and induction, and by the fact that the displacement currents are negligible with respect to the eddy ones. The former feature gives rise to a quasi-linear hyperbolic dissipative system, while the latter allows us to consider, instead, a reduced problem, which is of quasi-linear parabolic type. The complete system of Maxwell’s equations is the first order linear system, essentially derived from the so-called Amp`ere’s theorem and Faraday’s law, (7.1.1)

Dt − curl H = G − J ,

(7.1.2)

Bt + curl E = 0 ,

(7.1.3)

div D = ρ ,

(7.1.4)

div B = 0 ,

where E and H denote, respectively, the electric and the magnetic fields, and D, B, the corresponding inductions. The vector functions G and J in (7.1.1) represent, respectively, an external source, and the so-called eddy currents; the scalar function ρ in (7.1.3) is a measure of the total electric charge. We refer to equations (7.1.1), ... , (7.1.4) collectively as “system (M)”. System (M) consists of eight conditions on the four vector fields D, E, B and H; thus, on twelve scalar unknown functions. To make this system determined, one usually stipulates the validity of some so-called constituent relations, or material laws, between each field and its corresponding induction. For example, in ferromagnetic materials such relations usually take the form (7.1.5)

D = εE ,

H ∈ ζ(B) ,

where ε = ε(t, x) ∈ L∞ (Q; R>0 ) measures the effects of the displacement currents, and ζ is a monotone map, in general multi-valued, because of the presence of hysteresis. In addition, we assume that the eddy currents are everywhere present, and caused entirely by the conductivity of the medium; that is, that (7.1.6)

J = σE ,

where σ = σ(t, x) ∈ L∞ (Q; R>0 ) is a measure of the conductivity of the medium. In the sequel, we assume for simplicity that ε and σ are positive constants (the dielectric and resistivity constants), and that each half-cycle of the hysteresis loop induced by ζ in (7.1.5) can be approximated by a strongly monotone function, which we still denote by ζ; thus, the second relation in (7.1.5) takes the simpler form H = ζ(B). With these assumptions and simplifications, the complete system of Maxwell’s equations (M) can be

7.1. Maxwell’s Equations

337

written as (7.1.7)

ε Et − curl ζ(B) = G − σ E , Bt + curl E = 0 ,

(7.1.8)

1 ε

(7.1.9)

div E =

(7.1.10)

div B = 0 ,

ρ,

which we refer to as “system (FM)”. Note that (7.1.10) is redundant if it is satisfied at t = 0, because (7.1.8) implies that the function t → div B(t) is constant. Finally, to make system (FM) consistent, we need to assume the compatibility condition ρt + 1ε σ ρ = div G ,

(7.1.11)

which is derived by taking the divergence of (7.1.7) and using (7.1.9). 2. Potentials. By its very nature, system (FM) should be considered in a bounded domain of R3 , with suitable boundary conditions (usually on the normal component of B and the tangential component of E; see, e.g., Milani and Koksch [119, ch. 6, sct. 6.4]). In the present context, however, we formally consider system (FM) in the whole space R3 , our goal being to show how the results we presented in the previous chapters can be applied to the corresponding Cauchy problem, consisting of system (FM) and the initial conditions (7.1.12)

E(0) = E0 ,

B(0) = B0 .

To this end, we transform system (FM) into a second order evolution equation of the type (0.0.2), by introducing a suitable choice of electromagnetic potentials; of course, the first order nonlinear system (FM) can be solved directly, using techniques analogous to those presented, e.g., in Majda [108, ch. 2]. The introduction of electromagnetic potentials is usually justified by observing that (7.1.10) should imply the existence of a vector potential A such that (7.1.13)

B = curl A .

Then, (7.1.8) implies that (7.1.14)

At + E = − ∇ϕ ,

for some scalar potential ϕ. Replacing (7.1.13) and (7.1.14) into (FM), we obtain the second order equation (7.1.15)

ε Att + σ At + curl ζ(curl A) = − G − ∇(ε ϕt + σ ϕ) .

338

7. Maxwell and von Karman Equations

Clearly, neither A nor ϕ are uniquely determined; in fact, they are, in general, not mutually independent. As we shall see, we can choose A and ϕ so that (7.1.16)

ε ϕt + σ ϕ = − γ div A ,

γ >0,

(a condition usually known as a gauge relation); in this case, (7.1.15) becomes (7.1.17)

ε Att + σAt + curl ζ(curl A) − γ ∇ div A = − G .

Carrying out the differentiations in the term curl ζ(curl A) explicitly, we finally obtain from (7.1.17) a hyperbolic dissipative (vector) equation of the form (0.0.2). As we have mentioned, the effect of displacement currents in ferromagnetic media is usually negligible in comparison to that of the eddy currents; that is, typically, ε is negligible with respect to σ. It is then common, in applications, to neglect the term ε Et in (7.1.7), and to consider instead the reduced equations (7.1.18)

σ E − curl ζ(B) = G ,

(7.1.19)

Bt + curl E = 0 .

These equations are known as the quasi-stationary Maxwell’s equations; the corresponding equation, in terms of the potential A, is then (7.1.20)

σ At + curl ζ(curl A) − γ ∇ div A = − G ,

a parabolic (vector) equation of the form (0.0.15). 7.1.2. Solution Theory. In the sequel, if u is a three-dimensional vector field and X is a function space, we adopt the notation u ∈ X to mean that each component of u is in X; that is, we abuse notation and identify X 3 with X. 1. Assumptions. We assume that the function ζ : R3 → R3 , appearing in (7.1.7), is of class at least C 4 and satisfies the strong monotonicity condition (7.1.21)

∃ > 0

∀ p, q ∈ R3 ,

ζ  (p) q · q ≥  |q|2 .

This implies that the space operator (7.1.22)

u → M (u) := curl ζ(curl u) − γ ∇ div u

is uniformly strongly elliptic; in fact, noting that, for all u ∈ C0∞ (R3 ), (7.1.23)

ζ(0), curl u = curl ζ(0), u = 0 ,

7.1. Maxwell’s Equations

339

we deduce that M (u), u = ζ(curl u), curl u + γ |div u|22 ≥ min{, γ} (|curl u|22 + |div u|22 ) .

 

(7.1.24)

=: α0

In particular, when ζ is linear, that is, when ζ(u) = ζ0 u for some ζ0 ∈ R>0 , the choice γ = ζ0 in (7.1.16) shows that (7.1.25)

M (u) = ζ0 (curl2 u − ∇ div u) = − ζ0 Δu ,

where Δ denotes the vector Laplace operator; that is, the operator u = (u1 , u2 , u3 ) → (Δu1 , Δu2 , Δu3 ). In accord with the theory developed in the previous chapters, we fix s = 3 >  32  + 1, and assume that the source terms G and ρ of system (FM) satisfy (7.1.26)

G , ρ ∈ L2 (0, T ; H 3 ) ,

ρt ∈ L2 (0, T ; H 2 ) ,

for some given T > 0, as well as the compatibility condition (7.1.11). We also assume that the initial data E0 , B0 , are such that (7.1.27)

E0 ∈ H 3 ∩ L1 ,

div E0 =

(7.1.28)

B0 ∈ H ∩ L ,

div B0 = 0 ;

3

1

1 ε

ρ(0) ,

note that the second condition in (7.1.27), which is in accord with (7.1.9), is consistent, because, by the trace theorem, ρ ∈ C([0, T ]; H 2 ). 2. Main Result. Under these conditions, we claim: Theorem 7.1.1. Let G, ρ, E0 , and B0 satisfy (7.1.26), (7.1.11), (7.1.27) and (7.1.28). There exists τ ∈ ]0, T ] and a unique pair of vector fields (B, E) ∈ C([0, τ ]; H 3 × H 3 ) ∩ C 1 ([0, τ ]; H 2 × H 2 ), solution of the Cauchy problem (FM), with initial data (7.1.12). Sketch of Proof. (i) In conformity with (7.1.13), we first determine an initial potential A0 , such that (7.1.29)

curl A0 = B0 ,

div A0 = 0 .

We do so, by means of the Fourier transform; more precisely, following the proof of Theorem 6.2 of Duvaut and Lions [44, ch. 7, sct. 6.2], we let B0 = (B01 , B02 , B03 ), and for ξ ∈ R3 \ {0} we define i  ˆ3 ˆ 2 (ξ) , h1 (ξ) := ξ B B (7.1.30) (ξ) − ξ 2 3 0 0 |ξ|2 i  ˆ1 ˆ 3 (ξ) , h2 (ξ) := ξ B B (7.1.31) (ξ) − ξ 3 1 0 0 |ξ|2  i 2 1 ˆ ˆ h3 (ξ) := ξ B B (7.1.32) (ξ) − ξ (ξ) ; 1 0 2 0 |ξ|2

340

7. Maxwell and von Karman Equations

then, we set (7.1.33)

  A0 = (A10 , A20 , A30 ) := F − 1 (h1 ), F − 1 (h2 ), F − 1 (h3 ) .

Lemma 7.1.1. Let A0 be as in (7.1.33). Then, A0 ∈ H 4 , with A0 4 ≤ C (B0 3 + |B0 |1 ) ,

(7.1.34) and satisfies (7.1.29).

Proof. Consider for example the first component A10 of A0 . From (7.1.30), we formally deduce that  1 2 A0 4 = (1 + |ξ|2 )4 |h1 (ξ)|2 dξ  1 ˆ 3 (ξ) − ξ3 B ˆ 2 (ξ)|2 dξ = (1 + |ξ|2 )4 4 |ξ2 B (7.1.35) 0 0 |ξ|   = (· · · ) dξ + (· · · ) dξ =: α12 + α22 . |ξ|≥1

Since B0 ∈

H 3,



α12 ≤

|ξ|≥1



(7.1.36) ≤ 8

Since also B0 ∈ L1 ,  2 4 (7.1.37) α2 ≤ 2

|ξ|≤1

4(1 + |ξ|2 )4  2 ˆ3 2 2 ˆ2 2 2|ξ dξ | | B (ξ)| + 2|ξ | | B (ξ)| 2 3 0 0 (1 + |ξ|2 )2

ˆ0 (ξ)|2 dξ ≤ 8 B0 2 . (1 + |ξ|2 )3 |B 3

|ξ|≤1

1 ˆ ˆ 0 |2 |B0 (ξ)|2 dξ ≤ 24 |B ∞ |ξ|2



1 0

4 π dr ≤ C |B0 |21 .

Inserting (7.1.36) and (7.1.37) into (7.1.35) yields that A10 ∈ H 4 , and (7.1.38)

A10 4 ≤ C (B0 3 + |B0 |1 ) .

Arguing analogously for A20 and A30 , we conclude that Ao ∈ H 4 , and (7.1.34) holds. The verification of (7.1.29) is straightforward; for example, recalling (7.1.30), we compute that  F (∂2 A30 − ∂3 A20 ) = i ξ2 Aˆ30 − ξ3 Aˆ20 = i (ξ2 h3 − ξ3 h2 ) 1  ˆ 2 − ξ2 B ˆ 1 − ξ2 B ˆ 1 + ξ1 ξ3 B ˆ3 = − 2 ξ1 ξ2 B 0 2 0 3 0 0 |ξ|  1 ˆ 2 + ξ1 ξ3 B ˆ 3 + (ξ 2 − |ξ|2 ) B ˆ1 = − 2 ξ1 ξ2 B (7.1.39) 0 0 1 0 |ξ| ξ1  ˆ 1 ˆ1 ˆ2 ˆ3 = − 2 ξ1 B 0 + ξ 2 B0 + ξ 3 B0 + B0 |ξ| ξ1 ˆ1 = B ˆ1 , = − F (div B0 ) + B 0 0 i |ξ|2

7.1. Maxwell’s Equations

341

having recalled that div B0 = 0. Arguing analogously for the other components of curl A0 , we conclude that the first of (7.1.29) holds. Finally, we also compute that (7.1.40)

F (div A0 ) = i (ξ1 h1 (ξ) + ξ2 h2 (ξ) + ξ3 h3 (ξ)) = 0 ,

from which we deduce that div A0 = 0. This concludes the proof of Lemma 7.1.1.  (ii) We now define A1 := − E0 ∈ H 3 , and, arguing as in the first step of the proof of Theorem 4.5.2, we deduce that the quasi-linear dissipative hyperbolic Cauchy problem  εAtt + σAt + curl ζ(curl A) − ∇ div A = − G , (7.1.41) A(0) = A0 , At (0) = A1 , has a unique local solution A = Aε ∈ C([0, τε ]; H 4 ) ∩ C 1 ([0, τε ]; H 3 ), for some τε ∈ ]0, T ]. In fact, τε depends on the norms of A0 in H 4 and A1 in H 3 ; that is, recalling (7.1.34), on B0 3 , |B0 |1 and E0 3 . Since we assume that the data B0 and E0 are independent of ε, each Aε can be extended to a common interval [0, τ ] ⊆ [0, T ], with Aε ∈ X3 (τ ). Next, we solve the linear Cauchy problem  εϕtt + σϕt − Δ ϕ = 1ε ρ , (7.1.42) ϕ(0) = 0 , ϕt (0) = 0 , thereby obtaining a unique ϕ ∈ C([0, T ]; H 4 ) ∩ C 1 ([0, T ]; H 3 ). We then define (7.1.43)

B := curl A ,

E := − ∇ϕ − At ,

(compare to (7.1.13) and (7.1.14)), and proceed to verify that B and E are the desired solutions of (FM). To this end, we start to compute that ε Et + σ E − curl ζ(B) (7.1.44)

= ε(−∇ϕt − Att ) + σ(−∇ϕ − At ) − curl ζ(curl A) = −(εAtt + σAt + curl ζ(curl A)) − ∇(ε ϕt + σ ϕ) = G − ∇(div A + εϕt + σϕ) .

Taking the divergence of (7.1.41) and recalling (7.1.11) and (7.1.29), we see that the function u := div A solves the linear Cauchy problem ⎧ ε utt + σ ut − Δ u = − div G = − ρt − 1ε σ ρ , ⎪ ⎪ ⎨ (7.1.45) u(0) = divA0 = 0 , ⎪ ⎪ ⎩ ut (0) = div A1 = − div E0 .

342

7. Maxwell and von Karman Equations

On the other hand, differentiating (7.1.42) with respect to time, we see that the function v := −(εϕt + σϕ) satisfies the same Cauchy problem. Indeed, (7.1.46) ε vtt + σ vt − Δ v = − ε(εϕttt + σϕtt ) − σ(εϕtt + σϕt ) − (− Δ(εϕt + σϕ)) = − ε(εϕtt + σϕt − Δ ϕ)t − σ(εϕtt + σϕt − Δ ϕ) = − ε 1ε ρt − σ 1ε ρ , v(0) = 0, and, by the second of (7.1.27), (7.1.47)

vt (0) = −(εϕtt (0) + σϕt (0)) = −

1 ε

 ρ(0) + Δ ϕ(0) = − div E0 .

Consequently, u(t) ≡ v(t); that is, the gauge relation (7.1.16) with γ = 1, i.e., div A = − εϕt − σϕ, holds. Replacing this into (7.1.44), we deduce that (7.1.7), i.e., the first equation of system (FM), holds. The proof of the other three equations is simpler: (7.1.48)

Bt + curl E = curl At − curl(∇ϕ + At ) = 0 ,

(7.1.49)

div E = − Δ ϕ − div At = − Δϕ + εϕtt + σϕt =

(7.1.50)

div B = div(curl A) = 0 .

1 ε

ρ,

It is then immediate to verify that B and E satisfy the initial conditions (7.1.12). Indeed, by (7.1.29), (7.1.51)

B(0) = curlA(0) = curlA0 = B0 ,

and, recalling that A1 = − E0 , (7.1.52)

E(0) = − ∇ϕ(0) − At (0) = − A1 = E0 .

Thus, the fields B and E defined in (7.1.43) are the desired local solutions of system (FM), and the proof of Theorem 7.1.1 is complete.  Remarks. 1) As initial conditions for the function ϕ defined by (7.1.42), we can choose two arbitrary functions ϕ0 ∈ H 4 and ϕ1 ∈ H 3 , as long as they satisfy the gauge condition (7.1.53)

ε ϕ1 + σ ϕ0 = − γ divA0 ,

in accord with (7.1.16). 2) Clearly, the results of Chapters 4 and 5 also apply to equation (7.1.17), leading to corresponding almost global, global, and global bounded existence results if the data in (7.1.26), (7.1.27) and (7.1.28) are sufficiently small. Likewise, the results of Chapter 6 would allow us to study the corresponding singular perturbation problem, as ε → 0, thereby giving a justification to the neglect of the term εAtt in (7.1.17), common in applications, where the

7.2. von Karman’s Equations

343

reduced equation (7.1.20) is considered instead. For some results in this direction, we refer, e.g., to Milani [114]. 

7.2. von Karman’s Equations For our second application, we consider two types of evolution problems, one hyperbolic and the other parabolic, related to a highly nonlinear elliptic system of von Karman type equations on R2m , m ≥ 2. These equations do not fit exactly in the framework of the evolution equations we have presented so far (that is, of type (0.0.2)); however, we believe this example to be of particular interest, as it shows that the unified methods we presented in our previous chapters can be applied to a much wider class of systems. We call these equations “of von Karman type” because of a formal analogy with the well-known equations of the same name in the theory of elasticity (see, e.g., Ciarlet and Rabier [36]) in two dimensions of space (thus, m = 1). In this case, the usual von Karman equations model the dynamics of the vertical oscillations (buckling) of an elastic two-dimensional thin plate, due to both internal and external stresses, subject to appropriate boundary conditions. For more precise modeling issues related to the von Karman equations, as well as their physical motivations, we refer, e.g., to Ciarlet [34, 35], as well as to Chuesov and Lasiecka [33]. Our goal is to apply the methods of Chapters 3 and 4 to show the existence and uniqueness of a local strong solution to the Cauchy problem corresponding to each of the two systems, in a suitable class of function spaces. For both problems, we also establish an almost global existence result, in the spirit of Theorem 4.4.1, if the data are sufficiently small. In this presentation, we do not prove any result in detail; rather, and in a few cases only, we limit ourselves to point out the various steps in which the proof would proceed. A complete and detailed proof of all the claims and results we mention in this section, in particular of those for which we provide no proof at all, can be found at the Book Page hosted by the AMS for this book (www.ams.org/bookpages/gsm-135). 7.2.1. The Equations. 1. The operators. Let m ∈ N≥2 and u1 , . . . , um , um+1 , u ∈ C ∞ (R2m ). We define (7.2.1) (7.2.2) (7.2.3)

j1 jm im N (u1 , . . . , um ) := δji11 ··· ··· jm ∇i1 u1 · · · ∇im um ,

I(u1 , . . . , um , um+1 ) := N (u1 , . . . , um ) , um+1  , M (u) := N (u, . . . , u) = m! σm (∇2 u) ,

344

7. Maxwell and von Karman Equations

where, recalling the summation convention for repeated indices, we adopted im the following notations: for i1 , . . . , im , j1 , . . . , jm ∈ {1, . . . , 2m}, δji11 ··· ··· jm ,

denotes the Kronecker tensor; for 1 ≤ i, j ≤ 2m, ∇ji := ∂i ∂j ; and σm is the mth elementary symmetric function of the eigenvalues λj = λj (u), 1 ≤ j ≤ 2m, of the Hessian matrix H(u) := [∂i ∂j u], that is,  (7.2.4) σm (∇2 u) := λj 1 · · · λ j m . 1≤j1 2 require different kinds of assumptions on the functions uj , due to the restrictions imposed by the limit cases of the Sobolev imbeddings. In any case, we note that, by the density of C0∞ (R2m ) in H h , it is sufficient to establish these estimates when u1 , . . . , um ∈ C0∞ (R2m ). In the sequel, whenever a constant C appears in an estimate, it is understood that C is independent of each of the functions that appear in the estimate. ¯ m+1 . Then, N (Um ) ∈ L2 , and Lemma 7.2.1. 1) Case h = 0. Let Um ∈ H (7.2.21)

N (Um )0 ≤ C

m )

uj m+1 .

j=1

¯ m+2 ∩ H ¯ m+1 . Then, N (Um ) ∈ H 1 , and 2) Case h = 1. Let Um ∈ H (7.2.22)

∇N (Um )0 ≤ C

m 

uj m+2

j=1

m )

ui m+1 .

i=1 i=j

3) Case h ≥ 2. (i ) If m > 2 and h ≥ 2, or if m = 2 and h > 2, let ¯ m+h ∩ H ¯ m . Then, N (Um ) ∈ H h , and Um ∈ H (7.2.23)

∇h N (Um )0 ≤ C

m )

  max uj m , uj m+h .

j=1

¯5 ∩ H ¯ 2 and u2 ∈ H ¯4 ∩ H ¯ 2 . Then, (ii ) If m = h = 2, let u1 ∈ H 2 N (U2 ) ∈ H , and (7.2.24)

∇2 N (U2 )0 ≤ C max {u1 2 , u1 5 } max {u2 2 , u2 4 } .

7.2. von Karman’s Equations

347

¯ m+h−1 ∩ H ¯ m . Then, N (Um ) ∈ H h , 4) Case h > m > 2. Let Um ∈ H and (7.2.25)

∇h N (Um )0 ≤ C

m )

  max uj m , uj m+h−1 .

j=1

Remarks. 1) Estimate (7.2.21) does not require the special structure of N , as it is sufficient to use the fact that N is the sum of products of second order derivatives of the functions uj ’s. On the other hand, from (7.2.19) and (7.2.18), we deduce that (7.2.26)

⎞

⎛

m−1 )

|I(u1 , . . . , um , um+1 )| ≤ C ⎝

|∇2 uj |m ⎠ |∇um |2m |∇um+1 |2m

j=1

≤C

m+1 )

uj m ≤ C

j=1

m+1 )

uj m ,

j=1

and this estimate, which requires less regularity of the uj ’s, does use the special structure of N . 2) The improvement (7.2.25) of (7.2.23) if h > m > 2 is essentially due to the fact that H h is an algebra if h > m.  5. Elliptic Type Estimates on f . We now turn to equation (7.2.6), which defines f (u) in both problems (H) and (P), and show that estimate (7.2.23) can be somewhat improved, if u1 is replaced by f (u1 ). ¯ m . There exists a unique f ∈ H ¯ m , which is a Lemma 7.2.2. Let u ∈ H m ¯ weak solution of (7.2.6), in the sense that, for all ϕ ∈ H , f, ϕm = −M (u), ϕ0 .

(7.2.27) f satisfies the estimate

f m ≤ C um m,

(7.2.28) with C independent of u.

¯ m , then ∂ 2 u ∈ H ¯ m−2 → Lm ; thus, by Proof. We first note that if u ∈ H x 1 (7.2.18) and H¨older’s inequality, M (u) ∈ L at least, as we see from (7.2.29)

m |M (u)|1 ≤ C |∂x2 u|m m ≤ C um .

Consequently, the right side of (7.2.27) makes sense if ϕ ∈ L∞ . Note that, since the space dimension is N = 2 m, the imbedding H m → L∞ does not

348

7. Maxwell and von Karman Equations

¯ m → L∞ . On the other hand, by hold; hence, neither does the imbedding H ¯ m , the estimate (7.2.26), if ϕ ∈ C0∞ (R2m ) → L∞ ∩ H (7.2.30)

|M (u), ϕ| = |I(u, . . . , u, ϕ)| ≤ C um m ϕm

¯ m, holds, with C independent of u and ϕ. By the density of C0∞ (R2m ) in H m ¯ (7.2.30) also hold for all ϕ ∈ H ; hence, by Riesz’ representation theorem, ¯ m , solution of (7.2.27). Taking ϕ = f in (7.2.27), there is a unique f ∈ H ¯ m , and using (7.2.30), we obtain (7.2.28). which is admissible because f ∈ H  We now establish further regularity results for f . ¯ m+1 ∩ H ¯ m , and let f ∈ H ¯ m be the weak solution Lemma 7.2.3. 1) Let u ∈ H m+h ¯ of (7.2.6), given by Lemma 7.2.2. Then, f ∈ H for 0 < h ≤ m, and (7.2.31)

f m+h ≤ C um−h uhm+1 . m

¯ h+1 ∩ H ¯ m . Then, f ∈ H ¯ m+h , and 2) Let h > m, and u ∈ H  m (7.2.32) f m+h ≤ C max{um , uh , uh+1 } . ¯ m+h ∩ H ¯ m , h ≥ 0, then 3) If u ∈ H (7.2.33)

f m+h ≤ C um−1 um+h . m

In (7.2.31), (7.2.32) and (7.2.33), C is independent of u. Using (7.2.18) with h = 1, we deduce from Lemma 7.2.2 and part 1) of Lemma 7.2.3: ¯ m , and let f = f (u) be the corresponding Corollary 7.2.1. Let u ∈ H 2 ¯ m+1 ∩ H ¯ m , then ∂ 2 f ∈ L2m . solution of (7.2.6). Then, ∂x f ∈ Lm . If u ∈ H x In addition, the estimates (7.2.34)

|∂x2 f |m ≤ C f m ,

|∂x2 f |2m ≤ C f m+1

respectively hold, with C independent of u and f . 7.2.2. The Hyperbolic System. 1. Local Existence. In this section we give a local existence result for problem (H). For h, k ∈ N, with k ≤ h, and T > 0, we consider the space (7.2.35)

Hh,k (T ) := C([0, T ]; H h ) ∩ C 1 ([0, T ]; H k ) ,

endowed with its natural norm (7.2.36) We claim:

  u2Hh,k (T ) := max ut (t)2k + u(t)2h . 0≤t≤T

7.2. von Karman’s Equations

349

¯ 2m ) Theorem 7.2.1. Assume that u0 ∈ H 2m , u1 ∈ H m , and ϕ ∈ C([0, T ]; H ¯ 4+ε ), ε > 0, if m = 2. There exists τ∗ ∈ ]0, T ] and if m > 2, ϕ ∈ C([0, T ]; H 2m,m ¯ 2m ), which is a solution of a unique u ∈ H (τ∗ ), with f (u) ∈ C([0, τ∗ ]; H problem (H). This solution depends continuously on the data {u0 , u1 , ϕ}. Sketch of Proof. We loosely follow Cherrier and Milani [29], and proceed along the lines of the linearization and fixed point method described in section 3.2 of Chapter 3. For simplicity of exposition, whenever we write ¯ 2m ), we tacitly understand that, if m = 2, we mean to ϕ ∈ C([0, T ]; H ¯ 4+ε ), ε > 0. We proceed in five steps: linearization, write ϕ ∈ C([0, T ]; H contractivity, Picard’s iterations, continuity, and well-posedness. Linearization and Contractivity. Given τ ∈ ]0, T ] and R > 0, we define (7.2.37) Bm (τ, R) := {u ∈ H2m,m (τ ) | uH2m,m (τ ) ≤ R , u(0) = u0 , ut (0) = u1 } .  1/2 As we remarked after Proposition 3.2.1, for any R > u1 2m + u0 22m there exists τ = τ (R) ∈ ]0, T ] such that the ball Bm (τ, R) is not empty. Choose such a pair (R, τ ), and fix w ∈ Bm (τ, R). Then, Lemma 7.2.4. The function t → M (w(t)) is continuous from [0, τ ] into ¯ 2m ), and the maps L2 . Consequently, f (w) ∈ C([0, τ ]; H (7.2.38)

u → N (f (t), (w(t))(m−2), u) ,

u → N ((ϕ(t))(m−1), u)

are, for all t ∈ [0, τ ], well defined and continuous from H 2m into L2 . Again for fixed w ∈ Bm (τ, R), we consider then the linearized equation (7.2.39)

utt + Δm u = N (f (w), w(m−2) , u) + N (ϕ(m−1) , u) ,

with initial data (7.2.8). As a consequence of Lemma 7.2.4, the existence and uniqueness of a solution u ∈ H2m,m (τ ) to problem (7.2.39)+(7.2.8), ¯ 2m ), can be established by methods analogous to those with f ∈ C([0, τ ]; H of Chapter 2. As in section 3.3.1, this allows us to define the map w → u =: Γ(w) from Bm (τ, R) into H2m,m (τ ), which we then show to be a contraction: Proposition 7.2.2. 1) There exist τ0 ∈ ]0, T ] and R∗ > 0 such that, for all τ ∈ ]0, τ0 ], Γ maps the ball Bm (τ, R∗ ) into itself. 2) There exists τ∗ ∈ ]0, τ0 ] such that for all w, w ˜ ∈ Bm (τ∗ , R∗ ), (7.2.40)

Γ(w) − Γ(w) ˜ Hm,0 (τ∗ ) ≤

1 2

w − w ˜ Hm,0 (τ∗ ) .

Sketch of Proof. Part 1) is a consequence of the estimate (7.2.41)

   ut (t)2m + u(t)22m ≤ u1 2m + u0 22m exp C R2(m−1) + Cϕ t ,

350

7. Maxwell and von Karman Equations

where (7.2.42)

Cϕ :=

⎧ ⎪ max ϕ(t)m−1 ⎨ C 0≤t≤T 2m

if

m>2,

⎪ ⎩ C max ϕ(t)4+ε

if

m=2,

0≤t≤T

which is formally obtained by multiplying equation (7.2.39) in L2 by 2 Δm ut and 2 ut , and adding the resulting identities. Then, we define (for example) R∗ by    (7.2.43) R∗2 := max 1, 4 u1 2m + u0 22m , and then τ0 ∈ ]0, T ] by (7.2.44)



2

ln 4

τ0 := min τ (R∗ ),

2(m−1)

C R∗

, + Cϕ

to deduce from (7.2.41) that for all τ ∈ ]0, τ0 ], Γ maps Bm (τ, R∗ ) into itself. Part 2) is a consequence of the estimate (7.2.45)





zt (t)20 + z(t)2m ≤ γ(R∗ , ϕ)

max v(t)2m

0≤t≤τ

t exp(γ(R∗ , ϕ) t) ,

2(m−1)

where γ(R∗ , ϕ) := C R∗ + Cϕ , and z := u − u ˜, u := Γ(w), u ˜ := Γ(w), ˜ w, w ˜ ∈ Bm (τ, R∗ ), and v := w − w. ˜ Thus, choosing τ∗ ∈ ]0, τ0 ] such that (7.2.46)

γ(R∗ , ϕ) τ∗ exp(γ(R∗ , ϕ) τ∗ ) ≤

1 4

,

we deduce from (7.2.45) that, for all t ∈ [0, τ∗ ], (7.2.47)

zt (t)20 + z(t)2m ≤

1 max 4 0≤t≤τ

v(t)2m ,

from which (7.2.40) follows. Thus, Γ is a contraction on Bm (τ∗ , R∗ ), with respect to the weaker norm of Hm,0 (τ∗ ), as claimed.  Picard’s Iterations. We now consider the Picard’s iterations of Γ, that is, the sequence (un )n≥0 , defined recursively by un+1 = Γ(un ), starting from an arbitrary u0 ∈ Bm (τ∗ , R∗ ). Explicitly, the functions un+1 are defined, in terms of un , by the equation (7.2.48)

un+1 + Δm un+1 = N (f n , (un )(m−2) , un+1 ) + N (ϕ(m−1) , un+1 ) , tt

where f n := f (un ), and the initial conditions (7.2.49)

un+1 (0) = u0 ,

un+1 (0) = u1 . t

By Proposition 7.2.2, the sequence (un )n≥0 is bounded in H2m,m (τ∗ ), and a Cauchy one in Hm,0 (τ∗ ). Thus, there is a subsequence, still denoted (un )n≥0 ,

7.2. von Karman’s Equations

351

and a function u ∈ L∞ (0, τ∗ ; H 2m ), with ut ∈ L∞ (0, τ∗ ; H m ), such that (7.2.50)

un → u

(7.2.51)

unt → ut

in L∞ (0, τ∗ ; H 2m ) weak∗ and C([0, τ∗ ]; H m ) , in L∞ (0, τ∗ ; H m ) weak∗ and C([0, τ∗ ]; L2 ) .

By Proposition 1.7.1, (7.2.50) and (7.2.51) imply that u ∈ Cw ([0, τ∗ ]; H 2m ) and ut ∈ Cw ([0, τ∗ ]; H m ), and that the maps t → u(t)2m and t → ut (t)m are bounded on [0, τ∗ ]. We can then show that (7.2.52)

M (un ) → M (u)

in

C([0, τ∗ ]; L2 )

and (7.2.53)

N (f n , (un )(m−2) , un+1 ) → N (f, u(m−1) )

in

C([0, τ∗ ]; L2 ) .

From (7.2.53) and (7.2.50) it also follows that the sequence (untt )n≥0 is bounded in L∞ (0, τ∗ ; L2 ); consequently, we can suppose that (7.2.54)

untt → utt

in L∞ (0, τ∗ ; L2 )

weak∗ .

We can then let n → +∞ in (7.2.48); more precisely, by (7.2.50) and (7.2.53), (7.2.55) un+1 = − Δm un+1 + N (f n , (un )(m−2) , un+1 ) + N (ϕ(m−1) , un+1 ) tt → − Δm u + N (f, u(m−1) ) + N (ϕ(m−1) , u) in L∞ (0, τ∗ ; L2 ) weak∗ . Comparing (7.2.55) to (7.2.54), we deduce that u satisfies equation (7.2.7) in L2 , at least for a.a. t ∈ [0, τ∗ ]. Moreover, u satisfies the initial conditions (7.2.8), because, by (7.2.49) and the second part of (7.2.50) and (7.2.51), un (0) = u0 → u(0) in H m , and unt (0) = u1 → ut (0) in L2 .  Continuity and Well-Posedness. The claims u ∈ C([0, τ∗ ]; H 2m ) and ut ∈ C([0, τ∗ ]; H m ) are proven as in Lemma 2.3.2, first showing the continuity of the function (7.2.56)

t → E(t) := ut (t)2m + u(t)22m

on [0, τ∗ ], and then that, for all t, t0 ∈ [0, τ∗ ], the function (7.2.57)

F (t, t0 ) := ut (t) − ut (t0 )2m + u(t) − u(t0 )22m

vanishes as t → t0 . The continuous dependence of solutions of problem (H) on their data is shown in the following proposition, whose proof concludes that of Theorem 7.2.1.

352

7. Maxwell and von Karman Equations

¯ 2m ) Proposition 7.2.3. Let u0 , u ˜0 ∈ H 2m , u1 , u ˜1 ∈ H m , ϕ, ϕ˜ ∈ C([0, T ]; H ¯ 4+ε )). Assume that problem (H) has corre(if m = 2, ϕ, ϕ˜ ∈ C([0, T ]; H 2m,m sponding solutions u, u ˜∈H (τ ), for some τ ∈ ]0, T ]. Then, the difference u − u ˜ satisfies the estimate (7.2.58)

 u − u ˜H2m,m (τ ) ≤ h(ρ, δ, T ) u0 − u ˜0 2m + u1 − u ˜1 m + ϕ − ϕ ˜ C([0,T ];H¯ 2m ) ,

where h ∈ K, and (7.2.59) (7.2.60)

  ρ := max 1, uH2m,m (τ ) , ˜ uH2m,m (τ )   δ := max ϕC([0,T ];H¯ 2m ) , ϕ ˜ C([0,T ];H 2m ) .

In particular, there is at most one solution of problem (H) in H2m,m (τ ). 2. Higher Regularity. Higher regularity results for problem (H) can be established by a suitable generalization of Theorem 7.2.1. Theorem 7.2.2. Let k ≥ 0, and assume that u0 ∈ H 2m+k , u1 ∈ H m+k , ¯ 2m+k ). There is τk ∈ ]0, T ], such that problem (H) admits a ϕ ∈ C([0, T ]; H ¯ 2m+k ). unique solution u ∈ H2m+k,m+k (τk ), with f (u) ∈ C([0, τ∗ ]; H The proof of this theorem is based on the time-independent a priori estimates established in Proposition 7.2.4 below. Note that Theorem 7.2.1 corresponds to Theorem 7.2.2 when k = 0, with τ0 = τ∗ (and the additional ¯ 4+ε ) if m = 2). It is important to remark that assumption ϕ ∈ C([0, T ]; H the regularity result of Theorem 7.2.2 is uniform in k, in the sense that inf k>0 τk ≥ τ∗ . Roughly speaking, this means that increasing the regularity of the data does not decrease the life-span of the solution. Proposition 7.2.4. Let k > 0, and u0 , u1 , ϕ satisfy the assumptions of Theorem 7.2.2. Assume that problem (H) has a corresponding solution u ∈ H2m,m (τ ) ∩ H2m+k,m+k (τ  ), with 0 < τ  < τ ≤ T . There exists Λk , depending on τ but not on τ  , such that   (7.2.61) sup ut (t)2m+k + u(t)22m+k ≤ Λ2k . 0≤t≤τ 

Consequently, u ∈ H2m+k,m+k (τ ). 3. Almost Global Existence. The estimates established in the proof of Theorem 7.2.1 allow us to give an almost global existence result for problem (H), in the spirit of Theorem 4.4.1. More precisely, with E(t) as in (7.2.56), we claim:

7.2. von Karman’s Equations

353

Theorem 7.2.3. In the same assumptions of Theorem 7.2.1, given arbitrary T > 0, there is δ > 0 such that, if E(0) ≤ δ, then problem (H) admits a ¯ 2m ). unique solution u ∈ H2m,m (T ), with f (u) ∈ C([0, T ]; H Idea of Proof. We first show that, on any interval [0, τ ] ⊆ [0, T ] on which u is defined, E satisfies the differential inequality (7.2.62)

d ut m + Cϕ u2m ut m ≤ C E m + Cϕ E , E ≤ C u2m−1 2m dt

with Cϕ as in (7.2.42). The Bernoulli-type inequality (7.2.62) implies the exponential inequality d 1−m + (m − 1) Cϕ E 1−m ≥ − C (m − 1) , E dt

(7.2.63)

the integration of which leads to  (7.2.64) (E(t))1−m ≥ (E(0))1−m +

C Cϕ



e−(m−1)Cϕ t −

C Cϕ

.

Assuming that E(0) ≤ δ, we deduce from (7.2.64) that, for all t ∈ [0, τ ], (E(t))m−1 ≤

(7.2.65)

Cϕ e(m−1)Cϕ t δ m−1 . Cϕ + C δ m−1 − C δ m−1 e(m−1)Cϕ t

Thus, the life-span Tc of u satisfies the estimate Tc ≥ Tδ , where Tδ is the blow-up time of the right side of (7.2.65); that is,

Cϕ 1 (7.2.66) Tc ≥ Tδ := . ln 1 + (m − 1) Cϕ C δ m−1 In particular, since Tδ → +∞ as δ → 0, given T > 0 it is possible to have Tc > T , by choosing δ, and therefore E(0), sufficiently small, so that Tc ≥ Tδ > T .  7.2.3. The Parabolic System. 1. Local Existence. In this section we state a local existence result for problem (P). For h, k ∈ N, with k ≤ h, and T > 0, we consider the spaces (7.2.67)

P h,k (T ) := {u ∈ L2 (0, T ; H h ) | ut ∈ L2 (0, T ; H k )} ,

(7.2.68)

Y h,k (T ) := L2 (0, T ; H h ) ∩ C([0, T ]; H (h+k)/2 ) ,

(7.2.69)

Y

h,k

¯ h ) ∩ C([0, T ]; H ¯ (h+k)/2 ) , (T ) := L2 (0, T ; H

354

7. Maxwell and von Karman Equations

endowed with their natural norms  T   2 uP h,k (T ) := (7.2.70) ut 2k + u2h dt , (7.2.71) (7.2.72)

u2Y h,k (T ) := u2 h,k Y

(T )



0



0

T

0≤t≤T

T

:= 0

u2h dt + max u(t)2(h+k)/2 , u2h dt + max u(t)2(h+k)/2 . 0≤t≤T

Note that, by the trace Theorem 1.7.4, the fact that u ∈ P h,k (T ) implies that u ∈ C([0, T ]; H (h+k)/2 ), and, by (1.7.61), (7.2.73)

uC([0,T ];H (h+k)/2 ) ≤ C uP h,k (T ) .

In particular, (7.2.73) implies that P 2m,0 (T ) → Y 2m,0 (T ). We claim: Theorem 7.2.4. Assume that u0 ∈ H m , and ϕ ∈ Y 2m,0 (T ). There exists 2m,0 τ∗ ∈ ]0, T ], and a unique u ∈ P 2m,0 (τ∗ ), with f (u) ∈ Y (τ∗ ), solution of problem (P). This solution depends continuously on the data {u0 , ϕ}. Sketch of Proof. We loosely follow Cherrier and Milani [26, 27]. In contrast to the hyperbolic problem (H), we cannot prove Theorem 7.2.4 directly; rather, we first prove a higher regularity result for problem (P) and then use this result to prove Theorem 7.2.4 by means of an approximation argument. This roundabout procedure, which we do not know to what extent is necessary, is due to a rather drastic role played by the limit case of the Sobolev imbedding theorem H m−1 → L2m , which allows us to estimate N (u1 , . . . , um ) in L2 only in terms of the uj ’s in H m+1 (as opposed to H m ; see (7.2.21)). Thus, we first claim Theorem 7.2.5. Assume that u0 ∈ H m+1 , and ϕ ∈ Y 2m+1,1 (T ). There 2m+1,1 exists τ1 ∈ ]0, T ], and a unique u ∈ P 2m+1,1 (τ1 ), with f (u) ∈ Y (τ1 ), solution of problem (P). Proof. The proof of Theorem 7.2.5 proceeds along the lines of the linearization and fixed point method described in section 3.4 of Chapter 3, following essentially the same steps of the proof of Theorem 7.2.1 for problem (H). Linearization and Contractivity. Given τ ∈ [0, T ] and R > 0, we define (7.2.74)   Bm (τ, R) := u ∈ P 2m+1,1 (τ ) | uP 2m+1,1 (τ ) ≤ R , u(0) = u0 . Note that the condition on u(0) makes sense, because, by the trace theorem, u ∈ C([0, T ]; H m+1 ). For fixed w ∈ Bm (τ, R), we consider the linearized equation (7.2.75)

ut + Δm u = N (f (w), w(m−2) , u) + N (ϕ(m−1) , u) ,

7.2. von Karman’s Equations

355

with initial data (7.2.10). By (7.2.21) and (7.2.22) of Lemma 7.2.1, M (w) ∈ ¯ 2m+1 ); in turn, L2 (0, τ ; H 1 ), which implies that f := f (w) ∈ L2 (0, τ ; H (m−2) this implies that the maps u → N (f, w , u) and u → N (ϕ(m−1) , u) 2m+1,1 2 are continuous from P (τ ) into L (0, τ ; H 1 ). Thus, the existence and uniqueness of a solution u ∈ P 2m+1,1 (τ ) of problem (7.2.75)+ (7.2.10), with 2m+1,1 f ∈Y (τ ) can be established, by methods analogous to those of Chapter 2. As in section 3.4, this allows us to define the map w → u := Γ(w), from Bm (τ, R) into P 2m+1,1 (τ ), which we then show to be a contraction: Proposition 7.2.5. 1) There exists τ0 ∈ ]0, T ], and R1 ≥ 1, such that for all τ ∈ ]0, τ0 ], Γ maps the ball Bm (τ, R1 ) into itself. 2) There exists τ1 ∈ ]0, τ0 ] such that, for all w, w ˜ ∈ Bm (τ1 , R1 ), Γ(w) − Γ(w) ˜ P 2m,0 (τ1 ) ≤

(7.2.76)

1 2

w − w ˜ P 2m,0 (τ1 ) .

Remark. The proof of Proposition 7.2.5 cannot be adapted to show the existence of an invariant ball (for Γ) in P 2m,0 (τ ). It is for this reason that we need to first consider solutions in P 2m+1,1 (τ ).  Picard’s Iterations. We now consider the Picard’s iterations of Γ, that is, the sequence (un )n≥0 , defined recursively by un+1 = Γ(un ), starting from an arbitrary u0 ∈ Bm (τ1 , R1 ). Explicitly, the functions un+1 are defined, in terms of un , by the equation (7.2.77)

un+1 + Δm un+1 = N (f n , (un )(m−2) , un+1 ) + N (ϕ(m−1) , un+1 ) , t

where f n := f (un ), and the initial condition un+1 (0) = u0 . By Proposition 7.2.5, the sequence (un )n≥0 is bounded in P 2m+1,1 (τ1 ), and a Cauchy sequence in P 2m,0 (τ1 ). Thus, there is a subsequence, still denoted (un )n≥0 , and a function u ∈ P 2m+1,1 (τ1 ), such that (7.2.78)

un → u

(7.2.79)

unt → ut

in L2 (0, τ1 ; H 2m+1 ) weak∗ and L2 (0, τ1 ; H 2m ) , in L2 (0, τ1 ; H 1 ) weak∗ and L2 (0, τ1 ; L2 ) .

By the trace theorem, (7.2.78) and (7.2.79) imply that also un → u

(7.2.80)

in C([0, τ1 ]; H m ) ;

in turn, this allows us to show that M (un ) → M (u)

(7.2.81)

in

L2 (0, τ1 ; L2 ) ,

which implies that (7.2.82)

f n → f (u) =: f

in

¯ 2m ) , L2 (0, τ1 ; H

and, therefore, that (7.2.83)

N (f n , (un )(m−2) , un+1 ) → N (f, u(m−1) )

in L2 (0, τ1 ; L2 ) .

356

7. Maxwell and von Karman Equations

This allows us to conclude that u solves problem (P). Finally, since u ∈ C([0, τ1 ]; H m+1 ), ∂x2 u ∈ C([0, τ1 ]; H m−1 ) → C([0, τ1 ]; L2m ); thus, M (u) ∈ ¯ 2m ). By Lemma 7.2.2, using interpoC([0, τ1 ]; L2 ) and f (u) ∈ C([0, τ1 ]; H ¯ m+1 ). By means of (7.2.32) lation (see (7.2.17)), also f (u) ∈ C([0, τ1 ]; H ¯ 2m+1 ), so that of Lemma 7.2.3, we further deduce that f (u) ∈ L2 (0, τ1 ; H 2m+1,1 f (u) ∈ Y (τ1 ).  2. Proof of Theorem 7.2.4. We now go back to the proof of Theorem 7.2.4. We proceed in two steps, first proving the well-posedness of problem (P) in P 2m,0 (τ ), for any τ ∈ ]0, T ], and then the existence of such solutions, by means of an approximation argument, which uses the more regular solutions given by Theorem 7.2.5. Proposition 7.2.6. Let u0 , u ˜0 ∈ H m , and ϕ, ϕ˜ ∈ Y 2m,0 (T ). Assume that problem (P) has corresponding solutions u, u ˜ ∈ P 2m,0 (τ ), with f := f (u) 2m,0 and f˜ := f (˜ u) ∈ Y (τ ), for some τ ∈ ]0, T ]. Then, the difference u − u ˜ satisfies the estimate   (7.2.84) u − u ˜P 2m,0 (τ ) ≤ h1 (ρ1 , δ1 , T ) u0 − u ˜0 m + ϕ − ϕ ˜ Y 2m,0 (T ) , where h1 ∈ K, and (7.2.85) (7.2.86)

  ρ1 := max 1, uP 2m,0 (τ ) , ˜ uP 2m,0 (τ ) ,   δ1 := max 1, ϕY 2m,0 (τ ) , ϕ ˜ Y 2m,0 (τ ) .

In particular, there is at most one solution of problem (P) in P 2m,0 (τ ). Regular Approximations. In this step we resort to Theorem 7.2.5 to construct a sequence (un )n≥0 ⊂ P 2m+1,1 (τ ), for some τ ∈ ]0, T ], of approximate solutions to problem (P), which we will then show to have a limit u ∈ P 2m,0 (τ ), which is the desired solution of problem (P). Before doing this, we note that, if u ∈ P 2m,0 (τ ) is a solution of problem (P), we can use equation (7.2.9), together with (7.2.21) of Lemma 7.2.1, to estimate ut in L2 (0, τ ; L2 ), and thus deduce an inequality of the form   (7.2.87) uP 2m,0 (τ ) ≤ F uY 2m,0 (τ ) , ϕY 2m,0 (τ ) , with F ∈ K. This implies that, in order to show that a solution of problem (P) is in P 2m,0 (τ ) for some τ ∈ ]0, T ], it is sufficient to establish an a priori bound on the norm of u in Y 2m,0 (τ ). In fact, (7.2.87) implies that, if u is a solution of problem (P), then (7.2.88)

u ∈ P 2m,0 (τ ) ⇐⇒ u ∈ Y 2m,0 (τ ) .

If u0 = 0, the function u ≡ 0 is the only solution of problem (P), with f (0) ≡ 0, and there is nothing more to prove. Thus, we can assume that

7.2. von Karman’s Equations

357

u0 = 0, and set R := 4 u0 m . Denoting by C∗ the norm of the imbedding P 2m,0 (τ ) → Y 2m,0 (τ ) (recall (7.2.73)), and with h1 as in (7.2.84), we define   (7.2.89) κ(R) := C∗ h1 1 + 4 R, 1 + 2 ϕY 2m,0 (T ) , T . If ϕ ≡ 0, we fix γ ∈ ]0, 1[ such that   (7.2.90) 0 < γ ≤ min u0 m , ϕY 2m,0 (T ) ,

4γ 1−γ

≤

R κ(R)

,

and choose sequences (un0 )n≥0 ⊂ H m+1 and (ϕn )n≥0 ⊂ Y 2m+1,1 (T ), such that, for all n ≥ 0 (7.2.91)

un0 − u0 m ≤ γ n+1 ,

ϕn − ϕY 2m,0 (T ) ≤ γ n+1 ;

note that the first of (7.2.90) and (7.2.91), together with the fact that γ ∈ ]0, 1[, imply that un0 ≡ 0 for all n. If instead ϕ ≡ 0, we replace the first of (7.2.90) with 0 < γ ≤ u0 m , and take ϕn ≡ 0 for all n. In the sequel, we assume ϕ ≡ 0. With un0 and ϕn as data, we resort to Theorem 7.2.5 to determine local solutions un ∈ P 2m+1,1 (τn ) of problem (P ), for some τn ∈ ]0, T ]. The crucial step in the proof of Theorem 7.2.4 consists then in showing that there exists τ ∈ ]0, T ], independent of n, such that, if τn < τ , un can be extended to [0, τ ], with un ∈ P 2m,0 (τ ), and that the sequence (un )n≥0 is bounded in P 2m,0 (τ ). Convergence. The remaining steps for the rest of the argument are then the convergence of the nonlinear terms M and N as in (7.2.81) and (7.2.83), which follow from the strong convergence un → u in Y 2m,0 (τ1 ), as per the second claim of (7.2.78), and (7.2.80). Here, we have that (7.2.92)

∞ 

u − u n

n−1

Y 2m,0 (τ ) ≤ 4 κ(R)

n=1

∞  n=1

γ n = 4 κ(R)

γ , 1−γ

2m,0 (τ ) and this does imply that the sequence n≥0 converges strongly in Y to a limit u. We can then proceed as in the proof of Theorem 7.2.5, with τ1 replaced by τ and (7.2.83) replaced by

(un )

(7.2.93)

N (f n , (un )(m−1) ) → N (f, u(m−1) )

in L2 (0, τ ; L2 ) ,

and deduce that u is the desired solution of problem (P) in P 2m,0 (τ ). This concludes the proof of Theorem 7.2.4.  3. Higher Regularity. Just as for problem (H), higher regularity results for problem (P) can be established by a suitable generalization of Theorem 7.2.5. Theorem 7.2.6. Let k ≥ 0, and assume that u0 ∈ H m+k , ϕ ∈ Y 2m+k,k (T ). There is τk ∈ ]0, T ], such that problem (P) admits a unique solution u ∈ 2m+k,k P 2m+k,k (τk ), with f (u) ∈ Y (τk ).

358

7. Maxwell and von Karman Equations

The proof of this theorem is based on the time-independent a priori estimates established in the proof of Proposition 7.2.7 below. Again, note that Theorems 7.2.4 and 7.2.5 correspond to Theorem 7.2.6 when k = 0 and k = 1, with τ0 = τ , and that it is important to realize that the regularity result of Theorem 7.2.6 is uniform in k, in the sense that inf k≥0 τk ≥ τ . Roughly speaking, this means that increasing the regularity of the data does not decrease the life-span of the solution. Proposition 7.2.7. Let k ≥ 0, and u0 ∈ H m+k , ϕ ∈ Y 2m+k,k (T ). Assume that problem (P) has a corresponding solution u ∈ P 2m,0 (τ ) ∩ P 2m+k,k (τ  ), with 0 < τ  < τ ≤ T . There exists Λk , depending on τ but not on τ  , such that (7.2.94)

uP 2m+k,k (τ  ) ≤ Λk .

Consequently, u ∈ P 2m+k,k (τ ). 4. Almost Global Existence. We conclude with an almost global existence result for problem (P), in the same spirit of the one given in Theorem 7.2.3 for problem (H); the proof, however, is somewhat different. Theorem 7.2.7. In the same assumptions of Theorem 7.2.4, given arbitrary T > 0 there is δ > 0, depending only on ϕY 2m,0 (T ) , such that, if u0 m ≤ δ, then problem (P) admits a unique solution u ∈ P 2m,0 (T ), with f (u) ∈ 2m,0 Y (T ). Proof. 1) We follow a standard ODE extension method, similar to the one described in the first sections of Chapter 4. The local existence Theorem 7.2.4 yields a solution u ∈ P 2m,0 (τ ) of problem (P), for some τ ∈ ]0, T ]. In order to show that we can extend the local solution of problem (P) to a global one, it is sufficient to prove that we can bound the norm of u in C([0, τ ]; H m ), τ ∈ ]0, T ], independently of τ . To this end, we set Cϕ := C ϕm−1 C([0,T ];H m ) , where C is the largest between the universal constants appearing in (7.2.101) and (7.2.102) below, we define

 T 2 1 (7.2.95) Mϕ := exp 2 Cϕ T + Cϕ ϕ2m dθ , 0

and we claim that there is δ ∈ ]0, 1] such that, for all τ ∈ ]0, T ] for which problem (P) has a solution u ∈ P 2m,0 (τ ), with u(0)m ≤ δ, u satisfies the estimate (7.2.96)

max u(t)m ≤ Mϕ δ .

0≤t≤τ

Since the right side of (7.2.96) is independent of τ , this yields the desired time-independent estimate on u( · )m .

7.2. von Karman’s Equations

359

2) We prove our claim by contradiction. Thus, assume that for all δ ∈ ]0, 1] there is τδ ∈ ]0, T ] such that problem (P) has a solution u ∈ P 2m,0 (τδ ), with u(0)m ≤ δ but (7.2.97)

Mδ := max u(t)m > Mϕ δ . 0≤t≤τδ

If Mδ > 2 Mϕ δ, noting that u(0)m ≤ δ < Mϕ δ we deduce by continuity that there is θδ ∈ ]0, τδ ] such that, for all t ∈ [0, θδ ], (7.2.98)

u(t)m ≤ 2 Mϕ δ = u(θδ )m .

If instead Mδ ≤ 2 Mϕ δ, we set θδ := τδ , so that the first inequality of (7.2.98) still holds for all t ∈ [0, θδ ]. We now multiply equation (7.2.9) in L2 by 2 (Δm u + u), to obtain (7.2.99)

d u2m + 2 u22m + 2 u2m dt = 2N (f, u(m−1) ) + N (ϕ(m−1) , u), Δm u + u ,

where, as usual, f := f (u). By (7.2.21), (7.2.31) and interpolation, 2 |N (f, u(m−1) ), Δm u| ≤ 2 N (f, u(m−1) )0 u2m ≤ C f m+1 um−1 u2m m+1

(7.2.100)

≤ C um−1 um u2m m m+1 2(m−1)

≤ C um

u22m .

Likewise, (7.2.101) 2 |N (ϕ(m−1) , u), Δm u| ≤ 2 N (ϕ(m−1) , u)0 u2m ≤ C ϕm−1 m+1 um+1 u2m (m−1)2 m

≤ C ϕm

m−1

m−1

≤ Cϕ2 ϕ22m u2m + u22m . Next, recalling (7.2.2), by (7.2.26), 2 |N (f, u(m−1) ) + N (ϕ(m−1) , u), u| (7.2.102)

m+1

m m ϕ2m umm u2m

≤ 2 |I(f, u(m) )| + 2 |I(ϕ(m−1) , u, u)| m−1 u2 ≤ C f m um m + C ϕm m 2 ≤ C u2m m + Cϕ um .

360

7. Maxwell and von Karman Equations

Inserting (7.2.100), (7.2.101) and (7.2.102) into (7.2.99), and recalling the inequality of (7.2.98), we obtain that, for t ∈ [0, θδ ], and suitable constant C∗ , (7.2.103) d u2m + u22m + u2m dt     ≤ C∗ u2(m−1) u22m + u2m + Cϕ 1 + Cϕ ϕ22m u2m m     ≤ C∗ (2Mϕ δ)2(m−1) u2m + u22m + Cϕ 1 + Cϕ ϕ22m u2m . Thus, if we choose δ ∈ ]0, 1] so small that (7.2.104)

C∗ (2Mϕ δ)2(m−1) ≤ 1 ,

we deduce from (7.2.103) that, for all t ∈ [0, θδ ]   d (7.2.105) u2m ≤ Cϕ 1 + Cϕ ϕ22m u2m , dt and, by Gronwall’s inequality,

 T (7.2.106) u(t)2m ≤ u0 2m exp Cϕ T + Cϕ ϕ22m dθ ≤ Mϕ2 δ 2 . 0

If θδ = τδ , (7.2.106) contradicts (7.2.97), while if θδ < τδ , (7.2.106) for t = θδ contradicts (7.2.98). Consequently, (7.2.96) holds, and we conclude the proof of Theorem 7.2.7. 

List of Function Spaces

We report a list of all the function spaces we have introduced in this book. In the following, X is a real Banach space, Ω is a domain of RN , with boundary ∂Ω, and Q = ]a, b[ ×Ω (or Q = ]a, b[ ×RN ). When Ω = RN , the explicit reference to Ω is omitted; e.g., Lp := Lp (RN ). For each space, we indicate either the page where it has been first introduced or a reference where a definition of the space can be found.

*****

AC([a, b]; X) Space of absolutely continuous functions f : [a, b] → X (p. 62). C m (Ω) Space of functions f : Ω → R, which have continuous derivatives of order up to m (p. 2). Cbm (Ω) := {f ∈ C m (Ω) | max sup |∂ j f (x)| < +∞} (p. 2). 0≤j≤m x∈Ω

C0m (Ω) := {f ∈ C m (Ω) | supp(f ) is compact} (p. 2). C m (Ω): Space of functions which are restrictions to Ω of functions in C m (RN ) (p. 2). Cbm (Ω): Space of functions which are restrictions to Ω of functions in Cbm (RN ) (p. 3). C m,α (Ω) := {f ∈ C m (Ω) | Hα (∂ m f ) < +∞}, Hα defined in (1.3.1) (p. 7). 361

362

List of Function Spaces

C m,α (Ω) := C m,α (Ω) ∩ Cbm (Ω) (p. 7).  C m+α/2,2m+α(Q) := f ∈ C(Q) | ∂tk ∂xλ f ∈ C(Q) , 2k + |λ| ≤ 2m ,  ˜ α (∂ k ∂ λ f ) < +∞ , 2k + |λ| = 2m , H ˜ α defined in (1.3.4) (p. 8). H t x   C˜bm (Q) := f ∈ Cb (Q) | ∂tk ∂xλ f ∈ Cb (Q) , 2k + |λ| ≤ 2m (p. 8). C m+α/2,2m+α(Q) := C˜bm (Q) ∩ C m+α/2,2m+α(Q) (p. 8). C([a, b]; X): Space of strongly continuous functions f : [a, b] → X (p. 60). C m ([a, b]; X) := {u ∈ C([a, b]; X) | ∂tj u ∈ C([a, b]; X) , 0 ≤ j ≤ m} (p. 64). C m ([a, b]; X, Y ) := {u ∈ C([a, b]; X) | ∂tm ∈ C([a, b]; Y )} (p. 64). C([a, +∞[; X): Space of strongly continuous functions f : [a, +∞[ → X (p. 61). Cb ([a, +∞[; X): Space of strongly continuous and bounded functions f : [a, +∞[ → X (p. 61). Cbm ([a, +∞[; X): Space of continuously differentiable functions f : [a, +∞[ → X, with bounded derivatives, of order up to m (p. 64). Cbm ([a, +∞[; X, Y ) := {u ∈ Cb ([a, +∞[; X) | ∂tm u ∈ Cb ([a, +∞[; Y )} (p. 64). Cw ([a, b]; X): Space of weakly continuous functions f : [a, b] → X (p. 61). D(Ω): Space of test functions in Ω (Rudin [139, ch. 6]). D  (Ω): Space of distributions in Ω (Rudin [139, ch. 6]). D([a, b]; X): Space of test functions f : [a, b] → X (Lions and Magenes [101, ch. 1]). D  (]a, b[; X) := L(D(]a, b[); X) (p. 67). Gs (T ) := H s+1 × H s × Vs−1 (T ) (p. 122). Gs (∞) := H s+1 × H s × Vs−1 (∞) (p. 153). H m (Ω) = W m,2 (Ω) := {u ∈ L2 | ∂ α u ∈ L2 , |α| ≤ m} (p. 13). ˜ m (Ω) := {u ∈ L∞ (Ω) | ∇u ∈ H m−1 (Ω)} (m ≥ 1) (p. 14). H H s (Ω) := (H s+1 (Ω), H s (Ω))s− s (Lions and Magenes [101, ch. 1]). H s (RN ) := {f ∈ S  | (1 + | · |2 )s/2 fˆ ∈ L2 } (p. 15). H 1/2 (∂Ω): Space of traces on ∂Ω of functions in H 1 (Ω) (p. 16).

List of Function Spaces

363

H01 (Ω) := {u ∈ H 1 (Ω) | u |∂Ω = 0} (p. 16). H0m (Ω) := {u ∈ H m (Ω) |

∂j ∂ν j

u = 0,

0 ≤ j ≤ m − 1} (p. 16).

m (Ω) := {u ∈ H m (Ω) | (− Δ)k u ∈ H 1 (Ω) , HΔ 0

H∗m := {f ∈ H m | μr f ∈ H m−1 ,

0 ≤ k ≤  m−1 2 } (p. 32).

1 ≤ r ≤ N } (p. 53).

H m (a, b; X, Y ) := W m,2 (a, b; X, Y ) (p. 68). H m (a, b; X) := W m,2 (a, b; X, X) (p. 68). Hh,k (T ) := C([0, T ]; H h ) ∩ C 1 ([0, T ]; H k ) (p. 348). K := {f ∈ C 1 (Rm ≥0 ; R≥0 ) | ∂j f ≥ 0 ,

1 ≤ j ≤ m} (p. 4).

K0 := {f ∈ K | f (0) = 0} (p. 4). K∞ := {f ∈ C(R>0 ; R>0 ) | lim f (r) = +∞} (p. 4). r→0

L(X, Y ) := {f : X → Y | f is linear continuous} (p. 5). ' p Lp (Ω) := {f : Ω → R | f is measurable , Ω |f (x)| dx < +∞} (up to equivalence on sets of measure zero) (p. 9). Lp (Γ): Lp spaces on a (N − 1)-dimensional submanifold Γ ⊂ RN (p. 9). Lp (a, b; X) := {f : ]a, b[ → X | f is strongly measurable , 'b p a u(t)X dt < +∞} (p ∈ R≥1 ) (p. 60). L∞ (a, b; X) := {f : ]a, b[ → X | f is strongly measurable , sup ess u(t)X < +∞} (p. 60). a