Linear and Quasi Linear Evolution Equations in Hilbert Spaces: Exploring the Anatomy of Integers

398 Pages·2012·4.851 MB·English

by  Pascal Cherrier, Albert Milani

Linear and Quasi-linear
Evolution Equations
in Hilbert Spaces
Pascal Cherrier
Albert Milani
Graduate Studies
in Mathematics
Volume 1,33
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
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10987654321 171615141312
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 KA'ro,
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. Holder Spaces
§1.4. Lebesgue Spaces
§1.5. Sobolev Spaces
§1.6. Orthogonal Bases in H'(R N)
§1.7. Sobolev Spaces Involving Time
Chapter 2. Linear Equations
§2.1. Introduction
§2.2. The Hyperbolic Cauchy Problem
§2.3. Proof of Theorem 2.2.1
§2.4. Weak Solutions
§2.5. The Parabolic Cauchy Problem
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
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 233
§5.2. Convergence uhyp(t) -} usta 234
Convergence upar (t) -}
§5.3. 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) 6 utt + a ut - aij (t, x) 3 & u = f (t, x) ,
and their quasi-linear counterpart
(0.0.2) 6 utt + a ut - aij (t, x, u, ut, VU) 3Z3ju = f (t, x)
In (0.0.1) and (0.0.2), 6 and a are non-negative parameters; u = u(t, x),
t > 0, x E 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 E) aij (.. ) i j is positive definite.
We distinguish the following three cases.
(1) e > 0 and v = 0. Then, (0.0.1) and (0.0.2) are hyperbolic equations;
in particular, when 6 = 1, they reduce to
(0-0-3) utt - aij o9jo9ju = f ,
and when aij ( ) = S2j (the so-called Kronecker S, defined by
82j = 0 if i =,4 j, and Szj = 1 if i = j), (0.0.3) further reduces to the
classical wave equation
(0.0.4) utt - D u = f .
(2) 6 = 0 and v > 0. Then, (0.0.1) and (0.0.2) are parabolic equations;
in particular, when a = 1, they reduce to
ut-az;aza,u=f,
(0.0.5)
ix