Linear and Quasilinear Parabolic Systems: Sobolev Space Theory

241 Pages·2020·2.122 MB·English

by  David Hoff

Mathematical
Surveys
and
Monographs
Volume 251
Linear and Quasilinear
Parabolic Systems
Sobolev Space Theory
David Hoff
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
Linear and Quasilinear
Parabolic Systems
Sobolev Space Theory
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
1100..11009900//ssuurrvv//225511
Mathematical
Surveys
and
Monographs
Volume 251
Linear and Quasilinear
Parabolic Systems
Sobolev Space Theory
David Hoff
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
EDITORIAL COMMITTEE
Robert Guralnick, Chair Natasa Sesum
Bryna Kra ConstantinTeleman
Melanie Matchett Wood
2010 Mathematics Subject Classification. Primary 35K51, 35K61.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-251
Library of Congress Cataloging-in-Publication Data
Names: Hoff,DavidCharles,1948–author.
Title: Linearandquasilinearparabolicsystems: sobolevspacetheory/DavidHoff.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2021]
Series: Mathematicalsurveysandmonographs,0076-5376;volume251|Includesbibliographical
referencesandindex.
Identifiers: LCCN2020036612|ISBN9781470461614(paperback)|ISBN9781470463205(ebook)
Subjects: LCSH: Differential equations, Partial. | Differential equations, Parabolic. | AMS:
Partialdifferentialequations–Parabolicequationsandsystems[Seealso35Bxx,35Dxx,35R30,
35R35,58J35]–Initial-boundaryvalueproblemsforsecond-orderparabolicsystems. |Partial
differentialequations–Parabolicequationsandsystems[Seealso35Bxx,35Dxx,35R30,35R35,
58J35]–Nonlinearinitial-boundaryvalueproblemsfornonlinearparabolicequations.
Classification: LCCQA377.H622021|DDC515/.3534–dc23
LCrecordavailableathttps://lccn.loc.gov/2020036612
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10.1090/surv/251/01
To my family–may they all thrive!
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Contents
Preface ix
Chapter 1. Introduction 1
1.1 Weak forms: An Example 1
1.2 Basic Notations and Definitions 2
1.3 Weak Measurability 4
1.4 Vector-Valued Lp Spaces 5
1.5 Geometric Properties of Open Sets in Rn 6
Chapter 2. Differential Equations in Hilbert Space 11
2.1 Basic Existence and Uniqueness 11
2.2 Weak Absolute Continuity 16
2.3 Higher Order Regularity 24
2.4 Proof of Theorem 2.7 29
2.5 Incompatible Data and Initial Layer Regularization 40
2.6 Systems with Symmetry 43
2.7 Spectral Representations 49
Chapter 3. Linear Parabolic Systems: Basic Theory 59
3.1 Linear Parabolic Systems and Their Weak Forms 60
3.2 Preliminaries 63
3.3 The Basic Existence Theorem 67
3.4 Parabolic Systems with Symmetry 72
Chapter 4. Elliptic Systems: Higher Order Regularity 77
4.1 Statements of Results 78
4.2 Interior Regularity: Proof of Theorem 4.1 83
4.3 Global Regularity: Proof of Theorem 4.2 93
Chapter 5. Parabolic Systems: Higher Order Regularity 99
5.1 Boundary Conditions, Coefficients and Inhomogeneities 99
5.2 Higher Order Regularity for Parabolic Systems 111
5.3 Incompatible Data and Initial Layer Regularization 122
Chapter 6. Applications to Quasilinear Systems 127
6.1 A Quasilinear System in Hilbert Space 127
vii
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viii CONTENTS
6.2 Navier–Stokes Equations of Incompressible Flow 135
6.3 Nonlinearities with Polynomial Growth 139
6.4 Higher Order Regularity for Quasilinear Systems 148
6.5 Global Existence and Stability of Steady-States 166
6.6 Invariant Regions for Quasilinear Systems 172
6.7 A Global Existence Result for Systems in R2 and R3 177
Appendix. Selected Topics in Analysis 187
A.1 Density and Imbedding Theorems 187
A.2 Weak Solutions of Ordinary Differential Equations 190
A.3 Applications of Spectral Measures 193
A.4 Boundary Measures: Proofs of Theorems 3.1–3.3 197
A.5 The Spaces Lp,q 201
A.6 A Representation Theorem for Boundary Integrals 214
Notes and References 219
Bibliography 221
Index 225
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Preface
In this monograph I present a systematic theory of weak solutions in Hilbert-
Sobolev spaces of initial-boundary value problems for parabolic systems of partial
differential equations. The goal is to provide a practical reference for fundamental
results often considered well–known but often difficult to access in the desired
generality, as well as to provide instructional material for students preparing for
research in applied analysis.
The development begins in Chapter 2 with an abstract theory in which H and
0
H are separable Hilbert spaces with H dense and continuously included in H
1 1 0
and A is a mapping of the time axis into the set of bilinear forms on H ×H . The
1 1
problem is to construct a mapping u for which the equation
(cid:90) t
(cid:0) (cid:1)
(cid:104)u(t),v(cid:105) =(cid:104)ς ,v(cid:105) + −A(s)(u(s),v)+F(s)·v ds
H0 0 H0
0
holdsforalltestfunctionsv ∈H ,foralltimetandforgivenς ∈H andF mapping
1 0 0
time into the dual H∗. This abstract theory is then applied in Chapters 3 and 5
1
to initial–boundary value problems for linear parabolic systems with H a closed
0
subspace of [L2(Ω)]N for a given open set Ω ⊂ Rn (not assumed to be bounded)
and H a closed subspace of [H1(Ω)]N whose elements satisfy essential boundary
1
conditions. The basic theory of Chapter 3 addresses questions of measurability,
continuity and energy conservation in the absence of regularity, and includes special
considerations for problems with symmetry and representations of solutions in
terms of spectral measures. Higher-order regularity of solutions is discussed in
Chapter 5 in three related but categorically different formulations: the dynamical
systems (Bochner space) formulation in which solutions are mappings from the
time axis into various Sobolev spaces, the weak derivative formulation in which
solutions are locally integrable on the space-time cross product, and the continuous
derivative formulation familiar from calculus. Chapter 5 also includes a discussion
of instantaneous regularization of solutions whose data fail to satisfy compatibility
conditions. (Chapter 4 is a brief excursion into higher order regularity theory for
elliptic systems, required for the application in Chapter 5.)
This linear theory is then applied in chapter 6 to quasilinear systems in which
A and F may depend on u. A basic theory of solutions with minimal regularity
is given, corresponding to the basic theory for linear problems. This includes the
theorems of Leray and Hopf for the Navier-Stokes equations as well as results for
more general problems in which nonlinearities have specific, controlled growth as
|u|→∞. Results for more highly regular solutions of quasilinear systems are then
derived by iteration from the corresponding regularity theory for linear problems.
This includes local-in-time existence with large compatible data, global existence
for data near an attracting rest point, invariant regions for quasilinear systems, and
ix
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