Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems

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G.P. Galdi

An Introduction to the

Second Edition

Mathematical Theory of the

Navier-Stokes Equations

Steady-State Problems

ISSN -

ISBN 978-0-387-09619-3

DOI 10.1007/978- - - -

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e-ISBN 978-0-387-09620-9

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Department of Mechanical Engineering

and Materials Science

University of Pittsburgh

G.P. Galdi

O’Hara Street 3700

15261 Pittsburgh Pennsylvania

USA

galdi@pitt.edu

I dedicate this work to four generations of loving pe ople:

My parents, Gino and Elena;

My sister, Marisa;

My children, Adriana, Giovanni, Elena, Lisa,

Francesca and Sierra;

My grandchildren, Chiara, Davide, Soﬁa and Mir ko.

Preface

This boo k is the new edition of the origi nal two-volume book, published in

1994, and of its corrected second printing published in 1998. The ﬁrst volume

dealt with l inearized problems, while the second one was dedicated to fully

nonli near problems.

The current edition is new in many signiﬁcant ways.

In the ﬁrst place, the two volumes have been merged into one. This seems

to me a more appropriate choice that on the one hand, provides a na tural logi-

cal unity to the subject, and on the other hand, furnishes a better presentation

of the topics. In fact, nonlinear problems cannot be addressed properly with-

out a careful analysis of their suitable linear counterparts, and, conversely,

linearized problems can ﬁnd full justiﬁcation only as approximati ons of the

complete nonlinear model.

In the second place, I have added two entirely new chapters (Chapter VIII

and Chapter XI). The motivation for this additio n comes from the increasing

eﬀort that mathematicians, especially over the past decade, have devoted

to problems describing the interaction of a viscous liquid with rigid bodies.

For this reason, I dedicated the above chapters to a systematic and updated

analysis of a fundamental question of liquid–solid interaction, namely, the

steady ﬂow of a viscous liquid around a body that is allowed to translate and

to rotate. In the years 2003 through 2010, over ﬁfty relevant mathematical

papers, directly or indirectly dedicated to this subject, have been published.

Therefore, I deem it very useful for the interested researcher to have a place

where all signiﬁcant basic results are collected and treated in an orga nized

and detailed fashion.

Furthermore, several important new contributions to the ﬁeld that were

published after 1998 have led me to upda te numerous sections extensively, as

well as to add other new ones, not to mention the corresponding substantial

increase in the number of bibliographic items. Among the above contribu-

tions, I would like to point out especially those dedicated to the regularity

of solutions to the nonlinear problem in arbitrary dimension, as well as those

concerning the asymptotic behavior in two-dimensional exterior domains.

vii

viii

Another new feature of this edition is that most of the proofs given in

the previous editions, not only for the main results but also for those on the

periphery, have been clariﬁed either through a sim pliﬁcation or else through

an extended treatment. For completeness, I also have included the proof s of

several basic theorems that were not provided in the previous editions.

Finally, again for the reader’s sake, I have included an introductory section

collecting the basic properties of Banach spaces and related results that are

very often referenced in the text.

Last, but no t least, I take this opportunity to convey my warmest thanks

to all my colleagues from whose collaboration I have beneﬁted enormously in

writing this new edition, and in particular to Josef Bemelmans, Paul Deuring,

Reinhard Farwig, Mads Kyed, Anne Robertson (who is no t “just” a collabo-

rator), Ana Silvestre, Christian Simader, and Hermann Sohr.

In conclusion, I ho pe that the interested scientist will enjoy this b ook and

derive great beneﬁt from reading it just as I did while writing it.

Pittsburgh, Beechwo od Blvd

April 2011 G.P. Galdi

Preface

Contents

I Steady-State Solutions of th e Navier–Stokes Equations:

Statement o f the P roblem and Open Questions . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I.1 Flow in Bounded Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I.2 Flow in Exterior Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

I.2.1 Three-Dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

I.2.2 Plane Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

I.3 Flow in Regions with Unbounded Boundaries . . . . . . . . . . . 17

II Basic Function Spaces and Related Inequalities . . . . . . . . . . . 25

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

II.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

II.1.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

II.1.2 Banach Spaces and their Relevant Properties. . . . . . . . . . 29

II.1.3 Spaces of Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . 35

II.1.4 Classes of Domains and their Properties . . . . . . . . . . . . . . 36

II.2 The Lebesgue Spaces L

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

II.3 The So bolev Spaces W

m,q

and Embedding Inequalities . . . 48

II.4 Boundary Inequalities and the Trace of Functions of W

m,q

II.5 Further Inequalities a nd Compactness Criteria in W

m,q

. . 69

II.6 The Homogeneous Sobolev Spaces D

m,q

and Embedding

Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

II.7 Approximation of Functions f rom D

m,q

by Smooth

Functions and Characterization of Space D

m,q

. . . . . . . . . . 102

II.8 The Normed Dual of D

m,q

(Ω). The Spaces D

−m,q

. . . . . . . 109

II.9 Pointwise behavior at Large Distances o f Functions from

1,q

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

II.10 Boundary Trace of Functions from D

m,q

) . . . . . . . . . . . 121

II.11 Some Integral Transforms and Related Inequalities . . . . . . 125

II.12 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134