Graduate Studies in Mathematics
Volume: 192;
2018;
224 pp;
Hardcover
MSC: Primary 35; 76;
Print ISBN: 978-1-4704-3096-2
Product Code: GSM/192
List Price: $83.00
AMS Member Price: $66.40
MAA Member Price: $74.70
Electronic ISBN: 978-1-4704-4778-6
Product Code: GSM/192.E
List Price: $83.00
AMS Member Price: $66.40
MAA Member Price: $74.70
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Supplemental Materials
Lectures on Navier-Stokes Equations
Share this pageTai-Peng Tsai
This book is a graduate text on the
incompressible Navier-Stokes system, which is of fundamental
importance in mathematical fluid mechanics as well as in engineering
applications. The goal is to give a rapid exposition on the existence,
uniqueness, and regularity of its solutions, with a focus on the
regularity problem. To fit into a one-year course for students who
have already mastered the basics of PDE theory, many auxiliary results
have been described with references but without proofs, and several
topics were omitted. Most chapters end with a selection of problems
for the reader.
After an introduction and a careful study of weak, strong, and mild
solutions, the reader is introduced to partial regularity. The
coverage of boundary value problems, self-similar solutions, the
uniform \(L^3\) class including the celebrated
Escauriaza-Seregin-Šverák Theorem, and axisymmetric
flows in later chapters are unique features of this book that are less
explored in other texts.
The book can serve as a textbook for a course, as a self-study
source for people who already know some PDE theory and wish to learn
more about Navier-Stokes equations, or as a reference for some of the
important recent developments in the area.
Readership
Graduate students and researchers interested in incompressible Navier-Stokes equations.
Reviews & Endorsements
The book is an excellent contribution to the literature concerning the mathematical analysis of the incompressible Navier-Stokes equations. It provides a very good introduction to the subject, covering several important directions, and also presents a number of recent results, with an emphasis on non-perturbative regimes. The book is well written and both beginners and experts will benefit from it. It can also provide great material for a graduate course.
-- Vladimir Šverák, University of Minnesota
Table of Contents
Table of Contents
Lectures on Navier-Stokes Equations
- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Notation xi12
- Chapter 1. Introduction 114
- Chapter 2. Steady states 1932
- 2.1. Weak solutions 1932
- 2.2. Small-large uniqueness 2235
- 2.3. Existence for zero boundary data by the Galerkin method 2336
- 2.4. Existence for zero boundary data by the Leray-Schauder theorem 2538
- 2.5. Nonuniqueness 2942
- 2.6. 𝐿^{𝑞}-theory for the linear system 3245
- 2.7. Regularity 3851
- 2.8. The Bogovskii map 4558
- 2.9. Notes 4760
- Problems 4861
- Chapter 3. Weak solutions 5164
- Chapter 4. Strong solutions 6982
- Chapter 5. Mild solutions 7992
- Chapter 6. Partial regularity 93106
- Chapter 7. Boundary value problem and bifurcation 107120
- 7.1. Existence: A priori bound by a good extension 108121
- 7.2. Existence: A priori bound by contradiction 112125
- 7.3. The Korobkov-Pileckas-Russo approach for 2D BVP 116129
- 7.4. The bifurcation problem and degree 123136
- 7.5. Bifurcation of the Rayleigh-Bénard convection 128141
- 7.6. Bifurcation of Couette-Taylor flows 133146
- 7.7. Notes 139152
- Problems 140153
- Chapter 8. Self-similar solutions 141154
- Chapter 9. The uniform 𝐿³ class 173186
- Chapter 10. Axisymmetric flows 189202
- Bibliography 211224
- Index 223236
- Back Cover Back Cover1239