Recent developments in the Navier-Stokes problem / P.G. Lemarié-Rieusset.

Format:

Book

Author/Creator:

Lemarié, Pierre Gilles, 1960-

Contributor:

Alumni and Friends Memorial Book Fund.

Series:

Chapman & Hall/CRC research notes in mathematics series ; 431.

Chapman & Hall/CRC research notes in mathematics series ; 431

Language:

English

Subjects (All):

Navier-Stokes equations.

Physical Description:

xii, 395 pages ; 24 cm.

Place of Publication:

Boca Raton, Fla. : Chapman & Hall/CRC, [2002]

Summary:

The Navier-Stokes equations: fascinating, fundamentally important, and challenging,. Although many questions remain open, progress has been made in recent years. The regularity criterion of Caffarelli, Kohn, and Nirenberg led to many new results on existence and non-existence of solutions, and the very active search for mild solutions in the 1990's culminated in the theorem of Koch and Tataru that, in some ways, provides a definitive answer. Recent Developments in the Navier-Stokes Problem brings these and other advances together in a self-contained exposition presented from the perspective of real harmonic analysis. The author first builds a careful foundation in real harmonic analysis, introducing all the material needed for his later discussions. He then studies the Navier-Stokes equations on the whole space, exploring previously scattered results such as the decay of solutions in space and in time, uniqueness, self-similar solutions, the decay of Lebesgue or Besov norms of solutions, and the existence of solutions for a uniformly locally square integrable initial value. Many of the proofs and statements are original and, to the extent possible, presented in the context of real harmonic analysis. Although the existence, regularity, and uniqueness of solutions to the Navier-Stokes equations continue to be a challenge, this book is a welcome opportunity for mathematicians and physicists alike to explore the problem's intricacies from a new and enlightening perspective.

Contents:

Part 1 Some results of real harmonic analysis 13

Chapter 2 Real interpolation, Lorentz spaces and Sobolev embeddings 15

Chapter 3 Besov spaces and Littlewood-Paley decomposition 23

Chapter 4 Shift-invariant Banach spaces of distributions and related Besov spaces 31

Chapter 5 Vector-valued integrals 39

Chapter 6 Complex interpolation, Hardy space and Calderon-Zygmund operators 47

Chapter 7 Vector-valued singular integrals 57

Chapter 8 A primer to wavelets 67

Chapter 9 Wavelets and functional spaces 79

Chapter 10 The space BMO 91

Part 2 A general framework for shift-invariant estimates for the Navier-Stokes equations 103

Chapter 11 Weak solutions for the Navier-Stokes equations 105

Chapter 12 Divergence-free vector wavelets 115

Chapter 13 The mollified Navier-Stokes equations 123

Part 3 Classical existence results for the Navier-Stokes equations 133

Chapter 14 The Leray solutions for the Navier-Stokes equations 135

Chapter 15 The Kato theory of mild solutions for the Navier-Stokes equations 145

Part 4 New approaches to mild solutions 157

Chapter 16 The mild solutions of Koch and Tataru 159

Chapter 17 Generalization of the L[superscript p] theory: Navier-Stokes and local measures 171

Chapter 18 Further results for local measures 179

Chapter 19 Regular initial values 189

Chapter 20 Besov spaces of negative order 197

Chapter 21 Pointwise multipliers of negative order 205

Chapter 22 Further adapted spaces for the Navier-Stokes equations 221

Chapter 23 Cannone's approach of self-similarity 233

Part 5 Decay and regularity results for weak and mild solutions 245

Chapter 24 Solutions of the Navier-Stokes equations are space-analytical 247

Chapter 25 Space localization and Navier-Stokes equations 255

Chapter 26 Time decay for the solutions to the Navier-Stokes equations 267

Chapter 27 Uniqueness of L[superscript d] solutions 277

Chapter 28 Further results on uniqueness of mild solutions 289

Chapter 29 Stability and Lyapunov functionals 303

Part 6 Local energy inequalities for the Navier-Stokes equations on IR[superscript 3] 315

Chapter 30 The Caffarelli, Kohn, and Nirenberg regularity criterion 317

Chapter 31 On the dimension of the set of singular points 331

Chapter 32 Local existence (in time) of suitable local square-integrable weak solutions 341

Chapter 33 Global existence of suitable local square-integrable weak solutions 353

Chapter 34 Leray's conjecture on self-similar singularities 363

Chapter 35 Singular initial values 375.

Notes:

Includes bibliographical references (pages 383-390) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

1584882204

OCLC:

48857921