Recent developments in the Navier-Stokes problem

Recent developments in the Navier-Stokes problem

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Description

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.
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Product details

  • Hardback | 408 pages
  • 157.48 x 236.22 x 30.48mm | 703.06g
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • 1584882204
  • 9781584882206
  • 2,613,538

Table of contents

INTRODUCTION
What is this Book About?
SOME RESULTS OF REAL HARMONIC ANALYSIS
Real Interpolation, Lorentz Spaces, and Sobolev Embedding
Besov Spaces and Littlewood-Paley Decomposition
Shift-Invariant Banach Spaces of Distributions and Related Besov Spaces
Vector-Valued Integrals
Complex Interpolation, Hardy Space, and Calderon-Zygmund Operators
Vector-Valued Singular Integrals
A Primer to Wavelets
Wavelets and Functional Spaces
The Space BMO
A GENERAL FRAMEWORK FOR SHIFT-INVARIANT ESTIMATES FOR THE NAVIER-STOKES EQUATIONS
Weak Solutions for the Navier-Stokes Equations
Divergence-Free Vector Wavelets
The Mollified Navier-Stokes Equations
CLASSICAL EXISTENCE RESULTS FOR THE NAVIER-STOKES EQUATIONS
The Leray Solutions for the Navier-Stokes Equations
Kato's Mild Solutions for the Navier-Stokes Equations
NEW APPROACHES OF MILD SOLUTIONS
The Mild Solutions of Koch and Tataru: The Space BMO-1
Generalization of the Lp Theory: Navier-Stokes and Local Measures
Further Results on Local Measures
Regular Initial Values
Besov Spaces of Negative Order
Pointwise Multipliers of Negative Order
Further Adapted Spaces for the Navier-Stokes Equations
Cannone's Approach of Self-Similarity
DECAY AND REGULARITY RESULTS FOR WEAK AND MILD SOLUTIONS
Space-Analytic Solutions of the Navier-Stokes Equations
Space Localization and Navier-Stokes Equations
Time Decay for the Solutions to the Navier-Stokes Equations
Uniqueness of Ld Solutions
Further Results on Uniqueness of Mild Solutions
Stability and Lyapunov Functionals
LOCAL ENERGY INEQUALITIES FOR THE NAVIER-STOKES EQUATIONS ON R3
The Caffarelli, Kohn, and Nirenberg Regularity Criterion
On the Dimension of the Set of Singular Points
Local Existence (in Time) of Suitable Locally Square Integrable Weak Solutions
Global Existence of Suitable Locally Square Integrable Weak Solutions
Leray's Conjecture on Self-Similar Singularities
CONCLUSION
Singular Initial Values
REFERENCES
BIBLIOGRAPHY
INDEX NOMINUM
INDEX RERUM
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