Recent Developments in the Navier-Stokes Problem

ISBN-13: 9781584882206 / Angielski / Twarda / 2002 / 408 str.

P.G. Lemarie-Rieusset; Pierre Gilles Lemarie; P. G. Lemarie-Rieusset

Opis

Despite their importance and the amount of research published on the t heory of the Navier-Stokes equations, some very elementary and fundame ntal questions remain unanswered. In this book, the author explores tw o main topics: the search for mild solutions using the tools of real h armonic analysis, and the 1982 regularity criterion of Caffanelli et a l., which has produced many results on the existence or non-existence of solutions. The book includes a self-contained introduction to real harmonic analysis, addresses both strong solutions by direct methods a nd weak solutions by indirect methods, and presents new tools, includi ng bessor spaces and wavelets.

Spis treści

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

O autorze

Pierre Gilles Lemarie-Rieusset