1st Edition

Recent developments in the Navier-Stokes problem

By Pierre Gilles Lemarie-Rieusset Copyright 2002
    408 Pages
    by CRC Press

    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.

    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

    Biography

    Pierre Gilles Lemarie-Rieusset