Recent Developments in the Navier-Stokes Problem

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September 2000



Despite their importance and the amount of research published on the theory of the Navier-Stokes equations, some very elementary and fundamental questions remain unanswered. This book explores two main topics: the search for mild solutions using the tools of real harmonic analysis, and the 1982 regularity criterion of Caffarelli et al.


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
EAN: 9781584882206
ISBN: 1584882204
Untertitel: 'Chapman & Hall/CRC Research No'. Sprache: Englisch.
Erscheinungsdatum: September 2000
Seitenanzahl: 408 Seiten
Format: gebunden
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