Recent developments in the Navier-Stokes problem

Recent developments in the Navier-Stokes problem

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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 more

Product details

  • Hardback | 408 pages
  • 157.48 x 236.22 x 30.48mm | 703.06g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • black & white illustrations
  • 1584882204
  • 9781584882206
  • 1,852,445

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 RERUMshow more