Recent Developments in The Navier-Stokes Problem
Pierre Gilles Lemarié-Rieusset
Résumé
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. Explores important advances in the Cauchy problem on the whole of Rd for the Navier-Stokes equations
- Includes a detailed introduction to real harmonic analysis
- Discusses both strong solutions by direct methods and weak solutions by compactness methods
- Contains original material on the definition of weak solutions and on local energy inequalities
- Offers a simple, efficient introduction to Besov and Lorentz spaces using the discrete J-method of real interpolation
Contents
Introduction- What is this Book About?
- 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
- Weak Solutions for the Navier-Stokes Equations
- Divergence-Free Vector Wavelets
- The Mollified Navier-Stokes Equations
- The Leray Solutions for the Navier-Stokes Equations
- Kato's Mild Solutions for the Navier-Stokes Equations
- The Mild Solutions of Koch and Tataru
- 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
- Solutions of the Navier-Stokes Equations are Space-analytical
- 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
- 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
- Singular Initial Values
L'auteur - Pierre Gilles Lemarié-Rieusset
Pierre-Gilles Lemarié-Rieusset est professeur à l'université d'Evry. Il est l'auteur, avec Yves Meyer, du premier article paru sur la théorie mathématique des ondelettes (1986). Il est aujourd'hui reconnu comme un des meilleurs spécialistes de cette théorie dans le monde.
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Chapman and Hall / CRC |
| Auteur(s) | Pierre Gilles Lemarié-Rieusset |
| Parution | 09/12/2002 |
| Nb. de pages | 406 |
| Format | 15,5 x 23,5 |
| Couverture | Broché |
| Poids | 578g |
| Intérieur | Noir et Blanc |
| EAN13 | 9781584882206 |
| ISBN13 | 978-1-58488-220-6 |
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