1st Edition

The Navier-Stokes Problem in the 21st Century

ISBN 9781466566217
Published March 8, 2016 by Chapman and Hall/CRC
718 Pages

USD $115.00

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Book Description

Up-to-Date Coverage of the Navier–Stokes Equation from an Expert in Harmonic Analysis

The complete resolution of the Navier–Stokes equation—one of the Clay Millennium Prize Problems—remains an important open challenge in partial differential equations (PDEs) research despite substantial studies on turbulence and three-dimensional fluids. The Navier–Stokes Problem in the 21st Century provides a self-contained guide to the role of harmonic analysis in the PDEs of fluid mechanics.

The book focuses on incompressible deterministic Navier–Stokes equations in the case of a fluid filling the whole space. It explores the meaning of the equations, open problems, and recent progress. It includes classical results on local existence and studies criterion for regularity or uniqueness of solutions. The book also incorporates historical references to the (pre)history of the equations as well as recent references that highlight active mathematical research in the field.

Table of Contents

Presentation of the Clay Millennium Prizes
Regularity of the three-dimensional fluid flows: a mathematical challenge for the 21st century
The Clay Millennium Prizes
The Clay Millennium Prize for the Navier–Stokes equations
Boundaries and the Navier–Stokes Clay Millennium Problem

The physical meaning of the Navier–Stokes equations
Frames of references
The convection theorem
Conservation of mass
Newton's second law
The equations of hydrodynamics
The Navier–Stokes equations
Boundary terms
Blow up

History of the equation
Mechanics in the Scientific Revolution era
Bernoulli's Hydrodymica
Laplacian physics
Navier, Cauchy, Poisson, Saint-Venant, and Stokes
Oseen, Leray, Hopf, and Ladyzhenskaya
Turbulence models

Classical solutions
The heat kernel
The Poisson equation
The Helmholtz decomposition
The Stokes equation
The Oseen tensor
Classical solutions for the Navier–Stokes problem
Small data and global solutions
Time asymptotics for global solutions
Steady solutions
Spatial asymptotics
Spatial asymptotics for the vorticity
Intermediate conclusion

A capacitary approach of the Navier–Stokes integral equations
The integral Navier–Stokes problem
Quadratic equations in Banach spaces
A capacitary approach of quadratic integral equations
Generalized Riesz potentials on spaces of homogeneous type
Dominating functions for the Navier–Stokes integral equations
A proof of Oseen's theorem through dominating functions
Functional spaces and multipliers

The differential and the integral Navier–Stokes equations
Uniform local estimates
Heat equation
Stokes equations
Oseen equations
Very weak solutions for the Navier–Stokes equations
Mild solutions for the Navier–Stokes equations
Suitable solutions for the Navier–Stokes equations

Mild solutions in Lebesgue or Sobolev spaces
Kato's mild solutions
Local solutions in the Hilbertian setting
Global solutions in the Hilbertian setting
Sobolev spaces
A commutator estimate
Lebesgue spaces
Maximal functions
Basic lemmas on real interpolation spaces
Uniqueness of L3 solutions

Mild solutions in Besov or Morrey spaces
Morrey spaces
Morrey spaces and maximal functions
Uniqueness of Morrey solutions
Besov spaces
Regular Besov spaces
Triebel–Lizorkin spaces
Fourier transform and Navier–Stokes equations

The space BMO-1 and the Koch and Tataru theorem
Koch and Tataru's theorem
A special subclass of BMO-1
Further results on ill-posedness
Large data for mild solutions
Stability of global solutions
Small data

Special examples of solutions
Symmetries for the Navier–Stokes equations
Two-and-a-half dimensional flows
Axisymmetrical solutions
Helical solutions
Brandolese's symmetrical solutions
Self-similar solutions
Stationary solutions
Landau's solutions of the Navier–Stokes equations
Time-periodic solutions
Beltrami flows

Blow up?
First criteria
Blow up for the cheap Navier–Stokes equation
Serrin's criterion
Some further generalizations of Serrin's criterion

Leray's weak solutions
The Rellich lemma
Leray's weak solutions
Weak-strong uniqueness: the Prodi–Serrin criterion
Weak-strong uniqueness and Morrey spaces on the product space R × R3
Almost strong solutions
Weak perturbations of mild solutions

Partial regularity results for weak solutions
Interior regularity
Serrin's theorem on interior regularity
O'Leary's theorem on interior regularity
Further results on parabolic Morrey spaces
Hausdorff measures
Singular times
The local energy inequality
The Caffarelli–Kohn–Nirenberg theorem on partial regularity
Proof of the Caffarelli–Kohn–Nirenberg criterion
Parabolic Hausdorff dimension of the set of singular points
On the role of the pressure in the Caffarelli, Kohn, and Nirenberg regularity theorem

A theory of uniformly locally L2 solutions
Uniformly locally square integrable solutions
Local inequalities for local Leray solutions
The Caffarelli, Kohn, and Nirenberg ε-regularity criterion
A weak-strong uniqueness result

The L3 theory of suitable solutions
Local Leray solutions with an initial value in L3
Critical elements for the blow up of the Cauchy problem in L3
Backward uniqueness for local Leray solutions
Seregin's theorem
Known results on the Cauchy problem for the Navier–Stokes equations in presence of a force
Local estimates for suitable solutions
Uniqueness for suitable solutions
A quantitative one-scale estimate for the Caffarelli–Kohn–Nirenberg regularity criterion
The topological structure of the set of suitable solutions
Escauriaza, Seregin, and Šverák's theorem

Self-similarity and the Leray–Schauder principle
The Leray–Schauder principle
Steady-state solutions
Statement of Jia and Šverák's theorem
The case of locally bounded initial data
The case of rough data
Non-existence of backward self-similar solutions

Global existence, uniqueness and convergence issues for approximated equations
Leray's mollification and the Leray-α model
The Navier–Stokes α -model
The Clark- α model
The simplified Bardina model
Reynolds tensor

Other approximations of the Navier–Stokes equations
Faedo–Galerkin approximations
Frequency cut-off
Ladyzhenskaya's model
Damped Navier–Stokes equations

Artificial compressibility
Temam's model
Vishik and Fursikov's model
Hyperbolic approximation

Energy inequalities
Critical spaces for mild solutions
Models for the (potential) blow up
The method of critical elements

Notations and glossary



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Pierre Gilles Lemarié-Rieusset is a professor at the University of Evry Val d’Essonne. Dr. Lemarié-Rieusset has constructed many widely used bases, such as the Meyer-Lemarié wavelet basis and the Battle-Lemarié spline wavelet basis. His current research focuses on the application of harmonic analysis to the study of nonlinear PDEs in fluid mechanics. He is the author or coauthor of several books, including Recent Developments in the Navier-Stokes Problem.


"This monograph addresses a difficult question in the mathematical theory of a viscous incompressible fluid: global well-posedness of the Cauchy problem for the Navier-Stokes equations. … The author is an outstanding expert in harmonic analysis who has made important contributions. The book contains rigorous proofs of a number of the latest results in the field. I strongly recommend the book to postgraduate students and researchers working on challenging problems of harmonic analysis and mathematical theory of Navier-Stokes equations."
—Gregory Seregin, St Hildas College, Oxford University

"This is a great book on the mathematical aspects of the fundamental equations of hydrodynamics, the incompressible Navier-Stokes equations. It covers many important topics and recent results and gives the reader a very good idea about where the theory stands at present. The book contains an excellent overview of the history, a great modern exposition of many important classical theorems, and an outstanding presentation of a number of very recent results from the frontiers of research on the subject. The writing is flawless, the clarity of the presentation is exceptional, and the author’s choice of the topics is outstanding. I recommend the book very highly to anybody interested in the mathematics surrounding the PDEs of hydrodynamics, including the famous Navier-Stokes regularity problem. The book is perhaps even better than the author’s first book on the Navier-Stokes theory published about 10 years ago, which is regularly used by many mathematicians."
—Vladimir Sverak, University of Minnesota

"This book by Lemarié-Rieusset reports on recent fundamental progress toward understanding the existence, regularity, and stability of solutions to Navier-Stokes equations. This is a must-have book for all researchers working in fluid dynamics equations, and it will become a reference in the field. The very clear and self-contained presentation of this complex and deep subject makes it a very useful textbook for graduate and Ph.D. students as well."
—Marco Cannone, Professor, Université Paris-Est

"This is an outstanding reference and textbook on the Clay Millennium Problem of Navier–Stokes. The book brings together all the essential knowledge of this field in a unified, compact, and up-to-date way. Many noticeable results are stated and re-proved in a new manner, and some important results are even appearing in the literature for the first time. The book stresses on the usage of Morrey spaces, which has its internal advantage both in the framework of mild solutions and weak solutions, and the corresponding results have already been, and will be, very useful in the study of Navier–Stokes problem. The physical meaning, the history, and the classical solutions of the Navier–Stokes equations are included in the first chapters, which are helpful to newcomers of the field. In sum, the book is highly recommended and it will be immensely useful for scientists and students interested in the Navier–Stokes problem."
—Changxing Miao, Distinguished Professor, Institute of Applied Physics and Computational Mathematics

"I would strongly recommend this book to anyone seriously interested in developments on Navier–Stokes equations theory in the second half of the twentieth century, and in the first 17 years of the twenty-first century. The book is on a source of extremely valuable information on NSE for both the mathematicians, and the mathematically oriented theoretical physicists."

—Andrej Icha, Pure and Applied Geophysics