# The Navier-Stokes Problem in the 21st Century

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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 The book also incorporates historical references to the (pre)history of the equations as well as recent references that highlight active mathematical research in the field.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

Pressure

Strain

Stress

The equations of hydrodynamics

The Navier–Stokes equations

Vorticity

Boundary terms

Blow up

Turbulence

**History of the equation**

Mechanics in the Scientific Revolution era

Bernoulli's *Hydrodymica*

D'Alembert

Euler

Laplacian physics

Navier, Cauchy, Poisson, Saint-Venant, and Stokes

Reynolds

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 *L*^{3} 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

Q-spaces

A special subclass of

*BMO*

^{-1}

Ill-posedness

Further results on ill-posedness

Large data for mild solutions

Stability of global solutions

Analyticity

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

Vorticity

Squirts

**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 × R^{3}

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 L^{2} 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 L^{3} theory of suitable solutions**

Local Leray solutions with an initial value in

*L*

^{3}

Critical elements for the blow up of the Cauchy problem in

*L*

^{3}

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

Self-similarity

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

** α-models**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

Hyperviscosity

Ladyzhenskaya's model

Damped Navier–Stokes equations

**Artificial compressibility**

Temam's model

Vishik and Fursikov's model

Hyperbolic approximation

**Conclusion **

Energy inequalities

Critical spaces for mild solutions

Models for the (potential) blow up

The method of critical elements

**Notations and glossary**

Bibliography

Index

## Author(s)

### Biography

**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*.

## Reviews

"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