The Navier-Stokes Problem in the 21st Century

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 The book also incorporates historical references to the (pre)history of the equations as well as recent references that highlight active mathematical research in the field.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

About the Author :
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.

Review :
"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