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 equationsGregory 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 authors 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 authors first book on the Navier Stokes theory published aboutyears ago, which is regularly used by many mathematiciansVladimir Sverak, University of Minnesota This book by Lemari 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 PhD students as wellMarco Cannone, Professor, Universit Paris Est This is an outstanding reference and textbook on the Clay Millennium Problem of NavierStokes 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 NavierStokes problem The physical meaning, the history, and the classical solutions of the NavierStokes 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 NavierStokes problemChangxing Miao, Distinguished Professor, Institute of Applied Physics and Computational Mathematics I would strongly recommend this book to anyone seriously interested in developments on NavierStokes equations theory in the second half of the twentieth century, and in the firstyears 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 GeophysicsUp to Date Coverage of the NavierStokes Equation from an Expert in Harmonic AnalysisThe complete resolution of the NavierStokes equationone of the Clay Millennium Prize Problemsremains an important open challenge in partial differential equations PDEs research despite substantial studies on turbulence and three dimensional fluids The NavierStokes Problem in the st Century provides a self contained guide to the role of harmonic analysis in the PDEs of fluid mechanics The book focuses on incompressible deterministic NavierStokes 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

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