Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models: 3 (Oxford Lecture Series in Mathematics and Its Applications)

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9780198514879: Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models: 3 (Oxford Lecture Series in Mathematics and Its Applications)
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One of the most challenging topics in applied mathematics over the past decades has been the developent of the theory of nonlinear partial differential equations. Many of the problems in mechanics, geometry, probability, etc lead to such equations when formulated in mathematical terms. However, despite a long history of contributions, there exists no central core theory, and the most important advances have come from the study of particular equations and classes of equations arising in specific applications. This two volume work forms a unique and rigorous treatise on various mathematical aspects of fluid mechanics models. These models consist of systems of nonlinear partial differential equations like the incompressible and compressible Navier-Stokes equations. The main emphasis in Volume 1 is on the mathematical analysis of incompressible models. After recalling the fundamental description of Newtonian fluids, an original and self-contained study of both the classical Navier-Stokes equations (including the inhomogenous case) and the Euler equations is given. Known results and many new results about the existence and regularity of solutions are presented with complete proofs. The discussion contiatns many interesting insights and remarks. The text highlights in particular the use of modern analytical tools and methods and also indicates many open problems. Volume 2 will be devoted to essentially new results for compressible models. Written by one of the world's leading researchers in nonlinear partial differential equations, Mathematical Topics in Fluid Mechanics will be an indispensable reference for every serious researcher in the field. Its topicality and the clear, concise, and deep presentation by the author make it an outstanding contribution to the great theoretical problems in science concerning rigorous mathematical modelling of physical phenomena. Pierre-Louis Lions is Professor of Mathematics at the University of Paris-Dauphine and of Applied Mathematics at the Ecole Polytechnique.

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Review:

A lot of results are new, and for each of these recent results the complete and self-contained proofs are given..............Summarizing the impression of this interesting book, it is worth pointing out that it is written in an easy-reading fashion along with the deep and comprehensive analysis of the topics which are at the highest level of the importance in the modern theory of nonlinear differential equations (Zentralblatt für Mathematik 866-487)

Everybody who is interested in studying mathematical questions arisng in fluid mechanics should read this book. (joma)

...the material presented in the book is not only essential in the mathematical treatment of fluid mechanics but also in many other fields where the theory of non linear partial differential equations plays an important role. Its reading will become a must for researchers in this area of mathematics. (European Journal of Mechanics, B/Fluids, vol. 17, no. 3, May - June 1998)

The book in question goes far beyond the scope of the existing monographs on Euler and Navier Stokes equations, reaches the current frontiers of research, and points to numerous extensions or open problems... The text...is written in a clear and precise, but nevertheless lively style. (SIAM Review, vol. 40, no. 2, June 1998)

Without doubt the monograph will soon become one of the main references on the mathematical analysis of fluid mechanics and should be on the bookshelves of every researcher in the field. Having read the first volume I am looking forward to the second volume on compressible volumes. (SIAM Review, vol. 40, no. 2, June 1998)

the gathering of rigorous results in an easily accessible publication with an extensive bibliography will be of benefit to all workers in theoretical Fluid mechanics. (Mathematika, 44)

Everybody who is interested in studying mathematical questions arising in fluid mechanics should read this book. (European Mathematical Society, issue 27, March 1998)

About the Author:

Pierre-Louis Lions is a Professor of Partial differential equations and their applications at Collège de France in Paris and Professor in the Department of Applied Mathematics, Ecole Polytechnique. His work focuses on the theory of nonlinear partial differential equations and he received the Fields Medal for his work in 1994.

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