Overview

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 contains 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.

Full Product Details

Author:   Pierre-Louis Lions (, Professor of Partial Differential Equations and their Applications at Collège de France and Professor in the Department of Applied Mathematics, Ecole Polytechnique)
Publisher:   Oxford University Press
Imprint:   Oxford University Press
Dimensions:   Width: 15.60cm , Height: 1.50cm , Length: 23.20cm
Weight:   0.390kg
ISBN:  

9780199679218

ISBN 10:   0199679215
Pages:   252
Publication Date:   18 April 2013
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   To order  
Stock availability from the supplier is unknown. We will order it for you and ship this item to you once it is received by us.

Table of Contents

Preface Table of contents 1: Presentation of the models Part 1: Incompressible Models 2: Density-dependent Navier-Stokes equations 3: Navier-Stokes equations 4: Euler equations and other incompressible models Appendix A Truncation of divergence-free vectorfields Appendix B Two facts on D1,2(R2) Appendix C Compactness in time with values in weak topologies Appendix D Weak L1 estimates for solutions of the heat equation Appendix E A short proof of the existence of renormalized solutions for parabolic equations Intended Table of Contents of Volume 2 Part 2: Compressible Models 5: Compactness results for compressible isentropic Navier-Stokes 6: Stationary problems 7: Existence results 8: Related questions Part 3: Asymptotic limites 9: Asymptotic limits

Reviews

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 fur Mathematik 866-487 ...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

`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 `...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

Everybody who is interested in studying mathematical questions arising in fluid mechanics should read this book. * European Mathematical Society, issue 27, March 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 * 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 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 * ...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 * 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 fur Mathematik 866-487 *

`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 fur Mathematik 866-487 `...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

Author Information

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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