Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models

Hardcover | April 30, 1999

byPierre-Louis Lions

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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 variousmathematical 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 thefundamental 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 withcomplete 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 ofthe 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 outstandingcontribution 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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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, ...

Pierre-Louis Lions is a Professor of Mathematics, University of Paris-Dauphine, and of Applied Mathematics at Ecole Polytechnique.

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Hardcover|Jul 1 1998

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Format:HardcoverDimensions:252 pages, 9.21 × 6.14 × 0.75 inPublished:April 30, 1999Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0198514875

ISBN - 13:9780198514879

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Table of Contents

PrefaceTable of contents1. Presentation of the modelsPart 1: Incompressible Models2. Density-dependent Navier-Stokes equations3. Navier-Stokes equations4. Euler equations and other incompressible modelsAppendix A Truncation of divergence-free vectorfieldsAppendix B Two facts on D1,2(R2)Appendix C Compactness in time with values in weak topologiesAppendix D Weak L1 estimates for solutions of the heat equationAppendix E A short proof of the existence of renormalized solutions for parabolic equationsIntended Table of Contents of Volume 2Part 2: Compressible Models5. Compactness results for compressible isentropic Navier-Stokes6. Stationary problems7. Existence results8. Related questionsPart 3: Asymptotic limites9. Asymptotic limits

Editorial Reviews

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