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Full Description
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.
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
