Fundamental Directions In Mathematical Fluid Mechanics (advances In Mathematical Fluid Mechanics)

This volume consists of six articles, each treating an important topic in the theory ofthe Navier-Stokes equations, at the research level. Some of the articles are mainly expository, putting together, in a unified setting, the results of recent research papers and conference lectures. Several other articles are devoted mainly to new results, but present them within a wider context and with a fuller exposition than is usual for journals. The plan to publish these articles as a book began with the lecture notes for the short courses of G.P. Galdi and R. Rannacher, given at the beginning of the International Workshop on Theoretical and Numerical Fluid Dynamics, held in Vancouver, Canada, July 27 to August 2, 1996. A renewed energy for this project came with the founding of the Journal of Mathematical Fluid Mechanics, by G.P. Galdi, J. Heywood, and R. Rannacher, in 1998. At that time it was decided that this volume should be published in association with the journal, and expanded to include articles by J. Heywood and W. Nagata, J. Heywood and M. Padula, and P. Gervasio, A. Quarteroni and F. Saleri. The original lecture notes were also revised and updated. This Set Of Six Papers, Written By Eminent Experts In The Field, Is Concerned With That Part Of Fluid Mechanics That Seeks Its Foundation In The Rigorous Mathematical Treatment Of The Navier-stokes Equations. In Particular, An Overview Is Given On State Of Research Regarding The Global Existence Of Smooth Solutions, For Which Uniqueness And Continuous Dependence On The Data Can Be Proven. Then, The Book Moves On To A Discussion Of Recent Developments Of The Finite Element Galerkin Method, With An Emphasis On A Priori And A Posteriori Error Estimation And Adaptive Mesh Refinement. A Further Article Elaborates On Spectral Galerkin Methods And Their Extension To Domains With Complicated Geometries By Employing The Techniques Of Domain Decomposition. The Rigorous Explanation Of Bifurcation Phenomena In Fluids Has Long Been A Central Topic In The Theory Of Navier-stokes Equations. Here, Bifurcation Theory Is Introduced In A General Setting That Is Particularly Convenient For Application To Such Problems. Finally, The Extension Of Navier-stokes Theory To Compressible Viscous Flows, Studied In Two More Papers, Opens Up A Fascinating Panorama Of Theoretical And Numerical Problems. While Some Of The Contributions Are Expository, Others Primarily Present New Results Within A Wider Context And Fuller Exposition Than Is Usual For Research Papers. The Book Is Meant To Introduce Researchers And Advanced Students To The Research Level On Some Of The Most Important Topics Of The Field. Edited By Giovanni P. Galdi, John G. Heywood, Rolf Rannacher. Front Matter....Pages i-viii An Introduction to the Navier-Stokes Initial-Boundary Value Problem....Pages 1-70 Spectral Approximation of Navier-Stokes Equations....Pages 71-127 Simple Proofs of Bifurcation Theorems....Pages 129-148 On The Steady Transport Equation....Pages 149-170 On the Existence and Uniqueness Theory for Steady Compressible Viscous Flow....Pages 171-189 Finite Element Methods for the Incompressible Navier-Stokes Equations....Pages 191-293