Fundamental Directions in Mathematical Fluid Mechanics

by Galdi, Giovanni P.; Heywood, Malcolm I.; Rannacher, Rolf

• ISBN13: 9783764364144
• ISBN10: 3764364149
• Format: Hardcover
• Copyright: 2000-12-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface	vii
An Introduction to the Navier-Stokes Initial-Boundary Value Problem
Giovanni P. Galdi
Introduction	1	(2)
Some considerations on the structure of the Navier-Stokes equations	3	(2)
The Leray-Hopf weak solutions and related properties	5	(12)
Existence of weak solutions	17	(5)
The energy equality and uniqueness of weak solutions	22	(11)
Regularity of weak solutions	33	(12)
More regular solutions and the ``theoreme de structure''	45	(6)
Existence in the class Lr (0, T; Ls (Ω)), 2/ + n/s = 1, and further regularity properties	51	(22)
References	65	(6)
Spectral Approximation of Navier-Stokes Equations
P. Gervasio
A. Quarteroni
F. Saleri
Introduction	71	(2)
Mathematical foundation and different paradigms of spectral methods	73	(21)
Stokes and Navier-Stokes equations	94	(4)
Time-differentiation of Navier Stokes equations	98	(5)
Domain decomposition methods	103	(12)
Numerical results	115	(14)
References	122	(7)
Simple Proofs of Bifurcation Theorems
John G. Heywood
Wayne Nagata
Introduction	129	(2)
Bifurcation of equilibrium solutions	131	(3)
Bifurcation of periodic solutions	134	(5)
Generalizations	139	(10)
Appendix A: Proof of Proposition 3.1	144	(3)
References	147	(2)
On the Steady Transport Equation
John G. Heywood
Mariarosaria Padula
Introduction	149	(2)
Existence in W1,2 ↶Lq for the scalar transport equation	151	(6)
Existence in W1,2 ↶Lq for the vector transport equation	157	(5)
Estimates for σ2,2 Δσ and δσ-1,2	162	(2)
Existence in Wm,2 (Ω), for any fixed m	164	(2)
Integration along characteristics	166	(5)
References	169	(2)
On the Existence and Uniqueness Theory for the Steady Compressible Viscous Flow
John G. Heywood
Mariarosaria Padula
Introduction	171	(2)
Poisson-Stokes equation for isothermal flow	173	(1)
Main result	173	(1)
Iterative scheme	174	(2)
Regularity lemmas	176	(2)
Bounds for the iterates	178	(2)
Convergence of the iterates	180	(3)
Uniqueness in the ball of existence	183	(3)
Uniqueness reconsidered directly	186	(5)
References	188	(3)
Finite Element Methods for the Incompressible Navier-Stokes Equations
Rolf Rannacher
Introduction	191	(7)
Models of viscous flow	198	(9)
Spatial discretization by finite elements	207	(16)
Time discretization and linearization	223	(10)
Solution of the algebraic systems	233	(9)
A review of theoretical analysis	242	(15)
Error control and mesh adaptation	257	(21)
Extension to weakly compressible flows	278	(9)
References	287