Fundamental Directions 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.

An Introduction to the Navier-Stokes Initial-Boundary Value Problem.- 0 Introduction.- 1 Some considerations on the structure of the Navier-Stokes equations.- 2 The Leray-Hopf weak solutions and related properties.- 3 Existence of weak solutions.- 4 The energy equality and uniqueness of weak solutions.- 5 Regularity of weak solutions.- 6 More regular solutions and the “théorème de structure”.- 7 Existence in the class Lr (0,T; Ls(?), 2/r +n/s = 1, and further regularity properties.- References.- Spectral Approximation of Navier-Stokes Equations.- 1 Mathematical foundation and different paradigms of spectral methods.- 2 Stokes and Navier-Stokes equations.- 3 Time-differentiation of Navier Stokes equations.- 4 Domain decomposition methods.- 5 Numerical results.- References.- Simple Proofs of Bifurcation Theorems.- 1 Introduction.- 2 Bifurcation of equilibrium solutions.- 3 Bifurcation of periodic solutions.- 4 Generalizations.- Appendix A: Proof of Proposition 3.1.- References.- On The Steady Transport Equation.- 1 Introduction.- 2 Existence in W1,2 ? Lq for the scalar transport equation.- 3 Existence in W1,2 ? Lq for the scalar transport equation.- 4 Estimates for ???2,2, ???? and ?????1,2.- 5 Existence in Wm,2 (?), for any fixed m.- 6 Integration along characteristics.- References.- On the Existence and Uniqueness Theory for the Steady Compressible Viscous Flow.- 1 Introduction.- 2 Poisson-Stokes equations for isothermal flow.- 3 Main result.- 4 Iterative scheme.- 5 Regularity lemmas.- 6 Bounds for the iterates.- 7 Convergence of the iterates.- 8 Uniqueness in the ball of existence.- 9 Uniqueness reconsidered directly.- References.- Finite Element Methods for the Incompressible Navier-Stokes Equations.- 1 Introduction.- 2 Models of viscousflow.- 3 Spatial discretization by finite elements.- 4 Time discretization and linearization.- 5 Solution of the algebraic systems.- 6 A review of theoretical analysis.- 7 Error control and mesh adaptation.- 8 Extension to weakly compressible flows.- References.