Navier-Stokes Equations and Nonlinear Functional Analysis

This second edition, like the first, attempts to arrive as simply as possible at some central problems in the Navier-Stokes equations in the following areas: existence, uniqueness, and regularity of solutions in space dimensions two and three; large time behavior of solutions and attractors; and numerical analysis of the Navier-Stokes equations. Since publication of the first edition of these lectures in 1983, there has been extensive research in the area of inertial manifolds for Navier-Stokes equations. These developments are addressed in a new section devoted entirely to inertial manifolds.

Preface to the second edition; Introduction; Part I. Questions Related to the Existence, Uniqueness and Regularity of Solutions: 1. Representation of a Flow: the Navier-Stokes Equations; 2. Functional Setting of the Equations; 3. Existence and Uniqueness Theorems (Mostly Classical Results); 4. New a priori Estimates and Applications; 5. Regularity and Fractional Dimension; 6. Successive Regularity and Compatibility Conditions at t=0 (Bounded Case); 7. Analyticity in Time; 8. Lagrangian Representation of the Flow; Part II. Questions Related to Stationary Solutions and Functional Invariant Sets (Attractors): 9. The Couette-Taylor Experiment; 10. Stationary Solutions of the Navier-Stokes Equations; 11. The Squeezing Property; 12. Hausdorff Dimension of an Attractor; Part III. Questions Related to the Numerical Approximation: 13. Finite Time Approximation; 14. Long Time Approximation of the Navier-Stokes Equations; Appendix. Inertial Manifolds and Navier-Stokes Equations; Comments and Bibliography; Comments and Bibliography; Update for the Second Edition; References.