An Introduction to the Mathematical Theory of the Navier-Stokes Equations

This is the first of four volumes on the Navier-Stokes equations, specifically on Linearized Steady Problems. The volumes deal with the fundamental mathematical properties of the Navier-Stokes equations, such as existence, regularity and uniqueness of solutions, and, for unbounded domains, their asymptotic behavior. The work is an up-to-date and detailed investigation of these problems for motions in domains of different types: bounded, exterior and domain with noncompact boundaries. Throughout the work, main problems which, so far, remain open are pointed out and for some of these conjectures are offered. New results are presented throughout, while several classical subjects are treated in a completely original way. The work is mathematically self contained, requiring no specific background. The 200-plus exercises along with the chapter summaries and questions make this an excellent textbook for any theoretical Fluid Mechanics course; it is suitable as well for self teaching. It is set up to remain useful as a reference or dictionary. TOC: Contents: Steady-State Solutions of the Navier-Stokes Equations: Statement of the Problem and Open Questions.- Basic Function Spaces and Related Inequalities.- The Function Spaces of Hydrodynamics.- Steady Stokes Flow in Bounded Domains.- Steady Stokes Flow in Exterior Domains.- Steady Stokes Flow in Domains with Unbounded Boundaries.- Steady Oseen Flow in Exterioir Domains.