This book collects together a unique set of articles dedicated to several fundamental aspects of the NavierStokes equations. As is well known, understanding the mathematical properties of these equations, along with their physical interpretation, constitutes one of the most challenging questions of applied mathematics. Indeed, the Navier-Stokes equations feature among the Clay Mathematics Institute's seven Millennium Prize Problems (existence of global in time, regular solutions corresponding to initial data of unrestricted magnitude).

The text comprises three extensive contributions covering the following topics: (1) Operator-Valued H-calculus, R-boundedness, Fourier multipliers and maximal Lp-regularity theory for a large, abstract class of quasi-linear evolution problems with applications to NavierStokes equations and other fluid model equations; (2) Classical existence, uniqueness and regularity theorems of solutions to the NavierStokes initial-value problem, along with space-time partial regularity and investigation of the smoothness of the Lagrangean flow map; and (3) A complete mathematical theory of R-boundedness and maximal regularity with applications to free boundary problems for the NavierStokes equations with and without surface tension.

Offering a general mathematical framework that could be used to study fluid problems and, more generally, a wide class of abstract evolution equations, this volume is aimed at graduate students and researchers who want to become acquainted with fundamental problems related to the NavierStokes equations.

Provides an accessible introduction to the basic results and major open questions related to the NavierStokes initial-value problem Gives applications to difficult and still unresolved questions, like free boundary problems Describes the general theory of R-boundedness and maximal regularity for quasilinear evolution equations in Banach spaces

Inhalt
Giovanni P. Galdi, Yoshihiro Shibata: Preface .- Matthias Hieber: Analysis of Viscous Fluid Flows: An Approach by Evolution Equations.- James C. Robinson : Partial regularity for the 3D Navier-Stokes equations.- Yoshihiro Shibata : R Boundedness, Maximal Regularity and Free Boundary Problems for the Navier Stokes Equations.