This book collects together a unique set of articles dedicated to
several fundamental aspects of the Navier-Stokes 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 Navier-Stokes
equations and other fluid model equations; (2) Classical existence,
uniqueness and regularity theorems of solutions to the Navier-Stokes
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 Navier-Stokes
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
Navier-Stokes equations.
