This book compiles a distinctive set of articles focused on key aspects of the Navier-Stokes equations, a central topic in applied mathematics known for its complexity. The mathematical properties and physical interpretations of these equations pose significant challenges and are recognized as one of the Millennium Prize Problems by the Clay Mathematics Institute, particularly concerning the existence of global, regular solutions for arbitrary initial data. The text features three comprehensive contributions: (1) an exploration of Operator-Valued H∞-calculus, R-boundedness, Fourier multipliers, and maximal Lp-regularity theory applicable to a broad class of quasi-linear evolution problems, including Navier-Stokes and other fluid models; (2) a discussion on classical existence, uniqueness, and regularity theorems for the Navier-Stokes initial-value problem, addressing space-time partial regularity and the smoothness of the Lagrangean flow map; and (3) a thorough mathematical framework on R-boundedness and maximal regularity, with applications to free boundary problems for the Navier-Stokes equations, both with and without surface tension. This volume serves as a valuable resource for graduate students and researchers interested in fundamental issues related to these equations and fluid dynamics.
