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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.

### Chapter 1. Analysis of Viscous Fluid Flows: An Approach by Evolution Equations

Abstract
This course of lectures discusses various aspects of viscous fluid flows ranging from boundary layers and fluid structure interaction problems over free boundary value problems and liquid crystal flow to the primitive equations of geophysical flows. We will be mainly interested in strong solutions to the underlying equations and choose as mathematical tool for our investigations the theory of evolution equations. The models considered are mainly represented by semi- or quasilinear parabolic equations and from a modern point of view it is hence natural to investigate the underlying equations by means of the maximal L p-regularity approach.
Matthias Hieber

### Chapter 2. Partial Regularity for the 3D Navier–Stokes Equations

Abstract
These notes give a relatively quick introduction to some of the main results for the three-dimensional Navier–Stokes equations, concentrating in particular on ‘partial regularity’ results that limit the size of the set of (potential) singularities, both in time and in space-time.
James C. Robinson

### Chapter 3. Boundedness, Maximal Regularity and Free Boundary Problems for the Navier Stokes Equations

Abstract
In these lecture notes, we study free boundary problems for the Navier–Stokes equations with and without surface tension. The local well-posedness, global well-posedness, and asymptotics of solutions as time goes to infinity are studied in the L p in time and L q in space framework. To prove the local well-posedness, we use the tool of maximal L pL q regularity for the Stokes equations with nonhomogeneous free boundary conditions. Our approach to proving maximal L pL q regularity is based on the $${\mathcal R}$$-bounded solution operators of the generalized resolvent problem for the Stokes equations with non-homogeneous free boundary conditions and the Weis operator-valued Fourier multiplier.
Key to proving global well-posedness for the strong solutions is the decay properties of the Stokes semigroup, which are derived by spectral analysis of the Stokes operator in the bulk space and the Laplace–Beltrami operator on the boundary. We study the following two cases: (1) a bounded domain with surface tension and (2) an exterior domain without surface tension. In studying the latter case, since for unbounded domains we can obtain only polynomial decay in suitable L q norms in space, to guarantee the L p-integrability of solutions in time it is necessary to have the freedom to choose an exponent with respect to the time variable, thus it is essential to choose different exponents p and q.
The basic approach of this chapter is to analyze the generalized resolvent problem, prove the existence of $${\mathcal R}$$-bounded solution operators and determine the decay properties of solutions to the non-stationary problem. In particular, R-bounded solution operator and Weis’ operator valued Fourier multiplier theorem and transference theorem for the Fourier multiplier, we derive the maximal L pL q regularity for the initial boundary value problem, find periodic solutions with non-homogeneous boundary conditions, and generate analytic semigroups for systems of parabolic equations, including equations appearing in fluid mechanics. This approach is quite new and extends the Fujita–Kato method in the study of the Navier–Stokes equations.
Yoshihiro Shibata