Mathematical Analysis of the Navier-Stokes Equations -- Cetraro, Italy 2017

Description

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. Preface......Page 6 Contents......Page 7 1 Analysis of Viscous Fluid Flows: An Approach by Evolution Equations......Page 8 1.1 Basics on Distributions, Fourier Transforms and Sobolev Spaces......Page 10 1.2 The Vector Valued Setting......Page 18 1.3 Cauchy Problems and Semigroups......Page 24 1.4 Sectorial Operators and Bounded H∞-Functional Calculus......Page 32 1.5 Fractional Powers......Page 40 1.6 Operator-Valued H∞-Calculus, R-Boundedness, Fourier Multipliers and Maximal Lp-Regularity......Page 43 1.7 Quasilinear Evolution Equations......Page 55 1.8 Balance Laws......Page 61 1.9 The Stokes Equation in a Half Space......Page 64 1.10 The Stokes and Hydrostatic Stokes Equations on Domains......Page 78 1.11 Nonlinear Stability of Ekman Boundary Layers......Page 88 1.12 Fluid-Rigid Body Interaction Problems for Compressible Fluids......Page 100 1.13 Two-Phase Free Boundary Value Problems for a Class of Non-Newtonian Fluids......Page 106 1.14 Nematic Liquid Crystals......Page 118 1.15 Global Strong Well-Posedness of the Primitive Equations......Page 127 1.16 Justification of the Hydrostatic Approximation......Page 132 1.17 Notes......Page 139 References......Page 147 2.1 Introduction......Page 154 2.1.1 Notation and Preliminaries......Page 155 2.2.1 Weak Solutions......Page 157 2.2.2 Strong Solutions......Page 159 2.2.3 Weak-Strong Uniqueness......Page 163 2.2.4 Regular and Singular Times......Page 164 2.2.5.1 Dimensions......Page 166 2.2.5.2 Dimension of the Set of Singular Times......Page 168 2.3 Serrin's Local Regularity Result for uL5+(Q*)......Page 170 2.3.1 Step 1: Show that ωL∞(Qs*), s