The Steady Navier-Stokes System: Basics of the Theory and the Leray Problem (Advances in Mathematical Fluid Mechanics)

This book provides a successful solution to one of the central problems of mathematical fluid mechanics: the Leray’s problem on existence of a solution to the boundary value problem for the stationary Navier―Stokes system in bounded domains under sole condition of zero total flux. This marks the culmination of the authors' work over the past few years on this under-explored topic within the study of the Navier―Stokes equations. This book will be the first major work on the Navier―Stokes equations to explore Leray’s problem in detail. The results are presented with detailed proofs, as are the history of the problem and the previous approaches to finding a solution to it. In addition, for the reader’s convenience and for the self-sufficiency of the text, the foundations of the mathematical theory for incompressible fluid flows described by the steady state Stokes and Navier―Stokes systems are presented. For researchers in this active area, this book will be a valuable resource. Preface Abstract Contents 1 Preliminaries 1.1 Basic Notation, Elementary Inequalities, and Auxiliary Results 1.1.1 Basic Notations 1.1.2 Elementary Inequalities 1.2 Some Facts from Functional Analysis 1.2.1 Linear Operators in Banach and Hilbert Spaces 1.2.2 Results on Nonlinear Operator Equations 1.3 Hölder and Sobolev Function Spaces 1.3.1 Hölder Spaces 1.3.2 Boundaries 1.3.3 Sobolev Spaces 1.3.3.1 Definition of Sobolev Spaces 1.3.3.2 Traces 1.3.3.3 Inequalities 1.3.3.4 Sobolev Embedding Theorem 1.3.3.5 Green's and Stokes Formulas 1.3.3.6 Difference Quotients and Sobolev Spaces 1.4 Weakly Singular and Singular Integral Transforms in Lq-Spaces 1.5 Some Facts from Topology and Geometric Measure Theory 1.5.1 Definitions 1.5.2 Properties of Planar Continuous Functions: Kronrod's Graph 1.5.3 Hausdorff Content, Measure, and Dimension 1.5.4 Fine Properties of Sobolev Functions 1.5.5 Coarea Formula 1.6 On Morse-Sard and Luzin N-Properties of Sobolev Functions from W2,1(R2) 1.6.1 The Proof of the Morse–Sard Theorem: Preliminaries 1.6.2 On Images of Sets of Small Hausdorff Contents 1.6.3 Proof of the Morse–Sard Theorem: Main Part 1.7 Selected Topics from Harmonic Analysis 1.7.1 Stein's Regularized Distance 1.7.2 Hardy's Space H1(Rn), BMO(Rn) Spaces, and Their Basic Properties 1.7.3 The Div–Curl Lemma 1.8 Spaces of Divergence-Free Vector Fields 1.8.1 Basic Definitions 1.8.2 Some Problems of Vector Analysis 1.8.3 Spaces of Divergence-Free Vector Fields 1.8.4 About Problem I in Lipschitz Domains 1.8.5 Helmholtz–Weyl Decomposition 1.8.6 The Normal Trace 1.9 Basic Results on Elliptic Equations 1.9.1 The Maximum Principle for Elliptic Equations 1.9.2 Properties of Harmonic Functions 1.9.3 ADN-Elliptic Boundary Value Problems 1.9.3.1 Definition of General Elliptic Problems 1.9.3.2 The Fredholm Property of Elliptic Problems 1.10 Chapter Notes 2 Stokes Problem 2.1 Definitions of Weak Solutions 2.2 Existence and Uniqueness of Weak Solutions 2.3 The Case of Nonhomogeneous Boundary Conditions 2.4 Regularity of Weak Solutions 2.4.1 Interior Regularity 2.4.2 Regularity Up to the Boundary 2.5 ADN-Ellipticity of the Stokes System and Local Estimates 2.6 The Stokes Operator 2.7 Chapter Notes 3 The Stationary Navier–Stokes Problem in Bounded Domains 3.1 The Stationary Navier–Stokes Problem with Homogeneous Boundary Conditions 3.1.1 Existence of Weak Solutions 3.1.2 Uniqueness of Weak Solutions 3.2 Stationary Navier–Stokes Problem with Nonhomogeneous Boundary Conditions 3.2.1 Formulation of Leray's Problem 3.2.2 Uniqueness of ``Small'' Solutions 3.2.3 Example of Nonuniqueness 3.2.4 Regularity of Weak Solutions 3.2.5 Existence of the Solution: General Scheme 3.3 Hopf's Lemma and Its Applications 3.3.1 The Case of Stringent Outflow Condition 3.3.2 Counterexample 3.4 Method of Proving an A Priori Bound by Contradiction 3.5 Existence of Solutions for Small Data 3.6 Chapter Notes 4 The Case of Symmetric Two-Dimensional Domains: General Outflow Condition 4.1 Hopf's Lemma in a Symmetric Planar Domain 4.2 Proof of the A Priori Estimate by Contradiction in a Symmetric Planar Domain 4.3 Notes for the Chapter 5 The Case of General Two-Dimensional Domains and General Outflow Condition 5.1 Formulation of Main Results 5.2 On Uniform Convergence of the Bernoulli Pressure on Almost All Curves 5.3 Obtaining a Contradiction 5.3.1 Case (a) (the Maximum of Φ Is Attained on the Boundary ∂Ω) 5.3.2 Case (b) (the Maximum of Φ Is Not Attained at ∂Ω) 5.4 Some Properties of Weak (Sobolev) Solutions to Euler Equations 5.4.1 Continuity of the Pressure 5.4.2 Bernoulli Law 5.4.3 Weak Maximum Principle for Bernoulli Pressure 5.5 An Original Proof of the General Existence Theorem 5.5.1 The Maximum of Φ Is Attained on the Boundary ∂Ω 5.5.2 The Maximum of Φ Is Not Attained on the Boundary ∂Ω 6 The Case of Axially Symmetric Three-Dimensional Domains 6.1 Formulation of the Problem 6.2 Some Properties of Axisymmetric Solutions to Euler Equations 6.3 Bernoulli Law for Weak Axisymmetric Solutions to the Euler System 6.4 Obtaining a Contradiction 6.4.1 The Case esssupxΩΦ(x)=0 6.4.2 The Case 0 maxj=0,...,Nj 6.5 Appendix A 6.6 Appendix B References