Lectures on Navier-stokes Equations (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 192)

The book is an excellent contribution to the literature concerning the mathematical analysis of the incompressible Navier-Stokes equations. It provides a very good introduction to the subject, covering several important directions, and also presents a number of recent results, with an emphasis on non-perturbative regimes. The book is well written and both beginners and experts will benefit from it. It can also provide great material for a graduate course. —Vladimir Šverák, University of Minnesota This book is a graduate text on the incompressible Navier-Stokes system, which is of fundamental importance in mathematical fluid mechanics as well as in engineering applications. The goal is to give a rapid exposition on the existence, uniqueness, and regularity of its solutions, with a focus on the regularity problem. To fit into a one-year course for students who have already mastered the basics of PDE theory, many auxiliary results have been described with references but without proofs, and several topics were omitted. Most chapters end with a selection of problems for the reader. After an introduction and a careful study of weak, strong, and mild solutions, the reader is introduced to partial regularity. The coverage of boundary value problems, self-similar solutions, the uniform $L^3$ class including the celebrated Escauriaza-Seregin-Šverák Theorem, and axisymmetric flows in later chapters are unique features of this book that are less explored in other texts. The book can serve as a textbook for a course, as a self-study source for people who already know some PDE theory and wish to learn more about Navier-Stokes equations, or as a reference for some of the important recent developments in the area. Cover......Page 1 Title page......Page 4 Contents......Page 6 Preface......Page 10 Notation......Page 12 1.1. Navier-Stokes equations......Page 14 1.2. Derivation of Navier-Stokes equations......Page 16 1.3. Scaling and a priori estimates......Page 19 1.4. Vorticity......Page 20 1.5. Pressure......Page 23 1.6. Helmholtz decomposition......Page 26 Problems......Page 30 2.1. Weak solutions......Page 32 2.2. Small-large uniqueness......Page 35 2.3. Existence for zero boundary data by the Galerkin method......Page 36 2.4. Existence for zero boundary data by the Leray-Schauder theorem......Page 38 2.5. Nonuniqueness......Page 42 2.6. ��^{��}-theory for the linear system......Page 45 2.7. Regularity......Page 51 2.8. The Bogovskii map......Page 58 2.9. Notes......Page 60 Problems......Page 61 3.1. Weak form, energy inequalities, and definitions......Page 64 3.2. Auxiliary results......Page 68 3.3. Existence for the perturbed Stokes system......Page 71 3.4. Compactness lemma......Page 73 3.5. Existence of suitable weak solutions......Page 75 3.6. Notes......Page 80 Problems......Page 81 Chapter 4. Strong solutions......Page 82 4.1. Dimension analysis......Page 83 4.2. Uniqueness......Page 84 4.3. Regularity......Page 88 Problems......Page 90 5.1. Nonstationary Stokes system and Stokes semigroup......Page 92 5.2. Existence of mild solutions......Page 96 5.3. Applications to weak solutions......Page 102 Problems......Page 105 Chapter 6. Partial regularity......Page 106 6.1. The set of singular times......Page 107 6.2. The set of singular space-time points......Page 109 6.3. Regularity criteria in scaled norm......Page 110 6.4. Notes......Page 118 Problems......Page 119 Chapter 7. Boundary value problem and bifurcation......Page 120 7.1. Existence: A priori bound by a good extension......Page 121 7.2. Existence: A priori bound by contradiction......Page 125 7.3. The Korobkov-Pileckas-Russo approach for 2D BVP......Page 129 7.4. The bifurcation problem and degree......Page 136 7.5. Bifurcation of the Rayleigh-Bénard convection......Page 141 7.6. Bifurcation of Couette-Taylor flows......Page 146 7.7. Notes......Page 152 Problems......Page 153 8.1. Self-similar solutions and similarity transform......Page 154 8.2. Stationary self-similar solutions......Page 158 8.3. Backward self-similar solutions......Page 163 8.4. Forward self-similar solutions......Page 171 Problems......Page 184 Chapter 9. The uniform ��3 class......Page 186 9.1. Uniqueness......Page 187 9.2. Auxiliary results for regularity......Page 189 9.3. Regularity......Page 191 9.4. Backward uniqueness and unique continuation......Page 197 9.5. Notes......Page 200 10.1. Axisymmetric Navier-Stokes equations......Page 202 10.2. No swirl case......Page 208 10.3. Type I singularity: De Giorgi-Nash-Moser approach......Page 210 10.4. Type I singularity: Liouville theorem approach......Page 219 10.5. Connections between the two approaches......Page 222 10.6. Notes......Page 223 Bibliography......Page 224 Index......Page 236 Back Cover......Page 239 Cover 1 Title page 4 Contents 6 Preface 10 Notation 12 Chapter 1. Introduction 14 1.1. Navier-Stokes equations 14 1.2. Derivation of Navier-Stokes equations 16 1.3. Scaling and a priori estimates 19 1.4. Vorticity 20 1.5. Pressure 23 1.6. Helmholtz decomposition 26 1.7. Notes 30 Problems 30 Chapter 2. Steady states 32 2.1. Weak solutions 32 2.2. Small-large uniqueness 35 2.3. Existence for zero boundary data by the Galerkin method 36 2.4. Existence for zero boundary data by the Leray-Schauder theorem 38 2.5. Nonuniqueness 42 2.6. L^{q}-theory for the linear system 45 2.7. Regularity 51 2.8. The Bogovskii map 58 2.9. Notes 60 Problems 61 Chapter 3. Weak solutions 64 3.1. Weak form, energy inequalities, and definitions 64 3.2. Auxiliary results 68 3.3. Existence for the perturbed Stokes system 71 3.4. Compactness lemma 73 3.5. Existence of suitable weak solutions 75 3.6. Notes 80 Problems 81 Chapter 4. Strong solutions 82 4.1. Dimension analysis 83 4.2. Uniqueness 84 4.3. Regularity 88 4.4. Notes 90 Problems 90 Chapter 5. Mild solutions 92 5.1. Nonstationary Stokes system and Stokes semigroup 92 5.2. Existence of mild solutions 96 5.3. Applications to weak solutions 102 5.4. Notes 105 Problems 105 Chapter 6. Partial regularity 106 6.1. The set of singular times 107 6.2. The set of singular space-time points 109 6.3. Regularity criteria in scaled norm 110 6.4. Notes 118 Problems 119 Chapter 7. Boundary value problem and bifurcation 120 7.1. Existence: A priori bound by a good extension 121 7.2. Existence: A priori bound by contradiction 125 7.3. The Korobkov-Pileckas-Russo approach for 2D BVP 129 7.4. The bifurcation problem and degree 136 7.5. Bifurcation of the Rayleigh-Bénard convection 141 7.6. Bifurcation of Couette-Taylor flows 146 7.7. Notes 152 Problems 153 Chapter 8. Self-similar solutions 154 8.1. Self-similar solutions and similarity transform 154 8.2. Stationary self-similar solutions 158 8.3. Backward self-similar solutions 163 8.4. Forward self-similar solutions 171 8.5. Notes 184 Problems 184 Chapter 9. The uniform L3 class 186 9.1. Uniqueness 187 9.2. Auxiliary results for regularity 189 9.3. Regularity 191 9.4. Backward uniqueness and unique continuation 197 9.5. Notes 200 Chapter 10. Axisymmetric flows 202 10.1. Axisymmetric Navier-Stokes equations 202 10.2. No swirl case 208 10.3. Type I singularity: De Giorgi-Nash-Moser approach 210 10.4. Type I singularity: Liouville theorem approach 219 10.5. Connections between the two approaches 222 10.6. Notes 223 Bibliography 224 Index 236 Back Cover 239

The book is an excellent contribution to the literature concerning the mathematical analysis of the incompressible Navier-Stokes equations. It provides a very good introduction to the subject, covering several important directions, and also presents a number of recent results, with an emphasis on non-perturbative regimes. The book is well written and both beginners and experts will benefit from it. It can also provide great material for a graduate course.—Vladimir Šverák, University of MinnesotaThis book is a graduate text on the incompressible Navier-Stokes system, which is of fundamental importance in mathematical fluid mechanics as well as in engineering applications. The goal is to give a rapid exposition on the existence, uniqueness, and regularity of its solutions, with a focus on the regularity problem. To fit into a one-year course for students who have already mastered the basics of PDE theory, many auxiliary results have been described with references but without proofs, and several topics were omitted. Most chapters end with a selection of problems for the reader.After an introduction and a careful study of weak, strong, and mild solutions, the reader is introduced to partial regularity. The coverage of boundary value problems, self-similar solutions, the uniform $L^3$ class including the celebrated Escauriaza-Seregin-Šverák Theorem, and axisymmetric flows in later chapters are unique features of this book that are less explored in other texts.The book can serve as a textbook for a course, as a self-study source for people who already know some PDE theory and wish to learn more about Navier-Stokes equations, or as a reference for some of the important recent developments in the area.

This book is a graduate text on the incompressible Navier-Stokes system, which is of fundamental importance in mathematical fluid mechanics as well as in engineering applications. The goal is to give a rapid exposition on the existence, uniqueness, and regularity of its solutions, with a focus on the regularity problem. To fit into a one-year course for students who have already mastered the basics of PDE theory, many auxiliary results have been described with references but without proofs, and several topics were omitted. Most chapters end with a selection of problems for the reader. After an introduction and a careful study of weak, strong, and mild solutions, the reader is introduced to partial regularity. The coverage of boundary value problems, self-similar solutions, the uniform $L^3$ class including the celebrated Escauriaza-Seregin-Sverak Theorem, and axisymmetric flows in later chapters are unique features of this book that are less explored in other texts. The book can serve as a textbook for a course, as a self-study source for people who already know some PDE theory and wish to learn more about Navier-Stokes equations, or as a reference for some of the important recent developments in the area. The book is an excellent contribution to the literature concerning the mathematical analysis of the incompressible Navier-Stokes equations. It provides a very good introduction to the subject, covering several important directions, and also presents a number of recent results, with an emphasis on non-perturbative regimes. The book is well written and both beginners and experts will benefit from it. It can also provide great material for a graduate course. --Vladimir Sverák, University of Minnesota This book is a graduate text on the incompressible Navier-Stokes system, which is of fundamental