Navier–Stokes Equations (eBook)

Grzegorz Łukaszewicz is Professor of Mathematical Physics at the University of Warsaw. His research interests include partial differential equations, hydrodynamics, dynamical systems.Piotr Kalita works at the Faculty of Mathematics and Computer Science at Jagiellonian University in Krakow, Poland. His research interests include global dynamics for evolutionary partial differential equations, inequalities, and inclusions; modeling in mathematical physics; and numerical methods for evolutionary problems."

Preface 8
Contents 12
1 Introduction and Summary 16
2 Equations of Classical Hydrodynamics 25
2.1 Derivation of the Equations of Motion 25
2.2 The Stress Tensor 34
2.3 Field Equations 36
2.4 Navier–Stokes Equations 37
2.5 Vorticity Dynamics 38
2.6 Thermodynamics 40
2.7 Similarity of Flows and Nondimensional Variables 42
2.8 Examples of Simple Exact Solutions 45
2.9 Comments and Bibliographical Notes 50
3 Mathematical Preliminaries 52
3.1 Theorems from Functional Analysis 52
3.2 Sobolev Spaces and Distributions 57
3.3 Some Embedding Theorems and Inequalities 61
3.4 Sobolev Spaces of Periodic Functions 67
3.5 Evolution Spaces and Their Useful Properties 74
3.6 Gronwall Type Inequalities 80
3.7 Clarke Subdifferential and Its Properties 83
3.8 Nemytskii Operator for Multifunctions 87
3.9 Clarke Subdifferential: Examples 90
3.10 Comments and Bibliographical Notes 94
4 Stationary Solutions of the Navier–Stokes Equations 95
4.1 Basic Stationary Problem 95
4.1.1 The Stokes Operator 96
4.1.2 The Nonlinear Problem 98
4.1.3 Other Topological Methods to Deal with theNonlinearity 102
4.2 Comments and Bibliographical Notes 105
5 Stationary Solutions of the Navier–Stokes Equations with Friction 106
5.1 Problem Formulation 106
5.2 Friction Operator and Its Properties 107
5.3 Weak Formulation 109
5.4 Existence of Weak Solutions for the Case of Linear Growth Condition 114
5.5 Existence of Weak Solutions for the Case of Power Growth Condition 118
5.6 Comments and Bibliographical Notes 120
6 Stationary Flows in Narrow Films and the Reynolds Equation 122
6.1 Classical Formulation of the Problem 122
6.2 Weak Formulation and Main Estimates 125
6.3 Scaling and Uniform Estimates 132
6.4 Limit Variational Inequality, Strong Convergence, and the Limit Equation 136
6.5 Remarks on Function Spaces 139
6.6 Strong Convergence of Velocities and the Limit Equation 145
6.7 Reynolds Equation and the Limit Boundary Conditions 148
6.8 Uniqueness 152
6.9 Comments and Bibliographical Notes 153
7 Autonomous Two-Dimensional Navier–Stokes Equations 154
7.1 Navier–Stokes Equations with Periodic Boundary Conditions 154
7.2 Existence of the Global Attractor: Case of Periodic Boundary Conditions 162
7.3 Convergence to the Stationary Solution: The Simplest Case 168
7.4 Convergence to the Stationary Solution for Large Forces 170
7.5 Average Transfer of Energy 174
7.6 Comments and Bibliographical Notes 177
8 Invariant Measures and Statistical Solutions 179
8.1 Existence of Invariant Measures 179
8.2 Stationary Statistical Solutions 186
8.3 Comments and Bibliographical Notes 191
9 Global Attractors and a Lubrication Problem 192
9.1 Fractal Dimension 192
9.2 Abstract Theorem on Finite Dimensionality and an Algorithm 194
9.3 An Application to a Shear Flow in Lubrication Theory 202
9.3.1 Formulation of the Problem 202
9.3.2 Energy Dissipation Rate Estimate 205
9.3.3 A Version of the Lieb–Thirring Inequality 209
9.3.4 Dimension Estimate of the Global Attractor 210
9.4 Comments and Bibliographical Notes 213
10 Exponential Attractors in Contact Problems 215
10.1 Exponential Attractors and Fractal Dimension 215
10.2 Planar Shear Flows with the Tresca Friction Condition 219
10.2.1 Problem Formulation 219
10.2.2 Existence and Uniqueness of a Global in Time Solution 226
10.2.3 Existence of Finite Dimensional Global Attractor 230
10.2.4 Existence of an Exponential Attractor 237
10.3 Planar Shear Flows with Generalized Tresca Type Friction Law 239
10.3.1 Classical Formulation of the Problem 239
10.3.2 Weak Formulation of the Problem 241
10.3.3 Existence and Properties of Solutions 244
10.3.4 Existence of Finite Dimensional Global Attractor 249
10.3.5 Existence of an Exponential Attractor 254
10.4 Comments and Bibliographical Notes 256
11 Non-autonomous Navier–Stokes Equations and PullbackAttractors 259
11.1 Determining Modes 259
11.2 Determining Nodes 264
11.3 Pullback Attractors for Asymptotically Compact Non-autonomous Dynamical Systems 268
11.4 Application to Two-Dimensional Navier–Stokes Equations in Unbounded Domains 276
11.5 Comments and Bibliographical Notes 282
12 Pullback Attractors and Statistical Solutions 284
12.1 Pullback Attractors and Two-Dimensional Navier–Stokes Equations 284
12.2 Construction of the Family of Probability Measures 288
12.3 Liouville and Energy Equations 292
12.4 Time-Dependent and Stationary Statistical Solutions 295
12.5 The Case of an Unbounded Domain 298
12.6 Comments and Bibliographical Notes 302
13 Pullback Attractors and Shear Flows 303
13.1 Preliminaries 303
13.2 Formulation of the Problem 304
13.3 Existence and Uniqueness of Global in Time Solutions 307
13.4 Existence of the Pullback Attractor 311
13.5 Fractal Dimension of the Pullback Attractor 316
13.6 Comments and Bibliographical Notes 322
14 Trajectory Attractors and Feedback Boundary Control in Contact Problems 323
14.1 Setting of the Problem 323
14.2 Weak Formulation of the Problem 325
14.3 Existence of Global in Time Solutions 328
14.4 Existence of Attractors 335
14.5 Comments and Bibliographical Notes 341
15 Evolutionary Systems and the Navier–Stokes Equations 343
15.1 Evolutionary Systems and Their Attractors 343
15.2 Three-Dimensional Navier–Stokes Problem with Multivalued Friction 346
15.3 Existence of Leray–Hopf Weak Solution 348
15.4 Existence and Invariance of Weak Global Attractor, and Weak Tracking Property 357
15.5 Comments and Bibliographical Notes 362
16 Attractors for Multivalued Processes in Contact Problems 364
16.1 Abstract Theory of Pullback D-Attractors for Multivalued Processes 364
16.2 Application to a Contact Problem 371
16.3 Comments and Bibliographical Notes 381
References 382
Index 392