Initial-Boundary Value Problems and the Navier-Stokes Equations (eBook)

Front Cover 1
Initial-Boundary Value Problems and the Navier-Stokes Equations 4
Copyright Page 5
Contents 6
Introduction 10
Chapter 1. The Navier-Stokes Equations 14
1.1 Some Aspects of Our Approach 15
1.2 Derivation of the Navier-Stokes Equations 22
1.3 Linearization and Localization 31
Chapter 2. Constant-Coefficient Cauchy Problems 36
2.1 Pure Exponentials as Initial Data 37
2.2 Discussion of Concepts of Well-Posedness 47
2.3 Algebraic Characterization of Well-Posedness 57
2.4 Hyperbolic and Parabolic Systems 68
2.5 Mixed Systems and the Compressible N-S Equations Linearized at Constant Flow 75
2.6 Properties of Constant-Coefficient Equations 79
2.7 The Spatially Periodic Cauchy Problem: A Summary for Variable Coefficients 86
Notes on Chapter 2 92
Chapter 3. Linear Variable-Coefficient Cauchy Problems in 1D 94
3.1 A Priori Estimates for Strongly Parabolic Problems 95
3.2 Existence for Parabolic Problems via Difference Approximations 100
3.3 Hyperbolic Systems: Existence and Properties of Solutions 113
3.4 Mixed Hyperbolic-Parabolic Systems 124
3.5 The Linearized Navier-Stokes Equations in One Space Dimension 126
3.6. The Linearized KdV and the Schrodinger Equations 128
Notes on Chapter 3 131
Chapter 4. A Nonlinear Example: Burgers’ Equation 134
4.1 Burgers’ Equation: A Priori Estimates and Local Existence 135
4.2 Global Existence for the Viscous Burgers’ Equation 144
4.3 Generalized Solutions for Burgers’ Equation and Smoothing 148
4.4 The Inviscid Burgers’ Equation: A First Study of Shocks 154
Notes on Chapter 4 169
Chapter 5. Nonlinear Systems in One Space Dimension 172
5.1 The Case of Bounded Coefficients 173
5.2 Local Existence Theorems 178
5.3 Finite Time Existence and Asymptotic Expansions 180
5.4 On Global Existence for Parabolic and Mixed Systems 185
Notes on Chapter 5 188
Chapter 6. The Cauchy Problem for Systems in Several Dimensions 190
6.1 Linear Parabolic Systems 190
6.2 Linear Hyperbolic Systems 194
6.3 Mixed Hyperbolic-Parabolic Systems and the Linearized Navier-Stokes Equations 201
6.4 Short-Time Existence for Nonlinear Systems 203
6.5 A Global Existence Theorem in 2D 211
Notes on Chapter 6 215
Chapter 7. Initial-Boundary Value Problems in One Space Dimension 216
7.1 A Strip Problem for the Heat Equation 217
7.2 Strip Problems for Strongly Parabolic Systems 224
7.3 Discussion of Concepts of Well-Posedness 235
7.4 Half-Space Problems and the Laplace Transform 241
7.5 Mildly 111-Posed Half-Space Problems 261
7.6 Initial-Boundary Value Problems for Hyperbolic Equations 266
7.7 Boundary Conditions for Hyperbolic-Parabolic Problems 275
7.8 Semibounded Operators 281
Notes on Chapter 7 285
Chapter 8. Initial-Boundary Value Problems in Several Space Dimensions 288
8.1 Linear Strongly Parabolic Systems 288
8.2 Symmetric Hyperbolic Systems in Several Space Dimensions 296
8.3 The Linearized Compressible Euler Equations 315
8.4 The Laplace Transform Method for Hyperbolic Systems 319
8.5 Remarks on Mixed Systems and Nonlinear Problems 335
Notes on Chapter 8 336
Chapter 9. The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 338
9.1 The Spatially Periodic Case in Two Dimensions 338
9.2 The Spatially Periodic Case in Three Dimensions 350
Chapter 10. The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions 358
10.1 The Linearized Equations in 2D 358
10.2 Auxiliary Results for Poisson’s Equation 362
10.3 The Linearized Navier-Stokes Equations under Boundary Conditions 368
10.4 Remarks on the Passage from the Compressible to the Incompressible Equations 372
Appendix 1: Notations and Results from Linear Algebra 374
Appendix 2: Interpolation 378
Appendix 3: Sobolev Inequalities 384
Appendix 4: Application of the Arzela-Ascoil Theorem 402
References 406
Author Index 412
Subject Index 414