Initial-boundary value problems and the Navier-Stokes equations

Initial-Boundary Value Problems and the Navier-Stokes Equations provides an introduction to the vast subject of initial and initial-boundary value problems for PDEs. Applications to parabolic and hyperbolic systems are emphasized in this text. The Navier-Stokes equations for compressible and incompressible flows are taken as an example to illustrate the results.

Researchers and graduate students in applied mathematics and engineering will find Initial-Boundary Value Problems and the Navier-Stokes Equations invaluable. The subjects addressed in the book, such as the well-posedness of initial-boundary value problems, are of frequent interest when PDEs are used in modeling or when they are solved numerically. The book explains the principles of these subjects. The reader will learn what well-posedness or ill-posedness means and how it can be demonstrated for concrete problems. There are many new results, in particular on the Navier-Stokes equations.

When the book was written, the main intent was to write a text on initial-boundary value problems that was accessible to a rather wide audience. Therefore, functional analytical prerequisites were kept to a minimum or were developed in the book. Boundary conditions are analyzed without first proving trace theorems, and similar simplications have been used throughout. The direct approach to the subject still gives a valuable introduction to an important area of applied analysis.

Author Notes

Jens Lorenz is a Professor of Mathematics at the University of New Mexico.

Preface to the Classics Edition	p. xi
Errata	p. xiii
Introduction	p. xv
Chapter 1. The Navier-Stokes Equations	p. 1
1.1 Some Aspects of Our Approach	p. 2
1.2 Derivation of the Navier-Stokes Equations	p. 9
1.3 Linearization and Localization	p. 18
Chapter 2. Constant-Coefficient Cauchy Problems	p. 23
2.1 Pure Exponentials as Initial Data	p. 24
2.2 Discussion of Concepts of Well-Posedness	p. 34
2.3 Algebraic Characterization of Well-Posedness	p. 44
2.4 Hyperbolic and Parabolic Systems	p. 55
2.5 Mixed Systems and the Compressible N-S Equations Linearized at Constant Flow	p. 62
2.6 Properties of Constant-Coefficient Equations	p. 66
2.7 The Spatially Periodic Cauchy Problem: A Summary for Variable Coefficients	p. 73
Notes on Chapter 2	p. 79
Chapter 3. Linear Variable-Coefficient Cauchy Problems in 1D	p. 81
3.1 A Priori Estimates for Strongly Parabolic Problems	p. 82
3.2 Existence for Parabolic Problems via Difference Approximations	p. 87
3.3 Hyperbolic Systems: Existence and Properties of Solutions	p. 100
3.4 Mixed Hyperbolic-Parabolic Systems	p. 111
3.5 The Linearized Navier-Stokes Equations in One Space Dimension	p. 113
3.6 The Linearized KdV and the Schrodinger Equations	p. 115
Notes on Chapter 3	p. 118
Chapter 4. A Nonlinear Example: Burgers' Equation	p. 121
4.1 Burgers' Equation: A Priori Estimates and Local Existence	p. 122
4.2 Global Existence for the Viscous Burgers' Equation	p. 131
4.3 Generalized Solutions for Burgers' Equation and Smoothing	p. 138
4.4 The Inviscid Burgers' Equation: A First Study of Shocks	p. 141
Notes on Chapter 4	p. 156
Chapter 5. Nonlinear Systems in One Space Dimension	p. 159
5.1 The Case of Bounded Coefficients	p. 160
5.2 Local Existence Theorems	p. 165
5.3 Finite Time Existence and Asymptotic Expansions	p. 167
5.4 On Global Existence for Parabolic and Mixed Systems	p. 172
Notes on Chapter 5	p. 175
Chapter 6. The Cauchy Problem for Systems in Several Dimensions	p. 177
6.1 Linear Parabolic Systems	p. 177
6.2 Linear Hyperbolic Systems	p. 181
6.3 Mixed Hyperbolic-Parabolic Systems and the Linearized Navier-Stokes Equations	p. 188
6.4 Short-Time Existence for Nonlinear Systems	p. 190
6.5 A Global Existence Theorem in 2D	p. 198
Notes on Chapter 6	p. 202
Chapter 7. Initial-Boundary Value Problems in One Space Dimension	p. 203
7.1 A Strip Problem for the Heat Equation	p. 204
7.2 Strip Problems for Strongly Parabolic Systems	p. 211
7.3 Discussion of Concepts of Well-Posedness	p. 222
7.4 Half-Space Problems and the Laplace Transform	p. 228
7.5 Mildly Ill-Posed Half-Space Problems	p. 248
7.6 Initial-Boundary Value Problems for Hyperbolic Equations	p. 253
7.7 Boundary Conditions for Hyperbolic-Parabolic Problems	p. 262
7.8 Semibounded Operators	p. 268
Notes on Chapter 7	p. 272
Chapter 8. Initial-Boundary Value Problems in Several Space Dimensions	p. 275
8.1 Linear Strongly Parabolic Systems	p. 275
8.2 Symmetric Hyperbolic Systems in Several Space Dimensions	p. 283
8.3 The Linearized Compressible Euler Equations	p. 302
8.4 The Laplace Transform Method for Hyperbolic Systems	p. 306
8.5 Remarks on Mixed Systems and Nonlinear Problems	p. 322
Notes on Chapter 8	p. 323
Chapter 9. The Incompressible Navier-Stokes Equations: The Spatially Periodic Case	p. 325
9.1 The Spatially Periodic Case in Two Dimensions	p. 325
9.2 The Spatially Periodic Case in Three Dimensions	p. 337
Chapter 10. The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions	p. 345
10.1 The Linearized Equations in 2D	p. 348
10.2 Auxiliary Results for Poisson's Equation	p. 349
10.3 The Linearized Navier-Stokes Equations under Boundary Conditions	p. 355
10.4 Remarks on the Passage from the Compressible to the Incompressible Equations	p. 359
Appendix 1 Notations and Results from Linear Algebra	p. 361
Appendix 2 Interpolation	p. 365
Appendix 3 Sobolev Inequalities	p. 371
Appendix 4 Application of the Arzela-Ascoil Theorem	p. 389
References	p. 395
Author Index	p. 399
Subject Index	p. 401

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