This book introduces graduate students and researchers in mathematics and the sciences to the multifaceted subject of the equations of hyperbolic type, which are used, in particular, to describe propagation of waves at finite speed.
Among the topics carefully presented in the book are nonlinear geometric optics, the asymptotic analysis of short wavelength solutions, and nonlinear interaction of such waves. Studied in detail are the damping of waves, resonance, dispersive decay, and solutions to the compressible Euler equations with dense oscillations created by resonant interactions. Many fundamental results are presented for the first time in a textbook format. In addition to dense oscillations, these include the treatment of precise speed of propagation and the existence and stability questions for the three wave interaction equations.
One of the strengths of this book is its careful motivation of ideas and proofs, showing how they evolve from related, simpler cases. This makes the book quite useful to both researchers and graduate students interested in hyperbolic partial differential equations. Numerous exercises encourage active participation of the reader.
The author is a professor of mathematics at the University of Michigan. A recognized expert in partial differential equations, he has made important contributions to the transformation of three areas of hyperbolic partial differential equations: nonlinear microlocal analysis, the control of waves, and nonlinear geometric optics.
Readership: Graduate students and research mathematicians interested in partial differential equations.
Author(s): Jeffrey Rauch
Series: Graduate Studies in Mathematics 133
Publisher: American Mathematical Society
Year: 2012
Language: English
Pages: xx+363
Preface
P.1. How this book came to be, and its peculiarities
P.2. A bird's eye view of hyperbolic equations
Acknowledgments.
Chapter 1 Simple Examples of Propagation
1.1. The method of characteristics
1.2. Examples of propagation of singularities using progressing waves
1.3. Group velocity and the method of nonstationary phase
1.4. Fourier synthesis and rectilinear propagation
1.5. A cautionary example in geometric optics
1.6. The law of reflection
1.6.1. The method of images.
1.6.2. The plane wave derivation
1.6.3. Reflected high frequency wave packets.
1.7. Snell's law of refraction
Chapter 2 The Linear Cauchy Problem
2.1. Energy estimates for symmetric hyperbolic systems
2.1.1. The constant coefficient case
2.1.2. The variable coefficient case
2.2. Existence theorems for symmetric hyperbolic systems
2.3. Finite speed of propagation
2.3.1. The method of characteristics
2.3.2. Speed estimates uniform in space.
2.3.3. Time-like and propagation cones.
2.4. Plane waves, group velocity, and phase velocities
2.5. Precise speed estimate
2.6. Local Cauchy problems
Appendix 2.I. Constant coefficient hyperbolic systems
Appendix 2.II. Functional analytic proof of existence
Chapter 3 Dispersive Behavior
3.1. Orientation
3.2. Spectral decomposition of solutions
3.3. Large time asymptotics
3.4. Maximally dispersive systems
3.4.1. The L^1 --> L^\infty decay estimate
3.4.2. Fixed time dispersive Sobolev estimates.
3.4.3. Strichartz estimates.
Appendix 3.I. Perturbation theory for semisimple eigenvalues
Appendix 3.II. The stationary phase inequality
Chapter 4 Linear Elliptic Geometric Optics
4.1. Euler's method and elliptic geometric optics with constant coefficients
4.2. Iterative improvement for variable coefficients and nonlinear phases
4.3. Formal asymptotics approach
4.4. Perturbation approach
4.5. Elliptic regularity
4.6. The Microlocal Elliptic Regularity Theorem
Chapter 5 Linear Hyperbolic Geometric Optics
5.1. Introduction
5.2. Second order scalar constant coefficient principal part
5.2.1. Hyperbolic problems
5.2.2. The quasiclassical limit of quantum mechanics.
5.3. Symmetric hyperbolic systems
5.4. Rays and transport
5.4.1. The smooth variety hypothesis.
5.4.2. Transport for L = L1 (a)
5.4.3. Energy transport with variable coefficients
5.5. The Lax parametrix and propagation of singularities
5.5.1. The Lax parametrix
5.5.2. Oscillatory integrals and Fourier integral operators
5.5.3. Small time propagation of singularities
5.5.4. Global propagation of singularities
5.6. An application to stabilization
Appendix 5.I.. Hamilton-Jacobi theory for the eikonal equation
5.I.1. Introduction.
5.I.2. Determining the germ of 0 at the initial manifold
5.I.3. Propagation laws for $\phi$, d$\phi$
5.I.4. The symplectic approach
Chapter 6 The Nonlinear Cauchy Problem
6.1. Introduction
6.2. Schauder's lemma and Sobolev embedding
6.3. Basic existence theorem
6.4. Moser's inequality and the nature of the breakdown
6.5. Perturbation theory and smooth dependence
6.6. The Cauchy problem for quasilinear symmetrichy perbolic systems
6.6.1. Existence of solutions
6.6.2. Examples of breakdown
6.6.3. Dependence on initial data.
6.7. Global small solutions for maximally dispersive nonlinear systems
6.8. The subcritical nonlinear Klein-Gordon equation in the energy space
6.8.1. Introductory remarks
6.8.2. The ordinary differential equation and non-lipshitzean F
6.8.3. Subcritical nonlinearities
Chapter 7 One Phase Nonlinear Geometric Optics
7.1. Amplitudes and harmonics
7.2. Elementary examples of generation of harmonics
7.3. Formulating the ansatz
7.4. Equations for the profiles
7.5. Solving the profile equations
Chapter 8 Stability for One Phase Nonlinear Geometric Optics
8.1. The He(R^d) norms
8.2. He estimates for linear symmetric hyperbolic systems
8.3. Justification of the asymptotic expansion
8.4. Rays and nonlinear transport
Chapter 9 Resonant Interaction and Quasilinear Systems
9.1. Introduction to resonance
9.2. The three wave interaction partial differential equation
9.3. The three wave interaction ordinary differential equation
9.4. Formal asymptotic solutions for resonant quasilinear geometric optics
9.5. Existence for quasiperiodic principal profiles
9.6. Small divisors and correctors
9.7. Stability and accuracy of the approximate solutions
9.8. Semilinear resonant nonlinear geometric optics
Chapter 10 Examples of Resonance in One Dimensional Space
10.1. Resonance relations
10.2. Semilinear examples
10.3. Quasilinear examples
Chapter 11 Dense Oscillations for the Compressible Euler Equations
11.1. The 2 - d isentropic Euler equations
11.2. Homogeneous oscillations and many wave interaction systems
11.3. Linear oscillations for the Euler equations
11.4. Resonance relations
11.5. Interaction coefficients for Euler's equations
11.6. Dense oscillations for the Euler equations
11.6.1. The algebraic/geometric part.
11.6.2. Construction of the profiles
Bibliography
Index
