Hyperbolic Partial Differential Equations and Geometric Optics

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