双曲型偏微分方程和几何光学

本书介绍了双曲型方程的方方面面，这类方程特别适合描述以有限速度传播的波。本书的主题包括非线性几何光学、短波长解的渐近分析以及此类波的非线性相互作用。 作者详细论述了波的阻尼、共振、色散衰减、由共振相互作用引起的密集振荡的可压缩 Euler 方程的解。许多基本结果首次以教科书的形式呈现。除密集振荡外，本书还处理了传播的准确速度及三波相互作用方程组的存在性和稳定性等问题。 本书的特色之一是其关注提出思想和证明的动机，展示它们如何从相关的更简单情形演进而来。本书还提供了大量习题供读者进行练习。 作者是密歇根大学的数学教授，偏微分方程知名专家，为双曲型偏微分方程的三个领域（非线性微局部分析、波的控制和非线性几何光学）的变革做出了重要贡献。本书可供对双曲型偏微分方程感兴趣的研究生和研究人员使用参考。

   前辅文

 Preface

  §P.1. How this book came to be, and its peculiarities

  §P.2. A bird’s eye view of hyperbolic equations

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

 §2.2. 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 L1 → L∞ 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(∂)

  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 φ at the initial manifold

  5.I.3. Propagation laws for φ, dφ

  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 symmetric hyperbolic 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 nonlipshitzeanF

  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 Hs(Rd) norms

 §8.2. Hs 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