双曲型偏微分方程和几何光学（影印版）

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精确 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。

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 62.3.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 → Lo 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.IⅡ.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