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双曲型偏微分方程和几何光学（影印版）

样章
作者：
Jeffrey Rauch
定价：
169.00元
ISBN：
978-7-04-056980-3
版面字数：
580.000千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2022-02-25
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
偏微分方程
暂无
  • 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

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