图书信息
图书目录
双曲型偏微分方程和几何光学（影印版）
暂无简介
作者：
Jeffrey Rauch
定价：
169.00元
出版时间：
2022-02-25
ISBN：
978-7-04-056980-3
物料号：
56980-00
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
重点项目：
暂无
版面字数：
580.000千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
- 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
- Preface
- 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
