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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
版面字数：
580千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2022年
ISBN：
978-7-04-056980-3
物料号：
56980-00
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
  • 目录
    • 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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