Hyperbolic partial differential equations and geometric optics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2012
|
| Series: | Graduate studies in mathematics
133 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIX, 363 S. graph. Darst. |
| ISBN: | 9780821872918 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Titel: Hyperbolic partial differential equations and geometric optics
Autor: Rauch, Jeffrey
Jahr: 2012
Contents Preface xi §P.l. How this book came to be, and its peculiarities xi §P.2. A bird’s eye view of hyperbolic equations xiv Chapter 1. Simple Examples of Propagation 1 §1.1. The method of characteristics 2 §1.2. Examples of propagation of singularities using progressing waves 12 §1.3. Group velocity and the method of nonstationary phase 16 §1.4. Fourier synthesis and rectilinear propagation 20 §1.5. A cautionary example in geometric optics 27 §1.6. The law of reflection 28 1.6.1. The method of images 30 1.6.2. The plane wave derivation 33 1.6.3. Reflected high frequency wave packets 34 §1.7. Snell’s law of refraction 36 Chapter 2. The Linear Cauchy Problem 43 §2.1. Energy estimates for symmetric hyperbolic systems 44 §2.2. Existence theorems for symmetric hyperbolic systems 52 §2.3. Finite speed of propagation 58 2.3.1. The method of characteristics. 58 2.3.2. Speed estimates uniform in space 59 2.3.3. Time-like and propagation cones 64 §2.4. Plane waves, group velocity, and phase velocities 71 vii
Contents viii §2.5. Precise speed estimate §2.6. Local Cauchy problems Appendix 2.1. 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. 3.4.2. 3.4.3. Appendix 3.1. Appendix 3.II. Maximally dispersive systems 3.4.1. The L 1 — • L°° decay estimate Fixed time dispersive Sobolev estimates Strichartz estimates Perturbation theory for semisimple eigenvalues 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 = L {d) 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 79 83 84 89 91 91 93 96 104 104 107 111 117 120 123 123 125 127 131 132 136 141 141 143 143 149 151 161 161 166 173 177 177 180
Contents IX 5.5.3. Small time propagation of singularities 5.5.4. Global propagation of singularities 5.6. An application to stabilization Appendix 5.1. 5.1.1. 5.1.2. 5.1.3. 5.1.4. Hamilton-Jacobi theory for the eikonal equation Introduction Determining the germ of (ft at the initial manifold Propagation laws for / , d(j) 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 non- lipshitzean F 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 Hjl(B. d ) norms §8.2. #ƒ estimates for linear symmetric hyperbolic systems §8.3. Justification of the asymptotic expansion 188 192 195 206 206 207 209 212 215 215 216 222 224 227 230 231 237 239 242 246 246 248 250 259 259 262 263 265 270 277 278 281 282
X Contents §8.4. Rays and nonlinear transport 285 Chapter 9. Resonant Interaction and Quasilinear Systems 291 §9.1. Introduction to resonance 291 §9.2. The three wave interaction partial differential equation 294 §9.3. The three wave interaction ordinary differential equation 298 §9.4. Formal asymptotic solutions for resonant quasilinear geometric optics 302 §9.5. Existence for quasiperiodic principal profiles 307 §9.6. Small divisors and correctors 310 §9.7. Stability and accuracy of the approximate solutions 313 §9.8. Semilinear resonant nonlinear geometric optics 314 Chapter 10. Examples of Resonance in One Dimensional Space 317 §10.1. Resonance relations 317 §10.2. Semilinear examples 321 §10.3. Quasilinear examples 327 Chapter 11. Dense Oscillations for the Compressible Euler Equations 333 §11.1. The 2 — d isentropic Euler equations 333 §11.2. Homogeneous oscillations and many wave interaction systems 336 §11.3. Linear oscillations for the Euler equations 338 §11.4. Resonance relations 341 §11.5. Interaction coefficients for Euler’s equations 343 §11.6. Dense oscillations for the Euler equations 346 11.6.1. The algebraic/geometric part 346 11.6.2. Construction of the profiles 347 Bibliography 351 Index 359
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| dewey-ones | 535 - Light and related radiation |
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| dewey-sort | 3535 232 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
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| isbn | 9780821872918 |
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| publishDate | 2012 |
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| spelling | Rauch, Jeffrey 1945- Verfasser (DE-588)172326001 aut Hyperbolic partial differential equations and geometric optics Jeffrey Rauch Providence, RI American Math. Soc. 2012 XIX, 363 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 133 Includes bibliographical references and index Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd rswk-swf Geometrische Optik (DE-588)4020241-0 gnd rswk-swf Hyperbolische Differentialgleichung (DE-588)4131213-2 s Geometrische Optik (DE-588)4020241-0 s DE-604 Graduate studies in mathematics 133 (DE-604)BV009739289 133 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024780708&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rauch, Jeffrey 1945- Hyperbolic partial differential equations and geometric optics Graduate studies in mathematics Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Geometrische Optik (DE-588)4020241-0 gnd |
| subject_GND | (DE-588)4131213-2 (DE-588)4020241-0 |
| title | Hyperbolic partial differential equations and geometric optics |
| title_auth | Hyperbolic partial differential equations and geometric optics |
| title_exact_search | Hyperbolic partial differential equations and geometric optics |
| title_full | Hyperbolic partial differential equations and geometric optics Jeffrey Rauch |
| title_fullStr | Hyperbolic partial differential equations and geometric optics Jeffrey Rauch |
| title_full_unstemmed | Hyperbolic partial differential equations and geometric optics Jeffrey Rauch |
| title_short | Hyperbolic partial differential equations and geometric optics |
| title_sort | hyperbolic partial differential equations and geometric optics |
| topic | Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Geometrische Optik (DE-588)4020241-0 gnd |
| topic_facet | Hyperbolische Differentialgleichung Geometrische Optik |
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