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Hyperbolic equations and frequency interactions / Luis Caffarelli, Weinan E, editors.
Math/Physics/Astronomy Library QC381 .H96 1999
Available This item is available for access.
Math/Physics/Astronomy Library
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- Format:
- Contributor:
- Series:
-
- IAS/Park City mathematics series 1079-5634 ; v. 5.
- IAS/Park City mathematics series, 1079-5634 ; v. 5
- Language:
- English
- Subjects (All):
- Physical Description:
- xii, 466 pages : illustrations ; 27 cm.
- Place of Publication:
- Providence, R.I. : American Mathematical Society/Institute for Advanced Study, [1999]
- Contents:
-
- Jean Bourgain, Nonlinear Schrodinger Equations 3
- Lecture 1. Generalities and Initial Value Problems 7
- Lecture 2. The Initial Value Problem (continued) 15
- Proof of Theorem 1.36 20
- Comments on the proofs of Theorems 1.37, 1.40, 1.41 25
- Addition to Lecture 2, Remarks on the growth of higher Sobolev norms 29
- Lecture 3. A Digression: The Initial Value Problem for the KdV Equation 35
- Lecture 4. 1D Invariant Gibbs Measures 41
- Lecture 5. Invariant Measures (2D) 51
- Lecture 6. Quasi-Periodic Solutions of Hamiltonian PDE 69
- Lecture 7. Time Periodic Solutions 87
- Lecture 8. Time Quasi-Periodic Solutions 101
- Lecture 9. Normal Forms 117
- Lecture 10. Applications of Symplectic Capacities to Hamiltonian PDE 127
- Remarks on longtime behaviour of the flow of Hamiltonian PDE 141
- Ingrid C. Daubechies and Anna C. Gilbert, Harmonic Analysis, Wavelets and Applications 159
- Lecture 2. Constructing Orthonormal Wavelet Bases: Multiresolution Analysis 169
- Lecture 3. Wavelet Bases: Construction and Algorithms 175
- Lecture 4. More Wavelet Bases 183
- Lecture 5. Wavelets in Other Functional Spaces 191
- Lecture 6. Pointwise Convergence for Wavelet Expansions 199
- Lecture 7. Two-Dimensional Wavelets and Operators 205
- Lecture 8. Wavelets and Differential Equations 215
- Susan Friedlander, Lectures on Stability and Instability of an Ideal Fluid 227
- Lecture 1. Equations of Motion 233
- 1.1. Ideal fluid model 233
- 1.2. Euler equations 234
- 1.3. Lagrangian trajectories and streamlines 235
- 1.4. Vorticity 235
- Lecture 2. Initial-Boundary Value Problem 237
- 2.1. Existence and uniqueness theorems 238
- 2.2. Navier-Stokes equations 239
- Lecture 3. The Type of the Euler Equations 241
- 3.1. Linearized Euler equations 241
- 3.2. Computation of principal symbol 241
- 3.3. Wave motion supported by linearized Euler equations 244
- Lecture 4. Vorticity 247
- 4.1. Vorticity and stream function in two dimensions 248
- 4.2. Extra complications in three dimensions 250
- 4.3. Vorticity theorems 250
- 4.4. Helmholtz Vortex Theorems 252
- 4.5. Role of vorticity in PDE theory of Euler equations 255
- Lecture 5. Steady Flows 257
- 5.1. Two dimensional case 257
- 5.2. Three dimensional case 258
- Lecture 6. Stability/Instability of an Equilibrium State 263
- 6.1. Linear stability (instability) 263
- 6.2. Nonlinear (Lyapunov) stability 265
- Lecture 7. Two-Dimensional Spectral Problem 267
- Lecture 8. "Arnold" Criterion for Nonlinear Stability 271
- Lecture 9. Plane Parallel Shear Flow 273
- Lecture 10. Instability in a Vorticity Norm 277
- Lecture 11. Sufficient Condition for Instability 279
- Lecture 12. Exponential Stretching 285
- Lecture 13. Integrable Flows 289
- Lecture 14. Baroclinic Instability 291
- Lecture 15. Nonlinear Instability 295
- George Papanicolaou and Leonid Ryzhik, Waves and Transport 305
- 1.1. The geophysical problem 307
- 1.2. Radiative transport equations 309
- 1.3. Transport theory for electromagnetic waves 311
- 1.4. Transport theory for elastic waves 313
- 1.5. Brief outline 316
- Lecture 2. The Schrodinger Equation 317
- 2.1. The Schrodinger equation 317
- 2.2. Standard high frequency asymptotics 318
- 2.3. The Wigner distribution 320
- 2.4. General properties of the Wigner distribution 322
- 2.5. Convergence of energy 325
- Lecture 3. Symmetric Hyperbolic Systems 327
- 3.1. General symmetric hyperbolic systems 327
- 3.2. High frequency approximation for acoustic waves 332
- 3.3. Geometrical optics for electromagnetic waves 336
- 3.4. High frequency approximation for elastic waves 338
- Lecture 4. Waves in Random Media 343
- 4.1. The Schrodinger equation 343
- 4.2. Transport equations without polarization 346
- 4.3. Transport equations with polarization 350
- 4.4. Transport equations for acoustic waves 351
- 4.5. Transport equations for electromagnetic waves 351
- 4.6. Transport equations for elastic waves 353
- Lecture 5. The Diffusion Approximation 359
- 5.1. Diffusion approximation for acoustic waves 359
- 5.2. Diffusion approximation for electromagnetic waves 362
- 5.3. Diffusion approximation for elastic waves 363
- Lecture 6. The Geophysical Applications 367
- 6.2. Radiative transport equations 368
- 6.3. Transport theory for elastic waves 371
- Jeffrey Rauch with the assistance of Markus Keel, Lectures on Geometric Optics 383
- Lecture 2. Basic Linear Existence Theorems 389
- 2.1. Energy estimates for symmetric hyperbolic systems 389
- 2.2. Existence theorems for symmetric hyperbolic systems 393
- 2.3. Finite speed of propagation 395
- 2.4. Plane waves, characteristic variety and finite speed 397
- 2.5. Solutions on cones of determinacy 399
- Lecture 3. Examples of Propagation of Singularities and of Energy 401
- Lecture 4. Elliptic Geometric Optics 407
- 4.1. Constant coefficients and linear phases 407
- 4.2. Iterative improvement for variable coefficients and nonlinear phases 408
- 4.3. Formal asymptotics approach 410
- 4.4. Perturbation approach 413
- 4.5. Elliptic Regularity Theorem 414
- Lecture 5. Linear Hyperbolic Geometric Optics 417
- 5.1. Constant coefficients and linear phases 417
- 5.2. Scalar constant coefficient operators and linear phases 419
- 5.3. Variable coefficient systems and nonlinear phases 420
- 5.4. Rays and transport 427
- Lecture 6. Basic Nonlinear Existence Theorems 431
- 6.2. Schauder's Lemma and Sobolev Embedding 432
- 6.3. Basic existence theorem 436
- 6.4. Moser's inequality and the nature of the breakdown 438
- Lecture 7. One Phase Nonlinear Geometric Optics 441
- 7.1. Amplitudes and harmonics 441
- 7.2. More on the generation of harmonics 444
- 7.3. Formulating the ansatz 445
- 7.4. Equations for the profiles 446
- 7.5. Solving the profile equations 449
- 7.6. Rays and nonlinear transport 453
- Lecture 8. Justification of One Phase Nonlinear Geometric Optics 457
- 8.1. The spaces [characters not reproducible] (R[superscript d]) 457
- 8.2. [characters not reproducible] estimates for linear symmetric hyperbolic systems 460
- 8.3. Justification of the nonlinear asymptotics 461.
- Notes:
- Includes bibliographical references.
- ISBN:
- 0821805924
- OCLC:
- 39399253
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