Hyperbolic equations and frequency interactions / Luis Caffarelli, Weinan E, editors.

Format:

• Book

Contributor:

• Caffarelli, Luis A.
• E, Weinan, 1963-

Series:

• IAS/Park City mathematics series 1079-5634 ; v. 5.
• IAS/Park City mathematics series, 1079-5634 ; v. 5

Language:

English

Subjects (All):

• Geometrical optics--Mathematics.
• Differential equations, Partial.
• Schrödinger equation.
• Wavelets (Mathematics).
• Geometrical optics.

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