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