This volume is based on the AMS Special Session on Harmonic Analysis and Partial Differential Equations and the AMS Special Session on Nonlinear Analysis of Partial Differential Equations, both held March 12–13, 2011, at Georgia Southern University, Statesboro, Georgia, as well as the JAMI Conference on Analysis of PDEs, held March 21–25, 2011, at Johns Hopkins University, Baltimore, Maryland. These conferences all concentrated on problems of current interest in harmonic analysis and PDE, with emphasis on the interaction between them. This volume consists of invited expositions as well as research papers that address prospects of the recent significant development in the field of analysis and PDE. The central topics mainly focused on using Fourier, spectral and geometrical methods to treat wellposedness, scattering and stability problems in PDE, including dispersive type evolution equations, higher-order systems and Sobolev spaces theory that arise in aspects of mathematical physics. The study of all these problems involves state-of-the-art techniques and approaches that have been used and developed in the last decade. The interrelationship between the theory and the tools reflects the richness and deep connections between various subjects in both classical and modern analysis.

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Recent Advances in Harmonic Analysis and Partial Differential Equations: Ams Special Sessions, March 12-13, 2011, Statesboro, Georgia: the Jami ... Maryland (Contemporary Mathematics, 581)

Andrea R Nahmod; American Mathematical Society; Japan-US Mathematics Institute (Baltimore, Md)

American Mathematical Society; Japan-US Mathematics Institute (Baltimore, MD). Conference

Andrea R Nahmod; American Mathematical Society; JAMI Conference

Contemporary mathematics (American Mathematical Society), v. 581, Providence, Rhode Island, 2012

Contemporary mathematics -- 581, Providence, Rhode Island, Rhode Island, 2012

American Mathematical Society, [N.p.], 2012

United States, United States of America

FR, 2012

0

lg2110216

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Includes bibliographical references.

Preface 8
Combined list of speakers 10
The defocusing cubic nonlinear wave equation in the energy-supercritical regime 12
Introduction 12
Local theory 13
From local to global theory 14
Construction of a minimal counterexample 16
Precluding the minimal counterexample: properties of almost periodic solutions to (NLW) 18
Completion of the proof of Theorem 1 19
Completion of the proof of Theorem 2 19
Completion of the proof of Theorem 3 20
References 20
Stein’s extension operator on weighted Sobolev spaces on Lipschitz domains and applications to interpolation 24
1. Introduction 24
2. Extending weighted Sobolev spaces 26
3. The manifold setting 41
4. Two interpolation results 42
References 47
Multilinear Morawetz identities for the Gross-Pitaevskii hierarchy 50
1. Introduction 50
2. The nonlinear Schrödinger equation 57
3. Morawetz identities for the GP hierarchy 63
References 71
Elementary Proofs for Kato Smoothing Estimates of Schrödinger-like Dispersive Equations 74
1. Introduction 74
2. Statement of the Theorems 75
3. Proof of Theorem 2.1 76
4. Proof of Theorems 2.3 and 2.7 77
5. Acknowledgement 78
References 79
The conormal derivative problem for higher order elliptic systems with irregular coefficients 80
1. Introduction 80
2. Main results 83
3. Some auxiliary estimates 86
4. Systems on a half space 93
5. Systems on a Reifenberg flat domain 100
Acknowledgement 106
References 107
Localization of eigenfunctions of a one-dimensional elliptic operator 110
1. Introduction 110
2. Preliminaries: solution of the eigenvalue problem 115
3. Numerical results 119
4. Dependence of the localization coefficient on the amplitude of an eigenfunction at the juncture point: explicit asymptotic formula 122
References 126
Scattering and blow up for the two-dimensional focusing quintic nonlinear Schrödinger equation 128
1. Introduction 128
2. Preliminaries 132
3. Global versus Blow up Dichotomy 137
4. Outline of Scattering via Concentration Compactness 142
5. Profile decomposition 143
6. Proofs of claims in Step 1 and Step 2 for scattering 153
7. Weak blowup via Concentration Compactness 156
References 163
Transverse stability of periodic traveling waves in Kadomtsev-Petviashvili equations: A numerical study 166
1. Introduction 166
2. Cnoidal waves 167
3. Numerical Method 169
4. Gaussian perturbations 170
5. Periodic deformations 173
6. Conclusion 174
References 175
Time Decay for the Solutions of a Fourth-Order Nonlinear Schrödinger Equation 180
1. Introduction 180
2. Morawetz’ Radial Identity 181
3. Time decay 182
References 183
Self-adjoint realizations of Schrödinger operators on vector bundles over Riemannian manifolds 186
1. Introduction and the main results 186
2. Background of the problem 189
3. Sobolev spaces and quadratic form domination 191
4. Proofs of Theorem 1.2 and Corollary 1.4 192
5. A distributional inequality 194
6. Proofs Theorem 1.6 and Corollary 1.8 204
7. Proof of Remark 1.9 205
8. Proof of Theorem 1.5 206
References 207
Spectral properties of the reflection operator in two dimensions 210
1. Introduction 210
2. Preliminaries 211
3. The Mellin Symbol of the Reflection Operator 214
4. Spectral Properties of the Reflection Operator 217
References 225
Recent local and global solutions to the Lagrangian Averaged Navier-Stokes equation 228
1. Derivation of the Lagrangian Averaged Navier-Stokes equations 231
2. Proof of Theorem 1 239
3. Appendix: Results for the Projection operator P^{α} 242
References 243
Recent works on the Strauss conjecture 246
1. Introduction 247
1.1. Initial analysis of the equation: scaling 247
1.2. Strauss conjecture: the beginning of the story 247
1.3. First wave of study: 1979-1985 248
1.4. Second wave of study: 1990-1997 249
1.5. Third wave of study: 2008-now 250
1.6. Structure of the paper 252
2. Morawetz-KSS estimates and the global existence for n=4 252
2.1. A Proof of the Strauss conjecture: n=4 252
2.2. Adaption to the obstacle case: n=4 253
3. Weighted Strichartz estimates 254
3.1. A proof of the Strauss conjecture: 2\len\le4 254
3.2. Adaption to the obstacle case: 3\len\le4 255
4. Generalized Strichartz estimates 256
4.1. Generalized Strichartz estimates 256
4.2. A new proof of the Strauss conjecture for n=2 258
4.3. Adaption to the obstacle case 259
5. Appendix 259
5.1. Hardy’s inequality 259
5.2. Trace lemma 260
5.3. Morawetz type estimates 262
5.4. Morawetz-KSS estimates 263
5.5. Weighted Strichartz estimates 264
References 265
Wave operators for nonlinear wave equations with null structures 268
1. Introduction 268
2. Proof of Theorem 1.2: the first iteration method 271
3. Control of the first iterate 272
References 280
Fractional regularity for nonlinear Schrödinger equations with magnetic fields 282
1. Introduction 282
2. Strichartz estimates for L 284
3. H^{s} solution of NLS with magnetic potential 285
4. Identification of the H^{s,p}-Sobolev spaces 289
References 294

Fifteen papers from two March 2011 conferences explore harmonic analysis techniques for studying higher order elliptic systems with irregular coefficients, the behavior of solutions to the focusing quintic nonlinear Schrodinger equation, nonlinear wave equations with null structures, and the transverse stability of periodic traveling waves. Survey articles review recent progress on Strauss conjecture for wave equations on exterior domain, the defocusing cubic nonlinear wave equation in the energy-supercritical regime, and local and global solutions of the Lagrangian averaged Navier-Stokes equation. No index is provided. Annotation ©2013 Book News, Inc., Portland, OR (booknews.com)

2017-09-12