**Contemporary Mathematics**

Volume: 581;
2012;
285 pp;
Softcover

MSC: Primary 42; 35; 58;
Secondary 74; 76

Print ISBN: 978-0-8218-6921-5

Product Code: CONM/581

List Price: $97.00

Individual Member Price: $77.60

**Electronic ISBN: 978-0-8218-9402-6
Product Code: CONM/581.E**

List Price: $97.00

Individual Member Price: $77.60

# Recent Advances in Harmonic Analysis and Partial Differential Equations

Share this page *Edited by *
*Andrea R. Nahmod; Christopher D. Sogge; Xiaoyi Zhang; Shijun Zheng*

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.

#### Table of Contents

# Table of Contents

## Recent Advances in Harmonic Analysis and Partial Differential Equations

- Preface vii8 free
- Combined list of speakers ix10 free
- The defocusing cubic nonlinear wave equation in the energy-supercritical regime 112 free
- Introduction 112
- Local theory 213
- From local to global theory 314
- Construction of a minimal counterexample 516
- Precluding the minimal counterexample: properties of almost periodic solutions to (NLW) 718
- Completion of the proof of Theorem 1 819
- Completion of the proof of Theorem 2 819
- Completion of the proof of Theorem 3 920
- References 920

- Stein’s extension operator on weighted Sobolev spaces on Lipschitz domains and applications to interpolation 1324
- Multilinear Morawetz identities for the Gross-Pitaevskii hierarchy 3950
- Elementary Proofs for Kato Smoothing Estimates of Schrödinger-like Dispersive Equations 6374
- The conormal derivative problem for higher order elliptic systems with irregular coefficients 6980
- Localization of eigenfunctions of a one-dimensional elliptic operator 99110
- Scattering and blow up for the two-dimensional focusing quintic nonlinear Schrödinger equation 117128
- 1. Introduction 117128
- 2. Preliminaries 121132
- 3. Global versus Blow up Dichotomy 126137
- 4. Outline of Scattering via Concentration Compactness 131142
- 5. Profile decomposition 132143
- 6. Proofs of claims in Step 1 and Step 2 for scattering 142153
- 7. Weak blowup via Concentration Compactness 145156
- References 152163

- Transverse stability of periodic traveling waves in Kadomtsev-Petviashvili equations: A numerical study 155166
- Time Decay for the Solutions of a Fourth-Order Nonlinear Schrödinger Equation 169180
- Self-adjoint realizations of Schrödinger operators on vector bundles over Riemannian manifolds 175186
- 1. Introduction and the main results 175186
- 2. Background of the problem 178189
- 3. Sobolev spaces and quadratic form domination 180191
- 4. Proofs of Theorem 1.2 and Corollary 1.4 181192
- 5. A distributional inequality 183194
- 6. Proofs Theorem 1.6 and Corollary 1.8 193204
- 7. Proof of Remark 1.9 194205
- 8. Proof of Theorem 1.5 195206
- References 196207

- Spectral properties of the reflection operator in two dimensions 199210
- Recent local and global solutions to the Lagrangian Averaged Navier-Stokes equation 217228
- Recent works on the Strauss conjecture 235246
- Wave operators for nonlinear wave equations with null structures 257268
- Fractional regularity for nonlinear Schrödinger equations with magnetic fields 271282

#### Readership

Graduate students and research mathematicians interested in analysis and PDE.