## Recent Advances in Harmonic Analysis and Partial Differential Equations: AMS Special Sessions, March 12-13, 2011, Statesboro, Georgia : the JAMI Conference, March 21-25, 2011, Baltimore, MarylandThis 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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### Contents

1 | |

Steins extension operator on weighted Sobolev spaces on Lipschitz domains and applications to interpolation | 13 |

Multilinear Morawetz identities for the GrossPitaevskii hierarchy | 39 |

Elementary Proofs for Kato Smoothing Estimates of Schrödingerlike Dispersive Equations | 63 |

The conormal derivative problem for higher order elliptic systems with irregular coefficients | 69 |

Localization of eigenfunctions of a onedimensional elliptic operator | 99 |

Scattering and blow up for the twodimensional focusing quintic nonlinear Schrödinger equation | 117 |

A numerical study | 155 |

Time Decay for the Solutions of a FourthOrder Nonlinear Schrödinger Equation | 169 |

Selfadjoint realizations of Schrödinger operators on vector bundles over Riemannian manifolds | 175 |

Spectral properties of the reflection operator in two dimensions | 199 |

Recent local and global solutions to the Lagrangian Averaged NavierStokes equation | 217 |

Recent works on the Strauss conjecture | 235 |

Wave operators for nonlinear wave equations with null structures | 257 |

Fractional regularity for nonlinear Schrödinger equations with magnetic fields | 271 |

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American Mathematical Society Anal analysis argument Assume assumption asymptotic blow-up bounded Cauchy problem cnoidal waves coefficients conormal derivative boundary consider constant Corollary cubic decomposition deﬁned deﬁnition defocusing denote derivative boundary condition differential forms dimensions E-mail address eigenfunctions eigenvalues elliptic energy ﬁnd ﬁnite ﬁrst ﬁxed follows formula function G H1 global existence global solution Global well-posedness GP hierarchies holds inequality inﬁnite initial data integral interpolation Kadomtsev-Petviashvili equation KP-I KP-II L2 norm Lagrangian Averaged Lemma linear Lipschitz domain localization Math Mathematics Subject Classification Morawetz identities Navier-Stokes equation nonlinear Schrödinger equation nonlinear wave equations norm obtain Partial Differential Equations perturbations proﬁle proof of Theorem Proposition prove quadratic radial radiosity Reifenberg result Riemannian manifold satisﬁes satisfy scattering self-adjoint sequence smooth Sobolev spaces Sogge Strauss conjecture Strichartz estimates theory vector wave equations weighted Sobolev spaces