Title: Energy-critical Schrodinger equations on manifolds
Speaker: Sebastian Herr (Univ. Bielefeld)
Time: Feb. 27, 16:00-17:00
Place: 理科一号楼1479
Abstract
In this talk I will present recent small data global well-posedness results for energy-critical nonlinear Schroedinger equations on specific compact manifolds, such as flat tori and spheres. Key ingredients are certain multilinear estimates of Strichartz type which are based on $L^p$-estimates for exponential sums and spectral clusters. The classical dispersive estimate fails in this setup.
Prof. Herr will also give two talks in Beijing normal University. The following is the information
The Zakharov System, Part I & part II
报告人： Sebastian Herr 教授（Bielefeld University）
时间：2月26日下午3:00-5:00
2月28日下午3:00-5:00
摘要：The Zakharov system is a nonlinearly coupled system of a Schroedinger and a wave equation with a rich structure. In two talks I will give an introduction to the mathematical theory for the Zakharov system with a focus on local well-posedness results. In the seminal work of Ginibre-Tsutsumi-Velo the local well-posedness problem in subcritical spaces has been solved, with optimal results in dimension four and higher. In joint works of the speaker with Justin Holmer, Ioan Bejenaru and Daniel Tataru the corresponding problem in two and three dimensions has been addressed. After a thorough introduction I will present these results in detail and discuss relevant tools from harmonic analysis such as adapted function spaces and angular dyadic decompositions. Further, I will describe how resonant interactions can be controlled by using a nonlinear version of the Loomis-Whitney inequality, which states that the restriction of the convolution of two surface-carried measures can be bounded in an $L^2$-sense under a transversality condition. I will conclude with remarks on the long-time behaviour and open questions.