京区分析与偏微分方程青年学术研讨会通知

京区分析与偏微分方程青年学术研讨会将在11月20日在太阳成集团tyc7111cc举办。本次会议旨在围绕分析和偏微分方程方向最新的一些研究进展进行研讨，同时加强清华、北大及北航分析与偏微分方程方向青年学者之间的相互交流。

会议时间：2020年11月20日

会议地点：太阳成集团tyc7111cc沙河校区主楼E404会议室

会议报告人：

  金龙（清华大学）

  荆文甲（清华大学）

  刘保平（北京大学）

  杨诗武（北京大学）

  沈良明（太阳成集团tyc7111cc）

  郑孝信（太阳成集团tyc7111cc）

会议组织人：沈良明，徐丽

会议联系人：沈良明（lmshen@buaa.edu.cn），徐丽（xuliice@buaa.edu.cn）

太阳成集团tyc7111cc

太阳成集团tyc7111cc

报告题目和摘要

题目: Control of eigenfunctions on surfaces of negative curvature

报告人：金龙

摘要: In this talk, we present a uniform lower bound for the mass in any fixed nonempty open set of normalized Laplacian eigenfunctions on negatively curved surfaces, independent of eigenvalues. The result extends previous joint work with Semyon Dyatlov on surfaces with constant negative curvature. The proof relies on microlocal analysis, chaotic behavior of the geodesic flow and a new ingredient from harmonic analysis called Fractal Uncertainty Principle by Jean Bourgain and Semyon Dyatlov. Further applications include control for Schr\"{o}dinger equation and exponential decay of energy for damped waves. This is based on joint work with Semyon Dyatlov and St\'{e}phane Nonnenmacher.

题目: Layer potentials and homogenization in perforated domains

报告人：荆文甲

摘要:  In this talk I will present a unified homogenization method for Lamé systems in perforated domains with Dirichlet boundary conditions at the boundaries of the holes. The method is based on the layer potentials for Lamé system and a quantitative analysis of the rescaled cell problem. It treats various asymptotic regimes of the hole-cell ratio in a unified manner, and it provides natural correctors and quantitative estimates.

题目: Wellposedness for the KdV hierarchy

报告人：刘保平

摘要: The KdV hierarchy is a hierarchy of integrable equations generalizing the KdV equation. Using the modified Muria transform, we first relate it to the Gardner hierarchy, and by exploiting the idea of approximate flow, we show that the whole hierarchy is wellposed for initial data in H^{-1}. This is based on joint work with H.Koch and F. Klaus.

题目: The Calabi-Yau metric and complex Monge-Ampere equation in compact and noncompact settings

报告人：沈良明

摘要: We first recall Yau's folklore solution to the Calabi conjecture. Then we briefly discuss how to deduce the canonical metric problem to the complex Monge-Ampere equation and derive a priori estimates. After that we introduce the corresponding noncompact setting by Tian-Yau. Finally we talk a bit about some progress in the generalization to Tian-Yau's work.

题目: Asymptotic decay for semilinear wave equation

报告人: 杨诗武

摘要: In this talk, I will report recent progress on global behaviors for solutions of energy subcritical defocusing semilinear wave equations with pure power nonlinearity. We prove that in space dimension 1 and 2, the solution decays in time with an inverse polynomial rate, hence giving an affirmative answer to a conjecture raised by Lindblad and Tao. In higher dimension, we obtain improved scattering results for the solutions. These results are based on vector field method with new multipliers. Part of these works are jointed with Dongyi Wei.

题目：Existence and regularity of weak solutions to the generalized Leray equations

报告人：郑孝信

摘要：I will talk about global-in-space existence and regularity of weak solutions to the generalized Leray equations. The first part is the high regularity in the weighted space to weak solution. The second part is optimal decay in space to weak solution. This talk is based on a recent joint work with Baishun Lai and Changxing Miao.